A tool to approximate viability kernels, capture basins and resilience values

Kernel Approximation for VIAbility and Resilience Written in Java programming language – Regular grid / active learning algorithm – Capture basins and resilience values are computed in dim d – Heavy or optimal controllers – Two modes: GUI and batch mode r/wiki/download#Download 2

Installation and running wiki/download#Download Requires the java virtual machine (Sun’s JRE environment 5 or later compulsory) 2 set up –.jar file to test the models already implemented –.zip file to implement new models 3

GUI mode Display window Console window 4

GUI mode (cont) Dynamical system settings Viability constraint set Control bounds Time step Function of the size of the grid Study of the dynamics!!! 5

GUI mode (cont) Viability controller config Algorithm type Optimization settings General settings - Simple: gradient descent from the minimal values of the controls - Double: min and max values - Conjugate gradient - Double conjugate gradient - Newton method Visualize the individual trajectories 6

GUI mode (cont) SVM configuration Stopping criterion of the SVMs computation algorithm SVM algorithm - C-SMO: see libSVM - Simple SVM - Balk: automative bandwidth tuning - Soft-Balk: balk with soft margin Bandwidth - big C : hard margin(no misclassification) - small C: soft margin Only the gaussian kernel is implemented Control the “smoothness” of the SVM function - Small gamma: smooth - Big gamma: less smooth 7

GUI mode (cont) Execution and control 8

GUI mode (cont) Indicators + logs 9

Example on the population problem Viability kernel approximation – Play with dt, # time steps, # points (and show trajectories) To obtain a “good” approximation, the dt value must be chosen accordingly the number of points and time steps Inner approximation sometimes… – Save the results and reload them 10

Example on the population problem Controller – Kernel approx with dt = 0.05, 6 time steps, 31 points – A point out of the viability kernel approximation x 0 = 2, y 0 = 0.8, 20 time steps, 3 time steps anticipation, 3 distance SVM, dist(K) = – Inside the viability kernel x 0 = 2, y 0 = 0.5, 150 time steps – More time steps anticipation: 15 – Bigger SVM value: 30 – Same parameters, with 1 time step for the viability kernel approximation 11

Adding a dynamical system Creation of a new class file (for instance MyClass.java) – Extend Dynamic_System if viability kernel approximation – Extend Dynamic_System_Target if capture basins approximation – Extend Dynamic_System_Resilience if resilience values In this class, create a main method to add your model and launch the software Public static void main (String[]args){ //init Kaviar kaviar = new Kaviar(); //Optional: to add default models Kaviar.addModels(Kaviar.DEFAULT_MODEL); //Optional: to add one of the default models //Kaviar.addModels(Population.class); //replace my model by the name of your model Kaviar.addModels(MyModel.class); //Launch the GUI Kaviar.startGUI(); } 12

MyClass.java (extends Dynamical_System) 13

MyClass.java (extends Dynamical_System_Target) Previous + 14

MyClass.java (extends Dynamical_System_Resilience) Previous + 15

Example on the Abrams&Strogatz model Dynamics and constraints – 2 languages A and B in competition, no bilingual people σ A : density of speakers of language A (in % - [0;1]). Parameter a: volatility of language A (a > 1 leads a scenario of dominance of 1 language) Parameter s: prestige of language A (s = 0.5: the two languages are socially equivalent – [0;1]) – Government, institution etc. can play on the prestige of one language, but modifications take time – We consider that one language is endangered when its proportion of speakers is less that 20% – with 16

Example on the Abrams&Strogatz model Resilience values – Endangered language doesn’t mean that the language is dead. Is there any action policies that allows the system to recover? – At which cost? λ = 1: measure the time the system is deprived from its property of interest λ = c 1 *time + c 2 (distance(σ A from viability)) … 17

Example on the Abrams&Strogatz model Optimal control – Compute the resilience values with the following parameters: dt =0.2, dc = 0.5, double optimization, C0 = 1, C1 = 300, 31 points, 6 time steps, inner approx a = 2: dominance of one language a = 0.2: stable coexistence – Control of the system: x 0 =0.95, y 0 = 0.95 and x 0 =0.7, y 0 = 0.95 Optimal control outside the viability kernel Heavy control once the system is back to the kernel 18

Batch mode.simu files are needed Create them following a given template Use the GUI interface java -cp Kaviar-1.1.jar Appli/Batch Conso.simu 2 files:.svm +.log files, in the Conso… directory 19

Batch mode java -cp Kaviar-1.1.jar Appli/Batch Conso.simu -v 9*2 files:.svm +.log files, in the Conso… directory 20