dynamical systems 2 topological classification
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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