Dynamical Systems 2 Topological classification Ing. Jaroslav Jíra, CSc.

More Basic Terms Basin of attraction is the region in state space of all initial conditions that tend to a particular solution such as a limit cycle, fixed point, or other attractor. Trajectory is a solution of equation of motion, it is a curve in phase space parametrized by the time variable. The Flow of a dynamical system is the expression of its trajectory or beam of its trajectories in the phase space, i.e. the movement of the variable(s) in time Nullclines are the lines where the time derivative of one component of the state variable is zero. Separatrix is a boundary separating two modes of behavior of the dynamical system. In 2D cases it is a curve separating two neighboring basins of attraction.

A simple pendulum Differential equation After transformation into two first order equations

An output of the Mathematica program Phase portratit for the simple pendulum Used equations

A simple pendulum with various initial conditions Stable fixed point φ0=0° φ0=45° φ0=90° φ0=135° Unstable fixed point φ0=180° φ0=170° φ0=190° φ0=220°

A damped pendulum Differential equation After transformation into two first order equations Equation in Mathematica: NDSolve[{x'[t] == y[t], y'[t] == -.2 y[t] - .26 Sin[x[t]], … and phase portraits

A damped pendulum commented phase portrait Nullcline determination: At the crossing points of the null clines we can find fixed points.

A damped pendulum simulation

Classification of Dynamical Systems One-dimensional linear or linearized systems Time Derivative at x~ Fixed point is Continous f’(x~)<0 Stable f’(x~)>0 Unstable f’(x~)=0 Cannot decide Discrete |f’(x~)|<1 |f’(x~)|>1 |f’(x~)|=1

Verification from the bacteria example Bacteria equation Derivative 1st fixed point - unstable 2nd fixed point - stable

Classification of Dynamical Systems Two-dimensional linear or linearized systems Set of equations for 2D system Jacobian matrix for 2D system Calculation of eigenvalues Formulation using trace and determinant

Types of two-dimensional linear systems 1. Attracting Node (Sink) Equations Jacobian matrix Eigenvalues λ1= -1 λ2= -4 Eigenvectors Solution from Mathematica Conclusion: there is a stable fixed point, the node-sink

2. Repelling Node Equations Jacobian matrix Eigenvalues Eigenvectors λ1= 1 λ2= 4 Eigenvectors Solution from Mathematica Conclusion: there is an unstable fixed point, the repelling node

3. Saddle Point Equations Jacobian matrix Eigenvalues Eigenvectors λ1= -1 λ2= 4 Eigenvectors Solution from Mathematica Conclusion: there is an unstable fixed point, the saddle point

4. Spiral Source (Repelling Spiral) Equations Jacobian matrix Eigenvalues λ1= 1+2i λ2= 1-2i Eigenvectors Solution from Mathematica Conclusion: there is an unstable fixed point, the spiral source sometimes called unstable focal point

5. Spiral Sink Equations Jacobian matrix Eigenvalues Eigenvectors Solution from Mathematica Conclusion: there is a stable fixed point, the spiral sink sometimes called stable focal point

6. Node Center Equations Jacobian matrix Eigenvalues Eigenvectors Solution from Mathematica Conclusion: there is marginally stable (neutral) fixed point, the node center

Brief classification of two-dimensional dynamical systems according to eigenvalues

Special cases of identical eigenvalues A stable star (a stable proper node) Equations and matrix Eigenvalues + eigenvectors Solution An unstable star (an unstable proper node) Equations and matrix Eigenvalues + eigenvectors Solution

Special cases of identical eigenvalues A stable improper node with 1 eigenvector Equations and matrix Eigenvalues + eigenvectors Solution An unstable improper node with 1 eigenvector Equations and matrix Eigenvalues + eigenvectors Solution

Classification of dynamical systems using trace and determinant of the Jacobian matrix 1.Attracting node p=-5; q=4; Δ=9 2. Repelling node p=5; q=4; Δ=9 3. Saddle point p=3; q=-4; Δ=25 4. Spiral source p=2; q=5; Δ=-16 5. Spiral sink p=-2; q=5; Δ=-16 6. Node center p=0; q=5; Δ=-20 7. Stable/unstable star p=-/+ 2; q=1; Δ=0 8. Stable/unstable improper node

Example 1 – a saddle point calculation in Mathematica

Example 2 – an improper node calculation in Matlab function [t,y] = setequationsimnode tspan=[0,5]; for k=-10:10; for l=-10:20:10; init=[k;l]; [t,y]=ode45(@f,tspan,init); plot(y(:,1),y(:,2)); hold on; end init=[l;k]; %Plot annotation xlabel('x1') ylabel('x2') title('AN IMPROPER NODE'); grid on; function yprime=f(t,y) yprime=zeros(2,1); yprime(1)=-y(1)+y(2); yprime(2)=-y(2); end clc hold off

Classification of Dynamical Systems Linear or linearized systems with more dimensions Time Eigenvalues Fixed point is Continous all Re(λ)<0 Stable some Re(λ)>0 Unstable all Re(λ)<=0 some Re(λ)=0 Cannot decide Discrete all |λ|<1 some |λ|>1 all |λ|<=1 some |λ|=1