Nowadays, the whole world is being concerned with a major health problem, which is diabetes. A very common symptom of diabetes is the diabetic foot ulcer (DFU). The early detection of such foot complications can protect diabetic patients from any dangerous stages that develop later and may require foot amputation. This work aims at building a mobile thermal imaging system that can be used as an indicator for possible developing ulcers.The proposed system consists of a thermal camera connected to a Samsung smart phone, which is used to acquire thermal images. This thermal imaging system has a simulated temperature gradient of more than 2.2 °C, which represents the temperature difference (in the literature) than can indicate a possible development of ulcers. The acquired images are processed and segmented using basic image processing techniques. The analysis and interpretation is conducted using two techniques: Otsu thresholding technique and Point-to-Point mean difference technique.The proposed system was implemented under MATLAB Mobile platform and thermal images were analyzed and interpreted. Four testing images (feet images) were used to test this procedure; one image with any temperature variation to the feet, and three images with skin temperature increased to more than 2.2 °C introduced at different locations. With the two techniques applied during the analysis and interpretation stage, the system was successful in identifying the location of the temperature increase.This work successfully implemented a mobile thermal imaging system that includes an automated method to identify possible ulcers in diabetic patients. This may give diabetic patients the ability for a frequent self-check of possible ulcers. Although this work was implemented in simulated conditions, it provides the necessary feasibility to be further developed and tested in a clinical environment. Diabetes Mellitus (DM) is a metabolic chronic disease that is associated with abnormal glucose levels in the blood. There are two causes of Diabetes Mellitus, the first one is the abnormal production of insulin by the pancreas (Type I), while the second cause is related to inadequate cells action to insulin (Type II). Both types of Diabetes Mellitus can pose a serious threat to patients’ health concerning the cardiovascular system, kidneys, and extremities such as the feet . AccordiDiabetic foot ulcer can be avoided or delayed if adequately treated at an early stage. Currently, the assessment of such foot complications is done frequently by clinicians through analyzing blood circulations, plantar foot pressure, and foot neuropathy , 10. MorThe occurrence of diabetic foot complications is often related to the plantar region temperature distribution. Increased temperature may be present in the foot a week before a neuropathic ulcer appears . ResearcThe aim of this work is to build an ulcer detection/indication system based on a mobile thermal camera and a mobile application. The proposed system would serve as a self-monitoring tool with a mobile app giving diabetic patients the ability to self-check their extremities for any possible ulcer, without the need for frequent visits to the diabetic clinic. The proposed system was implemented using a mobile application where thermal images were acquired, processed, and analyzed for any possible ulceration. Two image-processing techniques were deployed to detect possible ulcers automatically: the Otsu thresholding techniques and the point-to-point difference techniques. Both techniques were tested on thermal images. The implementation of the image processing algorithms was done using MATLAB mobile . It was also complied into Java, and a mobile application was built for this purpose.The proposed system consists of a hardware part, which is mainly a mobile thermal image acquisition camera and a smart phone, along with image processing and analysis software. The entire software was run on MATLAB and was later implemented on a mobile phone through MATLAB Mobile Android application. After that, the entire software was compiled into Java code to build a complete and integrated application with user interface.The thermal image acquisition system, a Smartphone-based system, consists of an infrared (IR) thermal camera, the FLIR ONE (FLIR Systems), shown in Fig. Regarding the emissivity of objects, FLIR ONE cameras automatically calibrate the scene temperature in order to provide the highest possible accuracy. In this regard, the camera calculates the emissivity of objects, the distance, and the reflected temperature, and then it returns the actual temperature of the object. The accuracy of the FLIR ONE camera and the calibration process have been tested by researchers through placing a reference object with a known temperature and emissivity, such as a blackbody device (Omega BB701), which has 1.0 emissivity and a pre-assigned temperature, in the field-of-view of the camera while conducting a human subject test. Thus, the actual temperature measurements from the camera could be compared with much more accurate devices such as FLIR E60 and IR thermometer (Omega OSXL450) before and after manual calibration procedures. The results showed that both the FLIR E60 and Omega OSXL450 IR thermometer measurements were close to each other and were improved by the calibration process, while FLIR ONE measurements remained the same. FLIR ONE accuracy resulted in having ≈ 2 °C more temperature compared to the other accurate devices while observing the measurements , meaningThe images acquired by FLIR ONE camera were stored in the Smartphone in Joint Photographic Experts Group (JPEG) format. FLIR ONE fused the standard image with the thermal image through Multi Spectral Dynamic Technology (MSX). This was done to provide physical details to the raw thermal image, and therefore, resulted in a better vision as RGB scaled images . In addiThe recorded thermal image was a gray scale image and further analysis and interpretation was conducted. As shown in Fig. The purpose of image segmentation is the extraction of certain objects or segments from original images. In this work, image segmentation was performed twice. The first one was to extract the object of interest from the thermal image, which are the diabetic feet, while the second one was to extract and identify possible ulcer in the diabetic feet. The first segmentation procedure was applied on the raw thermal image with the purpose of extracting the feet from the background, which was in this case the cold towel behind the feet. The technique used for the segmentation process is called Histogram shape thresholding. This technique was applied indicating that thermal images acquired following the acquisition protocol mentioned above resulted in a bi-modal histogram with two peaks, as shown in Fig. i is represented by the following equation:ni is the number of pixels at level i and N is the total number of pixels.In the histogram representation of a sample image shown in Fig. 1 with gray levels  and R2 with gray levels , where t is the threshold value. The means of class R1 and R2 are given byw1 and w2 are the gray level probability distributions for the two regions. The total mean of gray levels is given byThen, the pixels of the images were separated into two regions: RThe within-region variance is given by:t by maximizing the between-region variance or minimizing the within-region variance. The total variance, which is the sum of the within-region variance and the between-region variance, is constant for different partitions.t, which can separate the two intensity histogram regions. This parameter is given as a normalized value between 0 and 1 of the corresponding scene temperature. The outcome from Otsu thresholding technique was a binary image, with values from 0 to 1, of the segments needed for further analysis, which are the plantar feet.The technique of Histogram shape segmentation resulted in a binary image of the segmented foreground, which were the plantar feet. However, some images are not easy to segment, especially if there are some parts within the image where the temperature of the background is close to the foreground temperature. As a result, dark objects are marked as bright objects and vise-versa. This affects any further image processing analysis of the plantar feet image. Therefore, image-smoothing techniques were performed to avoid such errors.At the beginning, borders clearing technique was used to remove any objects that are connected to the border of the image, or even separated in different places within the image. This was done by suppressing any light structures and removing them from the surrounding border of the image. Then, segments’ smoothing was performed to erode the resulting image with a diamond-structuring element. The element used had a single pixel distance from the origin of its structure to the points of the diamond. This prepares the feet segment to be smoothed at the edges and ensures that no unconnected objects are taking place in the image. Finally, the binary segment created might include some interior gaps, therefore, these gaps were filled with hole-filling objects . The plaAfter segmenting the plantar feet image, further processing was performed to identify any possible ulcers or occurrence of hyperthermia (a 2.2 °C difference) , 18, 19.Shown in Fig. The complete procedure for the proposed algorithm is illustrated on Fig. Another technique was applied for the detection of any possible ulcer, which is called the point-to-point mean difference shown in Fig. The processed image included both feet; therefore, it was automatically divided into two equal parts; one for the left foot and one for the right foot. As previously mentioned in the acquisition procedure, users are advised to maintain that their feet are in the center of the camera field-of-view for accurate cutting and analysis results. The left foot segment was chosen as the reference foot . The deployed technique requires both feet to be aligned together. Hence, two steps were performed; the first one was flipping the right foot to make both feet look identical and the second step was image registration to align both feet together. The adopted image registration technique was the intensity-based registration . Image rThe next step was implementing an edge detection technique to remove the edges from the left and right feet images. The edge detection was applied using the Sobel operators . The resThe complete procedure followed in this technique’s algorithm is shown in Fig. The proposed techniques were tested by acquiring thermal images from the feet of a healthy subject. The ulcers were simulated by heating a metal object (a coin) with a diameter of around 1.5 cm. The heated object was placed on the skin to raise its temperature and emulate the possible ulcer (hyperthermia). Eight test images were acquired, one without any skin heating (test image 1) and the seven other images with the feet heated into different locations. The thermal camera temperature measurement was used to make sure that the heated region temperature was more than 2.2 °C higher than the normal feet temperature. A test image illustrating the effect of image smoothing techniques is shown on Fig. Figure Furthermore, the algorithm was tested by adding additional six test images with various shapes simulating ulcer. The shapes used were a small coin of 0.5 cm diameter, a rectangle of 2 cm length, a small half-shaped ring, a 3.5 cm length rectangle, a 2.5 cm diameter coin, and a 1 cm diameter coin. Figure The same test images were used to test the proposed system with the technique of point-to-point difference for image analysis and interpretation. Figure The results of the point-to-point difference method implemented are shown in Fig. Moreover, the same six test images with the different shapes and sizes were analyzed by the point-to-point algorithm, and the results are shown on Fig. The main objective of this research was to build a thermal imaging system based on smart phone. The proposed system incorporated the hardware as well as the necessary image processing and interpretation software techniques. According to the obtained results, the proposed system has successfully identified regions with hyperthermia with temperature gradient greater than 2.2 °C which is considered as the value that can be used to identify possible ulcers , 19. TheThe second technique used for analysis and interpretation made use of both legs to be a reference for the background. This technique was used by Vilchuaman et al. . The onlThe entire system was implemented on a Smartphone under MATLAB mobile with processing on a cloud MATLAB server. The use of MATLAB mobile provides the flexibility to do further processing, store the data on a cloud account, and build the necessary interface.The proposed system provides a framework to build a complete mobile system that can help diabetic patients self-check their feet for any possible ulcers. The system provides only an indicative tool, not a diagnostic tool, as the final diagnosis should be done by the physician (the gold standard). The future work requires upgrading the system with an advanced thermal camera with higher image quality that can be connected to a mobile in order to perform the necessary processing. Further testing and validation of the system should be performed under clinical environment, which was not possible at this stage due to the strict regulations applied. Moreover, this work can be extended to other possible applications such as wound healing and trauma monitoring.


Nachhaltiges Landmanagement im Norddeutschen Tiefland (NaLaMa-nT), we investigated whether irrigation might need to be adapted to changing climatic conditions. To this end, results from regionalised climate change modelling were coupled with soil- and crop-specific evapotranspiration models to calculate potential amounts of irrigation to prevent crop failures. Three different runs of the climate change scenario RCP 8.5 were used for the time period until 2070.By example of a region in Northern Germany (County of Uelzen), this study investigates whether climate change is likely to require adaption of agricultural practices such as irrigation in Central Europe. Due to sandy soils with low water retention capacity and occasional insufficient rainfall, irrigation is a basic condition for agricultural production in the county of Uelzen. Thus, in the framework of the comprehensive research cluster The results show that the extent of probable necessary irrigation will likely increase in the future. For the scenario run with the highest temperature rise, the results suggest that the amount of ground water presently allowed to be extracted for irrigation might not be sufficient in the future to retain common agricultural pattern.The investigation at hand exemplifies data requirements and methods to estimate irrigation needs under climate change conditions. Restriction of ground water withdrawal by German environmental regulation may require an adaptation of crop selection and alterations in agricultural practice also in regions with comparable conditions. Current projections of global climatic change strongly support the expectation that during the next few decades, global temperatures will continue to increase as well as spatial and temporal patterns of temperature and precipitation will be shifting ,2. This 3 of water each year since the year 2000, i.e. 2% of annual precipitation over land and 17 mm of water spread evenly over the global land surface. This is a +75% increase from 1960 levels and a +400% increase from 1900 levels of irrigation. Out of the world’s croplands, 18%, i.e. about 2% of the total land surface, are irrigated and produced 40% of the world’s food. On average, the irrigated areas receive an addition of 800 mm of water each year . Du. Du35]. 2 and year of ground water are used for irrigation in the county of Uelzen [2 and year with a moving average over 7 years in order to avoid depletion [On average, 73 mm/mf Uelzen . The guaf Uelzen . The amoepletion . On averepletion . The couepletion .Climate change not only affects water availability but also the demand for water. If the climate in a given region gets drier and warmer, water availability will decrease and be exacerbated by increasing water demand . Soil wa2. Typical examples illustrating the variability of soil water content and required irrigation are shown in Figure Irrigation requirements were calculated for all years in the reference period 1991 to 2010 and in the scenario time span 2011 to 2070. Irrigation will be applied as soon as the soil water content is lower than 20% of the available water capacity. The amount of water applied in a single irrigation event was set to 20 mm/mAs climate conditions in terms of daily average temperature, precipitation and evapotranspiration show a high variance between different years Figure  in all sThe irrigation requirement is also related to the crop-specific evapotranspiration during the cultivation phase: crop species with a high irrigation requirement like sugar beet and potato show the highest total evapotranspiration amount (330 and 287 mm) during the irrigation phase compared to winter barley and winter rye (182 and 225 mm) with a lower irrigation requirement Figure .Figure 4b = +0.306, +0.266 and +0.262) and Tmax  as average temperatures increase. The longer the crop remains on the field, the higher the irrigation requirement is and will be, respectively. For field crops, which are harvested early (e.g. winter barley), the irrigation requirement remains on a comparatively low level . The irrigation requirement is decreasing in the scenario run Tmin for all field crops because of a decreasing evapotranspiration and a slight increase in precipitation . Because of a high variance, the Kendall’s tau b as correlation coefficient and the coefficients of determination (R2) show low or, at most, medium values for all different field crops and scenario runs. The irrigation requirement is directly linked to the climate conditions , which also show a high variance between the different years in all scenario runs , the moving average was slightly higher than during the reference period 1991 to 2010. In scenario run Tmin, the irrigation requirement is decreasing compared to the reference period 1991 to 2010 because of a slight increase in precipitation and a decreasing evapotranspiration.The crop rotation sugar beet-potato-winter rye-winter barley is most common in the county of Uelzen and in particular on areas labelled as soil type 31   and with higher amounts of irrigation water than 20 mm/mIn a few exceptional years, the soil water content might be lower at starting day 107 than the default start value due to a noticeably dry winter. In these cases the irrigation requirement could be higher than the calculated values indicate. Nevertheless, it could be shown that as by 2010, assuming the worst case scenario from down-scaled global climate change models, 62% to 80% of agricultural land within a Central European region could shift towards a new agroclimatic class and 98% by 2050, respectively .It is important to emphasise that irrigation requirements cannot be directly derived from climate modelling. In fact, outcomes of climate modelling need to be linked with data on soil conditions, cultivation patterns and eventually also economic considerations in order to assess future sustainability of particular crops to be grown and the resulting requirements of external water input. Calculations were done with evapotranspiration data for grassland  based onIn future research activities, it will be intended to expand the calculations to additional regions, soil types and crop rotations. This would be of high relevance since the European agriculture is characterised by high productivity  and accoFurthermore, the presented results for irrigation amounts obtained using the BewUe programme should be cross-checked with observation data for single field crops and different years. But such observation data are not available yet. Additionally, further climate change scenarios are of interest and should be used to improve understanding of the possible future irrigation requirement.The results show that it is reasonable to expect that regional implications of global climate change will affect evapotranspiration as an important aspect in crop cultivation. Ground water availability for irrigation allows, under the given conditions, a short-term buffering towards extremes. Based on the scenario calculations, it can be expected that the current agricultural practice in the county of Uelzen will not be directly limited by regional climatic alterations. However, in the more distant future, where climate change is on the one side to become more pronounced and more uncertain to be predicted on the other side, additional measures might be necessary to prevent higher frequencies of crop failures in some years. Changes in irrigation techniques or adaptation of crop rotation types are amongst these measures. The presented work shows how irrigation requirement can be calculated combining regional projections of climate conditions specified as long-term ‘scenario-weather’, combined with crop requirements and management, and soil conditions. The results show that it is likely that agriculture in the county of Uelzen has to be adapted to increasing irrigation requirement in the future, if conditions develop as assumed in the employed scenarios.However, further research will give evidence whether changes either in institutional and market conditions or in climatic conditions will dominantly influence the development of agriculture illustrated by a comparable example of adaptation for maize production in Switzerland . Since sHomogenised daily climate observation and scenario data for the climate change scenario Representative Concentration Pathways (RCP) 8.5 were provided by the Potsdam Institut für Klimafolgenforschung (PIK) to the NaLaMa-nT project consortium for a 10 × 10 km grid and the time periods 1951 to 2010  and 2011To calculate changes in soil humidity, we used additional data for evapotranspiration, which were available for grassland correspondent to the guidelines of the Food and Agriculture Organization of the United Nations (FAO) computed by the Penman-Monteith equation  but not The daily average temperature, precipitation and evapotranspiration show a high variance within the different years for the reference period 1991 to 2010 and in all scenario runs Figure . This meFigure Soil data of the land use-specific soil map of Germany  were useThe information on the current agricultural practice was provided by the Leibnitz Centre for Agricultural Landscape Research . The dominant crops are listed in Table 3), an available water capacity of 17% and a wilting point of 7%  in a soil depth of 0 to 60 cm [For the calculations we used data for soil type 31 , which is the soil type most common in the county’s agricultural area (47%) . The amount of water applied in a single irrigation event was set to 20 mm/m2. The main model functional is outlined in the pseudocode given in Equation Soil and climate data were used in the model BewUe written by B. Breckling using the programming language SIMULA . The mod3 (soil water content at field capacity) and the minimum soil water content as ≥ 70 l/m3 (amount of stagnant water).The maximum soil water content is defined as ≤ 240 l/m3 on the first day (1 January 1951) assuming that the soil is fully water saturated at this time. To compute the average development of the soil water content, the daily values were averaged over the time period 1951 to 2070  and in which the soil water content should be computed by the use of the SIMULA model. Therefore, the soil water content was calculated for average conditions with the climate observation data for the time period 1951 to 2010 and the Tmed scenario run for the time period 2011 to 2070. For this calculation, the soil water content was set to 240 l/m3. Starting time and start value are considered representative for average climate conditions in the county of Uelzen and were used to compute the irrigation requirement for all different scenario runs, years and crop species.The soil water content decreases noticeably after day 107 in an average year . The crop plants listed in Table Irrigation was considered unnecessary during the maturation phase of the field crops. In general, maturation takes between 2 and 4 weeks depending on the yearly course of the weather . In our calculations, we assumed that maturation starts 3 weeks before harvesting in all years and different scenario runs. Harvest dates differ between the relevant crops Table . Thus, tb as correlation coefficient (Figure Temporal trends for irrigation requirements are secured statistically by computing the Kendall’s tau t Figure . The Man


The triboelectric effect, charging by contact, is the working principle in a device called a triboelectric nanogenerator. They are used as efficient energy transducers in energy harvesting. In such generators the charging of surfaces at contact is followed by a separation of the surfaces increasing the electrical energy which can subsequently be used. Different materials have different triboelectric potentials leading to charging at contact. The temperature dependence of the charging has just recently been studied: the triboelectric effect is decreasing with temperature for a generator of Al-PTFE-Cu. Here, we suggest a mechanism to explain this effect assuming ion transfer using a two-level Schottky model where the two levels corresponds to the two surfaces. The difference in binding energy for ions on the two surfaces then enters the formula for charging. We fit the triboelectric power density as a function of temperature obtained from a two-level Schottky model to measured data for nanogenerators made of Al-PTFE-Cu found in three references. We obtain an average separation energy corresponding to a temperature of 365 K which is of the right magnitude for physically adsorbed atoms. We anticipate that this model could be used for many types of triboelectric nanogenerators. It has become important in a device called a triboelectric nanoenerator (TENG). They are used as efficient energy transducers in energy harvesting5. In the generators the charging Q of the surfaces with area A at contact is followed by a separation d of the surfaces increasing the electrical energy which can then be used. The energy is given byThe triboelectric effect, electrification by contact, has been studied for a long timeε0 is the permittivity of vacuum. Different materials have different triboelectric potentials leading to charging on contact6. Charge transfer can cause strong adhesive forces7. The mechanism of charging of metals are well known: the difference in work function between the two metals leads to electron transfer8.Here 9. There are loosely bound atoms on surfaces that can be be transfered to another surface on contact10. Triboelectric charging is affected by humidity11. The temperature dependence of charging has been studied by Wen et al.12, Su et al.13 and Lu et al.14. The charging energy was found to decrease with increasing temperature for a generator made of Al-PTFE-Cu.However for non-metals the charging is partially unknown and its temperature dependence has just recently been studied. Experiments suggest that the charging is done by ion transfer according to McCarty and Whitesideset al.12, Su et al.13 and Lu et al.14. We obtained a least square fit for a separation energy corresponding to an average temperature of 365 K. All three fitted values above are in the interval 365 ± 96 K.Here, we propose a model to explain this temperature dependence for the Al-PTFE-Cu system assuming charging by ions distributed between the contacting surfaces according to a simple two-level Schottky model. We have fitted our model of the power density or voltage of a triboelectric nanogenerator made of Al-PTFE-Cu to data from Wen First we take a look at the properties of the two-level Schottky model. Then we apply the model to triboelectic charging of triboelectric materials making up a capacitor working as a triboelectric nanogenerator on separation of the capacitor plates.15 is a well known model with only two energy levels 1 and 2 separated by the energy E. The ratio of the number of particles on the two levels is given by the Boltzmann factorA two-level Schottky systemN = N1 + N2 the number of particles on the lower energy level is given byIf the number of particles is fixed to T all the particles are on the lowest level N1 = N, N2 = 0 and at high temperature they are equally distributed N1 = N2 = N/2. From the partition function Z = 1 + exp(−E/kBT) for the two-level system many thermodynamical functions of state like internal energy, heat capacity and entropy can be calculated16.We see that at low temperature E in binding energy for ions are brought into contact a two levels system will develop. We assume that the materials have N ions in common that can be distributed between them. The net charge of the capacitor is then Q = q(N1 − N2), see Fig. When two materials with a difference U = Q/C where C = ε0A/d is the capacitance soThe capacitor voltage is given by W = Q2/(2C), see Eq.  independent of temperature and for high temperature T Eq.  can then be written using the law of mass action17How is adsorption affected by temperature? The Langmuir adsorption isotherm model is considering the adsorption of gas atoms on to a surface. This could be applied before surface contact. Neglecting interactions between the adsorbed atoms, for surface coverage 0 < AS] will be proportional to the surface coverage θ while [S] will be proportional to the number of vacant sites 1 − θ. The gas concentration [A] is denoted c. We obtainThe occupied surface sites concentration [K(T) is found to be decreasing with increasing temperature. This implies reduced surface coverage θ at higher temperatures. We see also that as the concentration c increases, the surface coverage θ approaches unity.The equilibrium constant 18. Atoms jumps on the surfaces and probably even between them at contact. The jump frequency Γ on a surface is given by19kBT/h is the thermal vibration frequency of the lattice, h is the Planck constant and EA is the activation energy to escape from the potential well. The higher temperature T the larger jumping frequency Γ. Higher temperature thus increase the ion transfer rate between the surfaces but should not affect the equilibrium distribution of ions on the two surfaces.However, in our triboelectric system we have two surfaces in temporary contact and most of the ion transfer probably happens when almost no gas is in between the surfaces so a better model of ion transfer should be that of surface diffusionet al., Su et al. and Lu et al.: decreasing voltage and power density with increasing temperature with a characteristic temperature of E/kB = 365 ± 96 K. In Wen et al. the general trend is also decreasing voltage with temperature. However, the experimentally found large peak in capacitor voltage between 200 K and 300 K can not be explained within our Schottky model, see Fig. et al. for the same temperature interval. Lu et al. studied the temperature interval 250 K–420 K, which covers half the interval where Wen et al. found the peak, but no peak was found in the experiment by Lu et al.Because the mechanism causing the temperature dependence for voltage and power density of a triboelectric generator is still unknown, we have here suggested a simple Schottky model. This model seems to be able to explain the main features found in experiments by Wen 9 are using a model where the charge of the ions on a dielectric plate is neutralized by bound counter ions close to the surface of this material. The ions thus “belong” to this plate making it neutral. When another plate is brought in contact some of the ions get transferred to the other plate. The ion transfer is then creating an electric field between the plates which counteract more ion transfer. In this way the number of ions on the two surfaces can be calculated and hence the net charge on the plates. In our model we do not assume that the ions belong to one plate: we are assuming that they come from both plates. They are then distributed between the plates depending on the difference in work function for ions for the two plates creating a charge difference between the electrodes. However, we can include the effect of the presence of bound net charges on one plate into our model by adding a constant charge QB to Eq.  where A is the plate area and ε0 is the permittivity of vacuum. The force on one charge carrier is then qQ/(ε0A) so the work required to transfer one particle the distance d between the plates becomes qQd/(ε0A), see McCarty and Whitesides9. The energy difference E should then be corrected to E + qQd/(ε0A). However, before the plate separation the plate distance d is small and the correction term should be possible to neglect. After the separation, as d increases further, the ions would quickly get trapped on the surfaces they sit on9. The approximation of a constant E should then be acceptable.How valid is our model in view of that the particles are charged and thus change the energy difference between the levels as the number of ions changes? The energy difference et al., Su et al. and Lu et al. uses frequencies ranging from 0.03 Hz to 3 Hz, see legend of Table How is the operating frequency of the triboelectric generator affecting the Schottky model? The frequency of charge/discharge of the triboelectric generators could affect the ratio of the number of ions on the two surfaces because the Schottky model assumes thermal equilibrium, and if the frequency is too high the system might not have time to equilibrate. The experiments of Wen 2-nanotubes the reverse temperature behavior was found with increasing charging with temperature13. This latter system involves however nanomaterials which may be similar to the materials in gecko feet with its long hair yielding a large effective contact area21. For gecko feet complex behavior of contact force on temperature and humidity has been found22. This may indicate the need for a more complex model for this latter system so that a simple Schottky model would not suffice. For a system not including nanomaterials the Schottky model should however work.Can the Schottky model be applied to all nanogenerators containing non-metals? No. For a generator with one sheet covered with TiO The Schottky model seems to work for a triboelectric nanogenerator made of Al-PTFE-Cu. We anticipate that the Schottky system could be used as a model of many types of normal triboelectric nanogenerators containing non-metals.


Campephilus woodpeckers may be modeled in terms of a harmonic oscillator with an impulsive forcing, and this hypothesis is consistent with audio and video recordings. Wingbeats are another type of periodic and transient motion of large woodpeckers. A model for the flap rate in cruising flight is applied to the Pileated Woodpecker (Dryocopus pileatus) and the Ivory-billed Woodpecker . During a brief transient just after taking off, the wing motion and flap rate of a large woodpecker may not be the same as in cruising flight. These concepts are relevant to videos that contain evidence for the persistence of the Ivory-billed Woodpecker.Two types of periodic and transient motions of large woodpeckers are considered. A drumming woodpecker may be modeled as a harmonic oscillator with a periodic forcing function. The transient behavior that occurs after the forcing is turned off suggests that the double knocks of  Campephilus principalis)3. Most woodpeckers signal by drumming, which consists of a rapid series of blows with the bill. Audio Dryocopus pileatus). Some members of the Campephilus genus do not engage in this type of drumming but instead signal with double knocks. Audio Campephilus woodpecker species. It is proposed that both of these behaviors may be modeled in terms of a harmonic oscillator4, and evidence is presented to support the hypothesis that double knocks correspond to a special case of drumming in which the driving force is impulsive rather than periodic, as it is in the case of most woodpeckers. The flap rate of a bird in cruising flight is amenable to statistical analysis7. In putative video footage of an Ivory-billed Woodpecker in cruising flight (Movie 3), the flap rate is about ten standard deviations greater than the mean flap rate of the Pileated Woodpecker. In a test of the consistency of this evidence, a flap rate model based on vortex shedding9 is applied to the Pileated and Ivory-billed Woodpeckers. Large woodpeckers fold their wings closed during the middle of the upstroke during cruising flight;7 an example of this behavior is shown in Movie 3. During the first several flaps after taking off, there may be a transient in which the wing motion and flap rate are not the same as in cruising flight.Drumming and wingbeats are two types of periodic motion of large woodpeckers. This paper discusses these behaviors and related transient motions that occur in evidence for the persistence of the Ivory-billed Woodpecker , the coefficient q is related to the natural frequency of the system, and R(t) is the forcing function. As shown in Fig. x = x0, it rebounds with a loss of energy that can be modeled with the change in velocity,α < 1. In order to illustrate some of the basic behaviors of harmonic oscillators, we consider the case p = 0.2, q = 0.3, α = 0.8, and x0 = 0.2, with the periodic forcing function,A drumming woodpecker may be modeled as a harmonic oscillator, which is a mechanical system in which an object experiences a restoring force when it is displaced from an equilibrium point. The object may have periodic or transient motions, depending on how it is forced. In the following example of a harmonic oscillatort = 2π and after the forcing function is turned off at t = 20π, with a periodic response between the transients. For the impulsive case, the response is a transient that is similar to the transient that follows the periodic forcing.Solutions for these cases appear in Fig. f of a bird in cruising flight is amenable to statistical analysis and limited to a narrow range of frequencies for each species. For most species that have been studied, the standard deviation of the flap rate has been found to be less than ten percent of the mean flap rate7. The flap rate model9,St is the Strouhal number, U is the flight speed, and b is the wingspan. Large woodpeckers have a flap style in which the wings are folded closed and briefly paused during the middle of each upstroke7. For such intermittent flight, an appropriate value for the Strouhal number is 0.259, and the period of a flap corresponds to the time between when the wings begin to open after a pause and when they close at the beginning of the next pause.The flap rate Campephilus woodpeckers correspond to a special case in which the driving force of the harmonic oscillator is impulsive rather than periodic, with the impacts analogous to the transient that occurs after the periodic driving force of a drumming woodpecker is turned off and the transient in the lower part of Fig. As discussed in Movie 10 (p. 62) and with the recordings of the transient drumming of three Campephilus woodpecker species in Audio Campephilus guatemalensis) that appears in Movie Campephilus magellanicus) appear in a documentary by David Attenborough12, and each of them results from a single thrust of the body. In putative video footage of an Ivory-billed Woodpecker giving a double knock (Movie 3), there is only one thrust of the body. These examples suggest that a double knock is the result of a single thrust rather than a pair of deliberate blows that are given in rapid succession.According to this hypothesis, the time interval between impacts and the number of impacts that would be detected at a distance should depend on the amount of tension in the body, the intensity of the impulsive driving force, the properties of the wood, and the amount of time that elapses before the bird releases the tension in its body. These considerations are consistent with Tanner’s account of the Ivory-billed Woodpecker giving both single and double knocksck Movie  of ref.3f, which appears in Eq. (7. It is possible to obtain an accurate estimate of f for a species with intermittent flaps (errors may be introduced in the estimates of when each flap begins and ends). When the bird was flying toward the camera in the cruising flight in Movie 3, there were 6 to 7 frames per flap, which corresponds to f = 8.5 to 10 Hz. In that part of the video, there were 15 flaps during an interval of 2.27 s13, which corresponds to f = 9.25 Hz, which is consistent with the range of values obtained by counting frames during single flaps.For birds that have intermittent flaps, the raw flap rate s in Eq. , is dete13. In Fig. 13, a 0.61 m reference object is compared with the wings of the bird in the video. The wingspan appears to be consistent with the 0.76 to 0.80 m wingspan of the Ivory-billed Woodpecker14 but greater than the 0.48 to 0.58 m wingspan of the Belted Kingfisher 15, which is the third largest species of the region (behind the two large woodpeckers) that has a wing motion in which the wings are folded closed (or nearly folded closed) during the flap cycle in cruising flight. As shown in Fig. 3. Using the mean value of 0.78 m for the wingspan of the Ivory-billed Woodpecker and 15.2 m/s for the flight speed in Eq. (7 and 0.705 m for the wingspan16 of the Pileated Woodpecker in Eq. (7.The bird in the video has a flight speed of 15.2 m/s7). In putative video footage of an Ivory-billed Woodpecker taking off (Movie 3), the wings are not folded closed and the flap rate is about 10 Hz (six frames per flap) just after takeoff, and this is consistent with an account by Christy of “deep and rapid strokes” at takeoff17. Historical accounts and video evidence suggest that the Ivory-billed Woodpecker has higher flap rates than the Pileated Woodpecker during takeoffs and cruising flights. In terms of flap rate, the relationship between these superficially similar species is evidently similar to the relationship between another pair of superficially similar species. Movies Mimus polyglottos) and the Loggerhead Shrike (Lanius ludovicianus). Those species are sometimes mistaken for one another when perched, but the difference in flap rate makes it easy to distinguish them immediately after takeoff.During a level takeoff that transitions into cruising flight, there may be a brief transient in which the wing motion and flap rate are not the same as in cruising flight. During the takeoffs by Pileated Woodpeckers that appear in Movie ff Movie  of ref.33. During upward swooping landings, woodpeckers typically ascend a short distance and land on a surface that faces the direction of approach. Some of the upward swooping landings in ref.3 involve long vertical ascents that provide time for maneuvering and landing on surfaces that do not face the direction of approach. Just after the 2:00 mark in a film of a pair of Magellanic Woodpeckers12, a close relative of the Ivory-billed Woodpecker, the female flies over to the male and ends the flight with a long vertical ascent that allows time for maneuvering. Just before landing, the bird is ascending and at the same time maneuvering to its left.A downward swooping takeoff on fixed wings is an alternative to a level takeoff with deep and rapid flaps. It is known from historical accounts that Ivory-billed Woodpeckers frequently take swooping flights. Putative video footage of swooping takeoffs and landings by Ivory-billed Woodpeckers is presented in Movies Drumming may be modeled as a harmonic oscillator with a periodic forcing. The transient behavior that occurs after the forcing is turned off suggests that double knocks are a transient corresponding to an impulsive forcing that results from a single thrust of the body, and the audio and video examples presented here support this hypothesis. A better understanding of the factors that may cause variations in the number of knocks and the time interval between knocks could be useful in interpreting evidence for the persistence of the Ivory-billed Woodpecker. Flap rate during cruising flight is amenable to statistical analysis, and the flap rate in putative footage of an Ivory-billed Woodpecker in cruising flight is consistent with that species in terms of the prediction of a flap rate model. In cruising flight, the Pileated Woodpecker closes its wings during the middle of the upstroke but uses a different wing motion and a higher flap rate during a transient just after taking off. Video evidence suggests that the Ivory-billed Woodpecker has the same types of wing motion during these types of flight but at higher flap rates.Movie S1Movie S2Movie S3Movie S4Movie S5Audio S1Audio S2Audio S3Supplementary information


Comparison of the presented results show that the sample-based approach offers comparable precision to the classical method of analysis. However, it is much simpler to apply and can be used despite the type of dynamical system and its dimensions. Moreover, the sample-based approach has some unique advantages and can be applied without the precise knowledge of parameter values.In this paper we show the first broad experimental confirmation of the basin stability approach. The basin stability is one of the sample-based approach methods for analysis of the complex, multidimensional dynamical systems. We show that investigated method is a reliable tool for the analysis of dynamical systems and we prove that it has a significant advantages which make it appropriate for many applications in which classical analysis methods are difficult to apply. We study theoretically and experimentally the dynamics of a forced double pendulum. We examine the ranges of stability for nine different solutions of the system in a two parameter space, namely the amplitude and the frequency of excitation. We apply the path-following and the extended basin stability methods (Brzeski  Therefore, also new tools to analyze multistable systems are being developed.There is a rich variety of different mathematical tools to analyze nonlinear dynamical systems. Still, more sophisticated methods are usually difficult to apply. For example, there are a number of different toolboxes that enable the path-following analysis but their functionality is strictly limited to the type of the investigated system and its dimensionality. The dynamical analysis is especially challenging for multistable systems, where we have to consider multiple steady states that coexist in the phase space. It is a challenging problem and multistability is widely studied in many disciplineset al. proposed a basin stability measure that uses Bernoulli trials to estimate the volume of a basin of attraction9. Despite it is new, the method was already successfully applied in numerous different applications13. In 2016 another new tools to investigate multistable systems were proposed, namely survivability14 that includes the analysis of the transient motion and basin entropy15 that measures the basin compactness.In 2013 Menck 16 or brain dynamics17. However, up to this moment there is a lack of experimental studies which enable to compare the accuracy of these methods and classical analysis.The growing interest in sample-based methods comes from two main advantages. They can be easily applied to all types of systems and reproduce the inherent uncertainty of perturbations.british To apply these methods, we just need to have a reliable direct numerical integration code for the mathematical model. Also the computational effort does not grow significantly with the increase of the phase space dimensions. This makes such sample-based methods even more appealing for the analysis of very high-dimensional systems such as large networks of oscillators, power grids18 we described an extended basin stability approach by taking into account mismatch in parameter values. This has practical fundamentals because all parameters values are measured or estimated with some finite precision and can slightly vary even during normal operation. In this paper, we expand our approach and perform an analysis in a two parameter space. We compare the results from sample-based analysis with detailed two parameter bifurcation diagrams obtained using the path-following method. Finally, we confront both methods with experimental data that we use as a benchmark. This is done for 9 different periodic solutions that coexist in a notably wide range of the parameter values. The results enable us to critically compare the accuracy of both methods and show their strengths and weaknesses. Apart from that, we show that the sample-based approach can be applied without a precise knowledge of parameter values and it gives sensible results.Recentlyx1 between both pin joints. The support is mounted on a shaker and excited kinematically in the vertical direction. This system is a modified version of the rig considered by Strzalko et al.19 and Dudkowski et al.1.We consider the specific type of a double pendulum see Fig.  which isφ1 and φ2 respectively. The shaker excites the system kinematically with a harmonic function of the amplitude A and frequency ω. The upper pendulum has the length l1, the mass m1, the moment of inertia J1 and the centre of gravity located at the distance d1 from its pin joint. The stiffness of the spring that supports the end of the pendulum is given by k. The second pendulum has the mass m2, the moment of inertia J2 and the centre of gravity located at d2 from its pin joint. Due to the relatively small damping in the system we neglect effects of dry friction and other non-linearities of damping characteristics and in both pivots we assume the viscoelastic damping. Hence, for the first pendulum the damping coefficient is given by c1 and for the second one by c2.The angular displacements of the first and the second pendulum are given by J1 = 4.524[10−3 kgm2], m1 = 0.5562 [kg], l1 = 0.315 [m], d1 = 0.180 [m], x1 = 0.153[m], c1 = 0.05 [Nms], J2 = 4.469 [10−5 kgm2], m2 = 0.02077[kg], d2 = 0.063[m], c1 = 7[10−6 Nms], g = 9.81[(m)/(s2)]. All the parameter values were determined in a series of dedicated experiments.The equations of motion of the system shown in Fig. A and the frequency ω of the external excitation. The system is multistable, hence we observe a coexistence of solutions for fixed parameter values. Different solutions stabilize and destabilize when varying the amplitude and/or frequency of excitation. The aim of our study is to investigate the ranges of attractors’ stability in the  plane. We start with the path-following method to get a precise boundaries of solutions’ stability. Then, for each solution we perform experiments. Finally, we apply the sample-based approach (the extended basin stability method18) to determine the ranges of parameters for which the given solution can be obtained. In the following subsections, we describe the applied methods and explain the scheme in which we present the results. This is crucial because we present and compare a large amount of results obtained with various methods. Due to limitations of the experimental rig, the maximum value of the amplitude of excitation is Amax = 7.7[10−3 m] (higher amplitudes of the shaker are impossible to achieve). Similarly, we consider the frequency of excitation in the range ωε〈0, 60〉[rad/s], because for higher values the accessible amplitude of the excitation Amax is rapidly dropping.We compare the results obtained by using two different approaches and validate them with experiments. As aforementioned, the considered system is a double pendulum forced kinematically in vertical direction Eq. . As contAε〈0, 7.7〉[10−3 m] and the frequency of excitation ωε〈0, 60〉[rad/s]. We perform path-following analysis for 9 different solutions. The upper pendulum always performs an oscillatory motion, while for the second pendulum we observe both oscillations and rotations with different locking ratios in respect to the frequency of the excitation. Therefore, we name each solution basing on the behaviour of the second pendulum and ratio n: m which means that for m periods of excitation we observe n full oscillations or rotations of the second pendulum. We detected and further consider: 1:1, 1:2, 1:3, 1:4, 1:6, 1:8 oscillations, and 1:1, 1:2, 1:3 rotations.In numerical study we consider the following ranges of the amplitude p software20. Each time we start with a direct numerical integration to prepare the initial periodic solution for continuation. Then, we start with one parameter continuation using A or ω as a continuation parameter. We detect the boundaries of stability and perform continuation of the bifurcation points in two parameter space. In Fig. For all 9 considered solutions we perform the path-following analysis by using the AUTO-07A, ω). The starting points for the experimental study are located far from the boundary of stability to ensure that the given solution is reachable experimentally. Depending on the shape and area of the stability range, we select from one to seven starting points and for each apply the following procedure:p software helps us to apply the correct initial state.We start the forcing with the amplitude and the frequency of the excitation taken from numerical results and try to reach the given solution by applying proper initial conditions to the second pendulum. The knowledge of the time trace of the solution from AUTO-07When the system reaches the presumed state we mark the amplitude and the frequency of excitation. These are the coordinates of a starting point in the two parameter space. Such points are marked with black dots in Fig. π × 10−3[rad/s] and in amplitude 0.01[m]). After each step we wait to check if the system remains on the presumed attractor. The range of the parameter value where the solution stays stable is marked as a trace with a black line in Fig. Then, we slowly change the value of the frequency or the amplitude of forcing with small steps , we repeat the above procedure for different starting points. We detect the ranges of stability for all 9 encountered solutions which required enormous effort and a lot of time.To estimate the position of a line that indicates the boundary of stability obtained by the path-following method in the two parameter space  we assume 23 subsets with equal width of 25[rad/s]. We intentionally omit extremely small values of ω and start from the range 〈1.25,3.75〉[rad/s] (for first column of the grid) and finish with 〈56.25, 58.75〉[rad/s] (for the last column). For the amplitude of forcing A∈〈0, 7.7〉[10−3m] we take 15 equally spaced subsets with the step of 0.5 [10−3 m]. Here, we also ignore values near the accessible boundaries and start the lower row of the grid with the range of 〈0.25, 0.75〉[10−3m] and finish the grid with 〈7.25, 7.75〉[10−3m]. By that, we receive a lattice of 345 boxes each covering the range of 2.5 [rad/s] and 0.5 [10−3 m] of the forcing frequencies and amplitudes respectively.Now we apply the extended basin stability approach described in our recent workA and ω from the ranges assigned to the grid box. For every trial we recognize the final attractor that is reached by the system. In a procedure described above, we obtain a large data set of 172, 500 Monte Carlo trials that we use to characterize the possible behaviours of the system.For each box in the grid we perform 500 trials of direct numerical integration. Each time we randomly pick the initial conditions from the accessible ranges and draw the values of A and ω) for which it has been approached. Then, we plot those values as a set of points in the  plane that indicates the region of stability of the investigated solution. In Fig. The data enable us to estimate the ranges of stability for each considered attractor. For that purpose, we detect all the trajectories that go to the investigated attractor and mark the parameters value  presents the one parameter continuation performed to detect the bifurcation points in which the stability of the solution changes. We perform two parameter continuation for these points, as shown in panel (d). We also mark the region of stability with a coloured area. In panel (b) we show the experimental results, while in panels (c) and (f) we present the outcome of the sample-based approach. We plot a single colour dot (here purple) each time the investigated solution is reached in a trial. The extended basin stability approach provides a basin stability map shown in panel (f). Such plot enables us to detect where the solution is more likely to appear due to the increased basin stability. Moreover, thanks to the lattice, we maintain a consistent level of error  because in each box we have 500 Bernoulli trials.To simplify the comparison between the results obtained with different analysis approaches, we combine the results from panels (d), (b) and (c) in a single plot such as Fig. The aim of our investigations is to compare different analysis approaches and experimentally validate the accuracy of the sample-based method. In this section we provide a thorough presentation of the results obtained using the path-following method, experimental investigations and the extended basin stability technique. We focus on the detection of the ranges of stability for 9 solutions mentioned in previous section. Moreover, we supplement the results with basin stability maps that enable us to quantify the stability and indicate the ranges of parameter values for which it is more likely that given solution occurs.We investigate the boundaries of stability for 9 different solutions that coexist in a wide range of the parameter values. The results are presented in Fig. ω = 20[rad/s]. The position of the branching bifurcation lines have been detected by path-following and confirmed using both experimental analysis and sample-based approach. The agreement between the results obtained using all three approaches is remarkably good.In panel (a) of Fig. ω = 20[rad/s]). The results are shown in Fig. A ≤ 5 [10−3 m]) or in a symmetry braking bifurcation (for A > 5 [10−3 m]) that is immediately followed by a cascade of period doubling bifurcations. The experimental analysis confirms that the position of the right-hand side border of the resonance tongue has been obtained with an excellent precision. However, the detected range of stability differs from the results obtained via the numerical continuation. The difference is especially visible for A = 4 [10−3 m]s  and for ω = 16[rad/s] in vertical direction. In these cases the sample-based method enables a better precision, especially when detecting the left-hand side border of stability. We see that the density of points decreases significantly before the bifurcation lines are reached. Thus using the sample-based approach we can better predict the range in which we are able to obtain 1:2 oscillations.The second analysed solution is 1:2 oscillations which corresponds to the resonance tongue  plane.The third analysed solution (panel (c)) is 1:3 oscillations. Here, the results from both methods of analysis are in good agreement with the experimental data. Similarly, for 1:6 oscillations (panel (e)) and 1:8 oscillations (panel (f)) we observe a concurrence of the results from all the applied methods. The difference between the experimental and numerical results is clearly distinguishable only in two cases. The experimental investigation revealed that the 1:6 oscillations can be achieved also below the line detected using the path-following method. Also, the position of a period doubling bifurcation which destroys the stability of 1:8 oscillations is different from what we observe experimentally. We think that the difference between the numerical results and the experiment mainly comes from the dissipation modelling approach which strongly simplifies this phenomenon. In panel (d) we show the results that correspond to 1:4 oscillations. For this solution, with the Monte Carlo approach can obtain the boundary of stability that is closer to the experimental results then the bifurcation lines obtained via numerical continuation. It is especially visible for the left and the lower boundaries in the  correspond to 1:1, 1:2 and 1:3 rotations respectively. For all rotary solutions we see a good convergence between the experiments and the results obtained using both numerical methods.21 or solutions with rather meager basins of attraction. In the investigated case we also found solutions that occurred only once for 172, 500 trials, such as for example 2:5 oscillations or 3:15 rotations. Moreover, analysing Fig. The results presented in Fig. 18. In Fig. The results presented in the previous subsection indicate that we are able to achieve comparable precision using the sample-based approach. Now, we show the additional information that can be obtained with extended basin stability methodPanel (a) refers to the 1:1 oscillations which has the largest range of stability see Fig. . The ploThe range where the semi-trivial solution has the dominant basin of attraction (see panel (a)).The resonance tongue of 1:2 oscillations in which this solution has the biggest volume of the basin (panel (b)).The region where, we observe the coexistence of many stable solutions and the 1:1 rotations has the dominant basin of attraction in most of that region (panel (g)).Comparing all the diagrams in Fig. Using the data obtained in a large series of trials we can use an interpolation and draw the continuous maps of basin stability (not using the lattice). We chose to present the data in the grid in order to maintain a similar level of error and to indicate the methodology of our approach. Despite the presentation method, basin stability maps enable to detect where the given solution has the biggest relative volume of the basin of attraction. This in consequence, refers to the ranges in parameter space for which the solution is more likely to appear. Therefore, such diagrams have strong practical significance.In practical applications we often cannot obtain the exact values of the system’s parameters. It is especially difficult when the model contains parameters whose values cannot be measured directly such as, for example, a value of the viscous damping coefficient in a pin joint. Apart from that, when modelling the mechanical and structural objects, we often simplify complex phenomena using simple models. This can decrease the accuracy of simulations and cause the divergence between numerical and experimental results. In this section we show that the extended basin stability method can be applied without the knowledge of actual parameter values and still maintain the high accuracy of the yielded results.m1, m2, l1, x1. This group contains the parameters that can be determined easily with good precision, namely masses and lengths of the physical objects. During calculations we assume that we determined values of these parameters precisely and do not draw them.Parameters that are easy to measure: J1, J2, d1, d2, k. The methods to measure these parameters values are not straight forward. In this group there are also parameters whose values are given by manufacturer with certain precision - as for example the stiffness of the spring. We assume that the error for the parameters from that group is ±5% and for each trial we draw the values of parameters from a certain range.Parameters that can be estimated with good precision: c1, c2. In this group we put parameters that require complex procedures or approach to infer their values and parameters that refers to phenomena that are difficult to model, such as energy dissipation. Values of parameters from this group are drawn from the range 〈−120%, 120%〉 of the estimated value.Parameters that are difficult to estimate: A, ω. The aim of the study is to analyse the influence of these parameters on the stability of solutions. For these parameters we do not estimate the error. Instead, we draw values of these parameters from a “grid” as described in the Methods section.Parameters whose special influence we investigate: Let us assume that we do not know precisely the values of some parameters of the system. In a classical approach, we have to set their values which may lead to wrong results. In the sample-based approach, instead of setting a value of the parameter, we can estimate the range to which this parameter belongs. Then, during a series of trials we draw the values of uncertain parameters. Before we apply that to our problem, we divide the parameters of our system Eq.  into fouApart from the above, we want to maintain the value of the natural frequency of the second pendulum which can be easily measured using a simple stopwatch. For that purpose, we draw the inertia of the second pendulum and recalculate the position of its centre of gravity to preserve the natural frequency.With the above assumptions we repeat all the calculations. In Fig. The difference in the results from the extended basin stability analysis are barely visible despite the fact, that the positions of the bifurcation lines change when we modify the parameter values within the assumed ranges. This shows that the sample-based approach can be applied even without time consuming detailed measurements of the system’s parameters and still ensures sensible results. Instead of precisely measure the parameter values, we only have to asses the precision for each of the determined parameter.In this paper we present a critical comparison between the results obtained using the extended basin stability method, the path-following bifurcation analysis and experimental data. We investigate the double pendulum system which is a model of an existing experimental rig. It is a multistable system in which 9 different periodic solutions have been detected.For each solution we use the path-following method to determine its range of stability in the two parameter space, namely the amplitude and the frequency of excitation. Then, we perform the experimental investigations for the same purpose. The obtained two parameter bifurcation diagrams allow us two determine ranges of stability and detect bifurcations that lead to destabilization of the investigated orbits. Finally, we apply the extended basin stability analysis to determine the ranges of stability and two parameters basin stability maps. Our results from both numerical methods are in a remarkably good agreement with the experimental data.In comparison to classical methods the advantage of the presented method is that it enables us to analyse the influence of several parameters simultaneously. The computational effort does not increase significantly with the dimensions of the system. Moreover, the method enables us to detect hidden attractors and solutions with rather meagre basins of attraction. The method only requires an efficient numerical integration algorithm. There is no need of specific knowledge or software to use this method. The usefulness of a sample-base method is especially visible in higher dimensional systems where the classical analysis requires a preliminary study to know the initial conditions, the shape of periodic solutions, the proper cross-sections of the phase space, while the extended basin stability method can be applied straightforward without such specific knowledge.Another advantage of the presented approach is that the collected data allow to quantify the stability of each attractor and prepare the two parameter basin stability maps. Such diagrams enable to detect the ranges in parameter space for which the solution is more likely to appear due to a large volume of the basin of attraction and have strong practical significance.In the last part of the work we repeat the basin stability analysis under uncertainties in the parameter values. The obtained results do not differ significantly from the original data, hence we claim that the proposed method can be applied even without the precise knowledge of the parameters values which is a unique feature.The above advantages predispose the sample-based approach for the analysis of multistable systems with large phase space dimensions and multiple co-existing attractors. Future application perspectives include especially analysis of large networks like power grids or simulations of brain dynamics.The presented results are the first broad comparison between the path-following method, the basin stability approach and the experimental investigation. We show that the sample-based methods are a reliable tool for the analysis of complex dynamical systems. Moreover, we prove that the extended basin stability method has significant advantages which make it robust and appropriate for many applications in which classical analysis methods are difficult to apply.Supplementary Info


Appropriate use criteria (AUC) for echocardiography based on clinical scenarios were previously published by an American Task Force. We determined whether members of the Dutch Working Group on Echocardiography (WGE) would rate these scenarios in a similar way.All 32 members of the WGE were invited to judge clinical scenarios independently using a blanked version of the previously published American version of AUC for echocardiography. During a face-to-face meeting, consensus about the final rating was reached by open discussion for each indication. For reasons of simplicity, the scores were reduced from a 9-point scale to a 3-point scale .n = 153). Seventy-one indications were rated as appropriate, 35 were rated as uncertain, and 47 were rated as inappropriate. In 5% of the cases the rating was opposite to that in the original (appropriate compared with inappropriate and vice versa), whereas in 20% judgements differed by 1 level of appropriateness. After the consensus meeting, the appropriateness of 7 (5%) cases was judged differently compared with the original paper.Nine cardiologist members of the WGE reported their judgment on the echo cases  contains supplementary material, which is available to authorized users. Sonography of the heart is one of the most widely used diagnostic tests in medicine. It has acquired a central role in the diagnosis, management, and follow-up of patients with any suspected or known heart diseases. The Dutch Working Group on Echocardiography (WGE) is an official working group within the Netherlands Society of Cardiology (NVVC) and one of its assignments is to guard the quality in clinical care, especially concerning echocardiography. The huge proportion of patients that nowadays undergo a first or follow-up echocardiographic evaluation in daily clinical care forced the WGE to search for tools to regulate the access to echocardiographic assessment according to clinical priority.In 2011, the American College of Cardiology published a revision of appropriate use criteria (AUC) regarding transthoracic and transoesophageal echocardiography and stress echocardiography [Several studies –7 have bIt is the aim of the current study to investigate whether the Dutch WGE would rate the clinical scenarios in a similar way as described in the original paper in order to determine the external validity.All 32 members of the WGE and members of the NVVC were invited to judge the 202 clinical scenarios of the original ACC paper  independNine cardiologists and members of the Dutch WGE judged the 154 clinical scenarios (see Supplementary online data). Two out of nine worked in a university hospital, five in a hospital with cardiac surgery. Less than 1% of the answers were missing.In the original paper, 52 of the scenarios were rated as inappropriate use, compared with 47 by the individual Dutch cardiologists. Uncertain and appropriate use were rated in 35 and 71 scenarios by our panel . The differences of the individual ratings compared with the original paper are presented in Fig. The 24 cases in which the mean judgement differed from the original paper were discussed extensively during the consensus meeting. Finally, after the consensus meeting, in only 7 cases the appropriateness was judged differently than in the original paper (see Supplementary online data). These cases were the following. Case 20 was judged as appropriate instead of uncertain. The majority of the group thought that the assessment of volume status in a critically ill patient is a meaningful diagnostic test and often performed in their clinical practice. However, the US authors could be led in their rating by uncertainties as ‘how to best assess volume status’ as it is not always straightforward and is open to differences in interpretation. In case 35 the score was judged as uncertain instead of inappropriate as there was consensus that echo can contribute to patient care in the evaluation of an asymptomatic murmur or click, although routine echocardiography without physical examination is not justified. Case 56 was judged as uncertain instead of inappropriate as it was believed that restrictive use of echocardiography in such a serious disease would give the wrong signal. Case 110 was considered inappropriate because a non-cardiac source had already been determined. Subsequent transoesophageal echocardiography is not of additional value. In case 185, the indication for stress echocardiography in asymptomatic severe mitral regurgitation and a left ventricle not meeting the surgical criteria was considered uncertain (as opposed to appropriate in the ACC paper). In case 187, stress echocardiography for asymptomatic moderate aortic regurgitation, the score was judged as inappropriate (as opposed to uncertain in the ACC paper). In case 188 the indication for stress echocardiography for asymptomatic severe aortic regurgitation and a left ventricle not meeting surgical criteria was also rated as uncertain (as opposed to appropriate in the ACC paper). These last three cases all concern patients who are asymptomatic, and in whom the stress echo would not change clinical management. In case 104 a remark was added that it should not be used as the initial test as transoesophageal echocardiography is not indicated as the initial diagnostic test in suspected aortic pathology including but not limited to dissection/transsection, because transoesophageal echocardiography might be harmful due to an increase in blood pressure. In our opinion, CT or MRI should be the initial test to explore the possibility of aortic dissection.In the Supplementary online data, the clinical scenarios are presented with the final judgement of the Dutch WGE.This is the first paper in which cardiologists dedicated to echocardiography compare their opinion about appropriateness of echocardiography in clinical scenarios with that of well-respected panellists of the original AUC publication. In summary, and in line with our American colleagues, echocardiography was rated appropriate when it is applied for an initial diagnosis, a change in clinical status or a change in patient management. Routine testing or a test without change in management are more likely to be inappropriate. However, an important finding was that in about 20%, the individual rating scores of clinical scenarios were different as compared with the ratings in the original paper. In addition, differences in opinion were reflected by lengthy discussions in the consensus meeting and by the remaining differences compared with the original paper. This illustrates the variation in clinical practice among dedicated cardiologists.The expanding use of imaging modalities in daily care and limited resources were the trigger to develop the AUC. The primary goal of the AUC presented in 2011 was to obtain a rational use of imaging services to deliver high-quality care . For exaThe current appropriateness criteria cover the majority of echocardiograpic studies in a general and university hospital. The current findings show the applicability of these criteria for the clinical practice of most hospitals. The clinical scenarios are easy to read, well structured, and are straightforward to interpret. This makes them a tool for every day practice, and applicable for everyone in the field. Their use will improve the responsible use of healthcare resources, motivate technicians by making echoes meaningful and reduce the burden of investigations for patients. Of note, the category of ‘uncertain’ is used when the panellists thought there were insufficient clinical data available for a definitive categorisation or there was substantial disagreement regarding the appropriateness of that indication. Therefore the rating ‘uncertain‘ should not be used as grounds for denial of reimbursement. However, an ‘uncertain’ appropriateness criterion may point to an undeveloped area of scientific research.Focused cardiac ultrasound is a simplified, clinician-performed application of echocardiography that is rapidly expanding in use, especially in emergency and critical care medicine. In 2014 clinical recommendations addressing focused cardiac ultrasound in clinical care were published . AlthougThis document is a complementary paper on the document on standard operating procedures of echocardiography from the WGE in the Netherlands in order to improve quality of care . MoreoveIn the present study, the rating of the clinical scenarios in the final paper was reduced to a 3-point scale rating system instead of a 9-point rating system for reasons of simplicity. However, this might have led to less information and make the paper less sensitive than the original one. Nevertheless, in our opinion the current document is a good reflection of cardiology healthcare in the Netherlands and can improve clinical decision-making resulting in a more effective use of resources.To improve the accuracy of our outcomes, 48 clinical scenarios on stress echocardiography in coronary artery disease were omitted because the use of this test for the purpose of ischaemia detection or viability is limited in our country. The current AUC for echocardiography are based on consensus statements making them subordinate to evidence-based recommendations. This type of recommendation evaluates all the information on a clinical subject in a systematic way by reviewing, rating, and synthesising the large amount of literature and then making an unbiased, evidence-based series of recommendations on clinical problems. In this manner, evidence-based recommendations yield a higher impact improvement of physician performance and patient outcomes.Echocardiography was rated appropriate when it is applied for an initial diagnosis, a change in clinical status or change in patient management. However, the appropriateness for echocardiography was rated differently from the original paper in 5% of the clinical scenarios. Further research is warranted to analyse causes of decreased external validity.Clinical scenarios rated by the Dutch Working Group echocardiography