Dynamical Systems 2 Topological classification

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. 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 of 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 attracting node (sink)

13. A quick preview by the Vectorplot function in the Mathematica

14. Meaning of the Eigenvectorexample of modified attracting node Equations Jacobian matrix Eigenvalues λ1= -3.62 λ2= -1.38 Eigenvectors Eigenvector directions are emphasized by black arrows

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

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

17. 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

18. 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 is sometimes called stable focal point

19. 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

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

21. 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

22. 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

23. 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

24. Several configurations of damped oscillator Equation of motion Rewritten into a set of 1st order equations Overdamped oscillator, δ=2 s-1, ω=1 s-1 Attracting node Underdamped oscillator, δ=1 s-1, ω=2 s-1 Spiral sink Critically damped osc. , δ=1 s-1, ω=1 s-1 Stable improper node Simple harmonic osc. , δ=0 s-1, ω=1 s-1 Node center

25. Example 1 – a saddle point calculation in Mathematica

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

28. Basic Types of 3D systems Node – all eigenvalues are real and have the same sign Attracting Node – all eigenvalues are negative λ1< λ2< λ3< 0 Repelling Node – all eigenvalues are positive λ1> λ2> λ3> 0

29. Basic Types of 3D systems Saddle point – all eigenvalues are real and at least one of them is positive and at least one is negative; Saddles are always unstable; λ1< λ2< 0 < λ3 λ1> λ2 > 0 > λ3

30. Basic Types of 3D systems Focus-Node – there is one real eigenvalue and a pair of complex-conjugate eigenvalues, and all eigenvalues have real parts of the same sign. Stable Focus-Node – real parts of all eigenvalues are negative Re(λ1)<Re(λ2)<Re(λ3)<0 Unstable Focus-Node – real parts of all eigenvalues are positive Re(λ1)>Re(λ2)>Re(λ3)>0

31. Basic Types of 3D systems Saddle-Focus Point – there is one real eigenvalue with the sign opposite to the sign of the real part of a pair of complex-conjugate eigenvalues; This type of fixed point is always unstable. Re(λ1)> Re(λ2) > 0 > λ3 Re(λ1) < Re(λ2) < 0 < λ3