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 http://trac.clermont.cemagref.fr/projets/Kaviar/wiki/download#Download

Installation and running http://trac.clermont.cemagref.fr/projets/Kaviar/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

GUI mode Display window Console window

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

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

Stopping criterion of the SVMs computation algorithm GUI mode (cont) SVM configuration Stopping criterion of the SVMs computation algorithm SVM algorithm 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 - C-SMO: see libSVM - Simple SVM - Balk: automative bandwidth tuning - Soft-Balk: balk with soft margin

GUI mode (cont) Execution and control

GUI mode (cont) Indicators + logs

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

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 x0 = 2, y0 = 0.8, 20 time steps, 3 time steps anticipation, 3 distance SVM, dist(K) = 0.025 Inside the viability kernel x0 = 2, y0 = 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

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(); }

MyClass.java (extends Dynamical_System)

MyClass.java (extends Dynamical_System_Target) Previous +

MyClass.java (extends Dynamical_System_Resilience) Previous +

Example on the Abrams&Strogatz model Dynamics and constraints 2 languages A and B in competition, no bilingual people σA: 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

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 = c1*time + c2(distance(σA from viability)) …

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: x0 =0.95, y0= 0.95 and x0 =0.7, y0= 0.95 Optimal control outside the viability kernel Heavy control once the system is back to the kernel

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

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