dynamical systems 2 topological classification

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 0sin

;0sin

20

20

L

g

L

g

After transformation into

two first order equations

sin20

An output of the Mathematica program

Phase portratit for the simple pendulum

sin26.0

Used equations

A simple pendulum with various initial conditions

Stable fixed point

φ0=0°

φ0=170°

φ0=45° φ0=90° φ0=135°

φ0=190° φ0=220°

Unstable fixed point

φ0=180°

A damped pendulum

Differential equation 0sin2

;0sin2

20

20

L

g

L

g

Equation in Mathematica: NDSolve[{x'[t] == y[t], y'[t] == -.2 y[t] - .26 Sin[x[t]], …and phase portraits

After transformation into

two first order equations

sin2 20

A damped pendulumcommented phase portrait

Nullcline determination:

2

sin0sin20

00202

0

At the crossing points of the null clines we can find fixed points.

A damped pendulumsimulation

Classification of Dynamical SystemsOne-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 Stable

|f’(x~)|>1 Unstable

|f’(x~)|=1 Cannot decide

Verification from the bacteria example

22 )(; pxbxxfpxbxdt

dxBacteria equation

Derivative

1st fixed point - unstable

2nd fixed point - stable

pxbxf 2)(

0)~(;0~11 bxfx

02)~(;~22 b

p

bpbxf

p

bx

Classification of Dynamical SystemsTwo-dimensional linear or linearized systems

212122

212111

),(

),(

dxcxxxfx

bxaxxxfx

dc

ba

x

f

x

fx

f

x

f

2

2

1

2

2

1

1

1

AJ

0)(0))((

0det0)det(

2

bcaddabcda

dc

ba

EA

Set of equations for 2D system

Jacobian matrix for 2D system

Calculation of eigenvalues

2

)(4)()(

0)()(

)(;)(

2

12

2

AAA

AA

AA

DetTrTr

DetTr

bcadDetdaTr

Formulation using trace and determinant

Types of two-dimensional linear systems1. Attracting Node (Sink)

22

11

4xx

xx

40

01A

Equations

Jacobian matrix

Eigenvalues

λ1= -1

λ2= -4

Conclusion: there is a stable fixed point, the node-sink

Solution from Mathematica

Eigenvectors

1

0

0

1

2. Repelling Node

22

11

4xx

xx

Equations

Jacobian matrix

Eigenvalues

λ1= 1

λ2= 4

40

01A

Conclusion: there is an unstable fixed point, the repelling node

Solution from Mathematica

Eigenvectors

1

0

0

1

3. Saddle Point

22

11

4xx

xx

Equations

Jacobian matrix

Eigenvalues

λ1= -1

λ2= 4

40

01A

Conclusion: there is an unstable fixed point, the saddle point

Solution from Mathematica

Eigenvectors

1

0

0

1

4. Spiral Source (Repelling Spiral)

212

211

2

2

xxx

xxx

Equations

Jacobian matrix

Eigenvalues

λ1= 1+2i

λ2= 1-2i

12

21A

Conclusion: there is an unstable fixed point, the spiral source sometimes called unstable focal point

Solution from Mathematica

Eigenvectors

11

ii

5. Spiral Sink

212

211

2

2

xxx

xxx

Equations

Jacobian matrix

Eigenvalues

λ1= -1+2i

λ2= -1-2i

12

21A

Conclusion: there is a stable fixed point, the spiral sink sometimes called stable focal point

Solution from Mathematica

Eigenvectors

11

ii

6. Node Center

212

211

4 xxx

xxx

Equations

Jacobian matrix

Eigenvalues

λ1= +1.732i

λ2= -1.732i

14

11A

Conclusion: there is marginally stable (neutral) fixed point, the node center

Solution from Mathematica

Eigenvectors

1

43.025.0

1

43.025.0 ii

Brief classification of two-dimensional dynamical systems according to eigenvalues

Special cases of identical eigenvalues

22

11

xx

xx

10

01A

0

1

1

0112

tt exxexx 202101 ;

Equationsand matrix

Eigenvalues +eigenvectors

22

11

xx

xx

10

01A

0

1

1

0112

Equationsand matrix

Eigenvalues +eigenvectors

tt exxexx 202101 ;Solution

Solution

A stable star (a stable proper node)

An unstable star (an unstable proper node)

Special cases of identical eigenvalues

22

211

xx

xxx

10

11A

0

0

0

1112

tt exxetxxx 20220101 ;)(

Equationsand matrix

Eigenvalues +eigenvectors

22

211

xx

xxx

10

11A

0

0

0

1112

Equationsand matrix

Eigenvalues +eigenvectors

tt exxetxxx 20220101 ;)(Solution

Solution

A stable improper node with 1 eigenvector

An unstable improper node with 1 eigenvector

2

4

2

)(4)()( 22

12

qppDetTrTr

AAA

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

Example 1 – a saddle point calculation in Mathematica

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; endend

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; endend

Example 2 – an improper node calculation in Matlab

%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);endclchold offend

Classification of Dynamical SystemsLinear 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)=0Cannot decide

Discrete

all |λ|<1 Stable

some |λ|>1 Unstable

all |λ|<=1

some |λ|=1Cannot decide