Download presentation
Published bySpencer Anderson Modified over 9 years ago
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
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.