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Dynamical Systems 2 Topological classification

Dynamical Systems 2 Topological classification. Ing. Jaroslav J í ra , 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.

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Dynamical Systems 2 Topological classification

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

  2. 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.

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

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

  5. 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°

  6. 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

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

  8. A damped pendulumsimulation

  9. Classification of Dynamical SystemsOne-dimensional linear or linearized systems

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

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

  12. Types of two-dimensional linear systems1.Attracting Node (Sink) Equations Jacobian matrix Eigenvalues λ1= -1 λ2= -4 Eigenvectors Solution from Mathematica Conclusion: there is a stablefixed point, the node-sink

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

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

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

  16. 5.Spiral Sink Equations Jacobian matrix Eigenvectors Eigenvalues λ1= -1+2i λ2= -1-2i Solution from Mathematica Conclusion: there is a stablefixed point, the spiral sink sometimes called stable focal point

  17. 6.Node Center Equations Jacobian matrix Eigenvectors Eigenvalues λ1= +1.732i λ2= -1.732i Solution from Mathematica Conclusion: there is marginally stable (neutral) fixed point, the node center

  18. Brief classification of two-dimensional dynamical systems according to eigenvalues

  19. 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

  20. 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

  21. Classification of dynamical systems usingtrace 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 p=-/+ 2; q=1; Δ=0

  22. Example 1 – a saddle point calculation in Mathematica

  23. 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 end for k=-10:10; for l=-10:20:10; init=[l;k]; [t,y]=ode45(@f,tspan,init); plot(y(:,1),y(:,2)); hold on; end end %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 end

  24. Classification of Dynamical SystemsLinear or linearized systems with more dimensions

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