Download Dynamical Systems 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.
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
g
g
2
sin   0; 0 
L
L
  0 2 sin   0
 
Differential equation
After transformation into
two first order equations
  
  0 2 sin 
An output of the Mathematica program
Phase portratit of the simple pendulum
Used equations
  
  0.26 sin 
A simple pendulum with various initial conditions
Stable fixed point
φ0=0°
φ0=170°
φ0=45°
φ0=190°
φ0=90°
φ0=220°
φ0=135°
Unstable fixed point
φ0=180°
A Damped Pendulum
Differential equation
After transformation into
two first order equations
g
g
2
sin   0; 0 
L
L
  2  0 2 sin   0
  2 
  
  2  0 2 sin 
Equation in Mathematica: NDSolve[{x'[t] == y[t], y'[t] == -.2 y[t] - .26 Sin[x[t]], …
and phase portraits
A Damped Pendulum
commented phase portrait
  0    0
Nullcline determination:
0 2 sin 
  0   2  0 sin   0    
2
2
At the crossing points of the null clines we can find fixed points.
A Damped Pendulum
simulation
Classification of Dynamical Systems
One-dimensional linear or linearized systems
Time
Continous
Discrete
Derivative at x~ Fixed point is
f’(x~)<0
Stable
f’(x~)>0
Unstable
f’(x~)=0
Cannot decide
|f’(x~)|<1
Stable
|f’(x~)|>1
Unstable
|f’(x~)|=1
Cannot decide
Verification from the bacteria example
Bacteria equation
dx
 bx  px 2 ;
dt
Derivative
f ( x)  b  2 px
1st fixed point - unstable
~
x1  0;
2nd fixed point - stable
b
~
x2  ;
p
f ( x)  bx  px 2
f (~
x1 )  b  0
b
~
f ( x2 )  b  2 p  b  0
p
Classification of Dynamical Systems
Two-dimensional linear or linearized systems
Set of equations for 2D system
x1  f1 ( x1 , x2 )  ax1  bx2
x2  f 2 ( x1 , x2 )  cx1  dx2
Jacobian matrix for 2D system
 f1
 x
JA 1
 f 2
 x1
Calculation of eigenvalues
b 
a  
det( A  E)  0  det 
0
c
d




(a   )( d   )  bc  0  2  (a  d )  ad  bc  0
Formulation using trace and
determinant
f1 
x2  a b 

f 2   c d 
x2 
Tr ( A)  a  d ; Det ( A)  ad  bc
2  Tr ( A)  Det ( A)  0
Tr ( A)  Tr ( A) 2  4 Det ( A)
12 
2
Types of two-dimensional linear systems
1. Attracting Node (Sink)
Equations
Jacobian matrix
Eigenvalues
λ1= -1
λ2= -4
x1   x1
x2  4x2
 1 0 
A

0

4


Eigenvectors
1   0 
 0  1 
   
Solution from Mathematica
Conclusion: there is a stable fixed point, the attracting node (sink)
A quick preview by the Vectorplot function in the Mathematica
Meaning of the Eigenvector
example of modified attracting node
x1   x1  x2
x2   x1  4x2
Equations
Jacobian matrix
Eigenvalues
λ1= -3.62
λ2= -1.38
 1 1 
A


1

4


Eigenvectors
 0.382  2.618
 1   1 

 

Eigenvector directions are
emphasized by black arrows
2. Repelling Node
Equations
Jacobian matrix
Eigenvalues
λ1= 1
λ2= 4
x1  x1
x2  4x2
1 0
A

0
4


Eigenvectors
1   0 
 0  1 
   
Solution from Mathematica
Conclusion: there is an unstable fixed point, the repelling node
3. Saddle Point
Equations
Jacobian matrix
Eigenvalues
λ1= -1
λ2= 4
x1   x1
x2  4x2
 1 0
A

0
4


Eigenvectors
1   0 
 0  1 
   
Solution from Mathematica
Conclusion: there is an unstable fixed point, the saddle point
4. Spiral Source (Repelling Spiral)
Equations
Jacobian matrix
Eigenvalues
λ1= 1+2i
λ2= 1-2i
x1  x1  2 x2
x2  2 x1  x2
 1 2
A

 2 1 
Eigenvectors
  i  i 
 1  1
  
Solution from Mathematica
Conclusion: there is an unstable fixed point, the spiral source
sometimes called unstable focal point
5. Spiral Sink
Equations
Jacobian matrix
Eigenvalues
λ1= -1+2i
λ2= -1-2i
x1   x1  2 x2
x2  2 x1  x2
 1 2 
A


2

1


Eigenvectors
  i  i 
 1  1
  
Solution from Mathematica
Conclusion: there is a stable fixed point, the spiral sink
is sometimes called stable focal point
6. Node Center
Equations
Jacobian matrix
Eigenvalues
λ1= +1.732i
λ2= -1.732i
x1   x1  x2
x2  4 x1  x2
 1  1
A

4 1
Eigenvectors
 0.25  0.43i   0.25  0.43i 

 

1
1

 

Solution from Mathematica
Conclusion: there is marginally stable (neutral) fixed point, the node center
Brief classification of two-dimensional dynamical systems according to
eigenvalues
Special cases of identical eigenvalues
A stable star (a stable proper node)
Equations
and matrix
x1   x1
x2   x2
Eigenvalues +
eigenvectors
12  1
Solution
x1  x10e t ; x2  x20e t
 1 0 
A

0

1


 0  1 
1   0 
   
An unstable star (an unstable proper node)
Equations
and matrix
x1  x1
x2  x2
Eigenvalues +
eigenvectors
12  1
Solution
x1  x10et ; x2  x20et
1 0
A

0 1
 0  1 
1   0 
   
Special cases of identical eigenvalues
A stable improper node with 1 eigenvector
Equations
and matrix
x1   x1  x2
x 2   x2
Eigenvalues +
eigenvectors
12  1
Solution
 1 1 
A

0

1


1   0 
0  0 
   
x1  ( x10  tx20 )e t ; x2  x20e t
An unstable improper node with 1 eigenvector
Equations
and matrix
x1  x1  x2
x2  x2
Eigenvalues +
eigenvectors
12  1
Solution
x1  ( x10  tx20 )et ; x2  x20et
1 1
A

0 1
1   0 
0  0 
   
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
p=-/+ 2; q=1; Δ=0
Tr ( A)  Tr ( A) 2  4 Det ( A) p  p 2  4q
12 

2
2
Several configurations of damped oscillator
Equation of
motion
Rewritten into a set of
1st order equations
x  2 x   2 x  0
x  v
v   2 x  2 v
Overdamped oscillator,
δ=2 s-1, ω=1 s-1
1
0
A

  1  4
Underdamped oscillator,
δ=1 s-1, ω=2 s-1
1
0
A


4

2


12  1  i 3
Spiral sink
Critically damped osc. ,
δ=1 s-1, ω=1 s-1
1
0
A

  1  2
12  1
Stable improper node
Simple harmonic osc. ,
δ=0 s-1, ω=1 s-1
 0 1
A


1
0


12  i
Node center
1  2  3
 2  2  3
Attracting node
Example 1 – a saddle point calculation in Mathematica
Classification of Dynamical Systems
Linear or linearized systems with more dimensions
Time
Eigenvalues
all Re(λ)<0
Continous
Discrete
Fixed point is
Stable
some Re(λ)>0
Unstable
all Re(λ)<=0
some Re(λ)=0
all |λ|<1
Cannot decide
Stable
some |λ|>1
Unstable
all |λ|<=1
some |λ|=1
Cannot decide
Basic Types of 3D systems
Node – all eigenvalues are real and have the same sign
Attracting Node – all eigenvalues
are negative
Repelling Node – all eigenvalues
are positive
λ1< λ2< λ3< 0
λ1> λ2> λ3> 0
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
Basic Types of 3D systems
Focus-Node – there is one real eigenvalue and a pair of complexconjugate 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
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