1. Dynamical Systems 2 Topological 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] Sin[x[t]], …
and phase portraits

7. A Damped Pendulum commented phase portrait
Nullcline determination:
At the crossing points of the null clines we can find fixed points.

8. A Damped Pendulum simulation

9. 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

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

11. 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

12. 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

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

14. Meaning of the Eigenvector example 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 Eigenvalues Eigenvectors
λ1= 1
λ2= 4
Eigenvectors
Solution from Mathematica
Conclusion: there is an unstable fixed point, the repelling node

16. 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

17. 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

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

19. 6. Node Center Equations Jacobian matrix Eigenvalues Eigenvectors
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 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

24. Example 1 – a saddle point calculation in Mathematica

26. 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

27. 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

28. 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

29. 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

30. 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