II. Towards a Theory of Nonlinear Dynamics & Chaos

3.

4.

5.

6.

7.

8.

Dynamics in State Space: 1- & 2- D 3-D State Space & Chaos Iterated Maps Quasi-Periodicity & Chaos Intermittency & Crises Hamiltonian Systems

3.

Dynamics in State Space: 1- & 2- D

Concepts to be introduced: State space / Phase Space H. Poincare J.W. Gibbs Fixed points ( equilibium / stationary / critical / singular ) points Limit Cycles Stability (attractor) / Instability (repellor) Bifurcations : Change of stability / Birth of f.p. or l.c.

State Space

Degrees of freedom : 1. Classical mechanics ( phase space ) : number of (q,p) pairs.

2. Dynamical systems ( state space ) : number of independent variables.

Spring obeying Hooke ’s law : 

x



t



x

  sin 

t m x

  

k x

Cycle : Closed periodic trajectory  

k m

Systems of 1

st

Order ODEs

u



u i

 

f i

Autonomous

i

 1, ,

n

DoF = n

u



u i

 

f i

Non Autonomous Dimension of state space = number of 1st order autonomous ODEs.

N-DoF non autonomous → (N+1)-DoF autonomous

u i

 

f i i

 1, ,

n

 1 Autonomous

u n

 1

u n

 1  

t



f n

 1  1 Non-crossing theorem is applicable only to autonomous systems

One n th order ODE ~ n 1 st order ODEs Mass spring:      2         2

u

 

f u





u u

1 , 2

f

 

f f

1 , 2   

y

,   2

x

 

u

2 ,   2

u

1  2 nd order ODE Two 1 st order ODEs

u

1 

u

2   2      2 1  

Given

u

  u* is a fixed point if 

0

Caution : Autonomous version of a non-autonomous system requires special treatment [ u n+1  = 1  0 ].

All dynamical systems can be converted to a set of 1st order ODEs.

For some systems this requires DoF = ∞, e.g., • PDEs • integral – differential eqs • memory eqs If the system is dissipative, only a few DoFs will remain active eventually.

No-Intersection Theorem

• A state space trajectory cannot cross itelf.

• 2 distinct state space trajectories cannot intersect in a finite amount of time.

Physical implication : Determinism Mathematical origin : Uniqueness solutions of ODE that satisfy the Lipschitz condition (f  bounded).

Apparent violations: • Asymptotic intersects.

• Projections

Dissipative Systems & Attractors

• Transients not important in dissipative systems ( long time final states independent of IC ) • Attractor : Region of state space to which some trajectories converge.

• Basin of an attractor : Region of state space through which all trajectories converge to that attractor.

• Separatrix : Boundary between the basins of two different attractors.

•Miscellaneous: –Fractal basin boundaries.

–Riddled basins of attraction.

–Dimension of the state space.

Evolution eq. : Fixed point :

X

1-D State Space

X

 Types of fixed points in 1-D state spaces: • Nodes / sinks / stable fixed points • Repellors / sources / unstable fixed points • Saddle points

Type Determination

Let X 0 be a fixed point:

X

0  

d f d X

= characteristic value ( eigenvalue ) of X 0 0 For λ > 0 For λ < 0 

X

0  

X

0 

X

  0

for X

 

X

0     

X

0     

X

0

X

  0

for X

 

X

0 

X

0 

X

0  

X

0 

X

0         

λ = 0 2

d f d X

2 0  0  0

for for X X

 

X

0

X

0

d X

2 0  0  0

for for X X

 

X

0

X

0

λ = 0 2

d f d X

2 0  0

X

0  

X

0 

X

0   

X

 0

for X

 

X

0        

X

0 Repells Attracts

Saddle points

λ = 0

d X

2 0  0

X

 0

for X

 

X

0

X

0  

X

0  

X

0          

X

0 Attracts Repells Convex Concave

Structural Instability

Saddle point is structurally unstable

Taylor series Expansion

X 0 = fixed point     0 

X



X

0 

d f d X

0  1 2 

X



X

0  2 2

d f d X

2 0 

X

  

X



X

0  

X

0 

X



x e



t

 

d f d X

0  0 

X

0

e



t

 

X

0 Lyapunov exponent  > 0  X 0 repellor  < 0  X 0 node  = 0  X 0 s.p. / node / rep.

Trajectories in 1-D State Space

  local behavior Global behavior determined by matching fixed point basins.

→ joining arrows pointing toward (away) from nodes (repellors).

f continuous  neighboring fixed points cannot be • both nodes or both repellors • saddle points of different types Exercises 3.8- 3,4

Bounded systems

: Outermost fixed points must be • nodes or • • type I saddle point on the left type II saddle point on the right.

A node must be on the repelling side of a saddle point.

When Is A System Dissipative ?

Defining characteristics of a dissipative system : Motion reduced asymptotically to a few active DoFs. Cluster of ICs ( those that lead to fixed points excluded ).

C.f., statistical ensembles.

d L



d t d d t



X B



X A

 

d L Ld t d f d X A X B



X A

  

B

  

A d f d X A



X B



X A

  System is dissipative near X A if df/dX < 0 .

e.g., near a node.

Divergence theorem

X

1 

X

2 

f

1

f

2  

X X

1 , 2

X X

1 , 2  

2-D State Space

0  0 

f

1

f

2  

X

10 ,

X

10 ,

X

20

X

20  

X

1  

f

1 

X

1 0 

X

1 

X

10    

f

1

X

2 0 

X

2 

X

20  

X

2  

f

2 

X

1 0 

X

1 

X

10   

f

2 

X

2 0 

X

2 

X

20  

x

1 

x

2 

f

1 , 1

f

2 , 1

x

1 

x

1 

f

1 , 2

f

2 , 2

x

2

x

2

x



f x

x j



X j



X j

0

f i

,

j

 

f i



x j

0 Fixed point

λ 1 < 0 λ 2 < 0 λ 1 > 0 λ 2 > 0

Special case

f

1 , 1

f

2 , 1   1  0

x

1

x

2   1   2

x

1

x

2

f

1 , 2

f

2 , 2  0   2 λ 1 > 0 λ 2 < 0 λ 1 < 0 λ 2 > 0

  

y

Saddle point at (x,y) = (-π/2 , 0 ) 1 0.5

0 -0.5

-1

Hyperbolic point : λ  0 • • • Invariant manifold: all trajectories along the principal axes of a hyperbolic saddle point.

Stable (invariant) manifold ( in-set ): Trajectories heading towards the hyperbolic saddle point.

Unstable (invariant) manifold ( out-set ): Trajectories heading away from the hyperbolic saddle point.

In-sets & outsets serves as separatrices.

1-D saddle point are non-hyperbolic since λ = 0

→ Fixed points:

Brusselator

X



B

 1



X

 2

X Y Y



BX

 2

X Y X A X A



X

 0

BX

 2

X Y

 0 

X

0 ,

Y

0 

A

,

B A

A,B > 0

x

1 

x

2 

f x

11 1 

f x

21 1 

f x

12 2

f x

22 2 General 2-D Case

x

1 

f x

11 1 

f x

12 2 

f x

11 1 

f

12 

f x

21 1 

f x

22 2    

f x

11 1 

f

11 

f

12  

f x

21 1 

f

22

f

22 

x

1  

f f

12 21 

x

1 

f

12

f x

11 1  

f f

11 22 

x

1

x

1 

Ce



t

→  2  

f

11 

f

22    

f f

11 22 

f f

12 21   0    1 2  

f

11 

f

22  

f

11 

f

22  2  4 

f f

11 22 

f f

12 21     1 2  

f

11 

f

22  

f

11 

f

22  2  4

f f

12 21  

x

1   

t

  

t

State variables can be any pair from

x

2 

B e

 

t

 

x x x x

1 , 1 , 2 , 2   

t

Ex 3.11

Complex Characteristic Values

   1 2  

f

11 

f

22  

f

11 

f

22  2  4  

i

21  

R

 1 2  

f

11 

f

22  Re 

f

11 

f

22  2  4

f f

12 21     Im



f

11 

f

22



2  4 21 Note :



f

11 

f

22



2  4 21 is either real or purely imaginary

x

1 

e R t C e i



t



x

2 

e R t B e i



t



t t

Spirals, inward if R < 0 ( focus ) outward if R > 0 Limit cycle if R = 0

C

 

C

 

A

2

B

  

B

 

A

2

i x

1 

Ae R t

cos 

t x

2 

Ae R t

sin 

t

R < 0 R > 0

Dissipation & Divergence Theorem 2-D state space Area :

A

 

X

1

C



X

1

B



X

2

C



X

2

B



d A



d t f

1 

X

1

C

,

X

2

B

 

f

1 

X

1

B

,

X

2

B

 

X

2

C



X

2

B f

1

f

2 

X

1

C

, 

X

1

B

,

X

2

B X

2

C

 

f

1

f

2 

X

1

B

, 

X

1

B

,

X

2

B X

2

B X

1

C



X

1

B



f

2 

X

1

B

,

X

2

C

 

X

1

C X

2

C

 

X

1

B X

2

B

   

f X

1 1  

f

2

X

2 

X

1

B

,

X

2

B

 

X

1

B

,

X

2

B

  

f

2 

X

1

B

,

X

2

B



d A



d t f

11 

X

1

B

,

X

2

B

 

X

1

C



X

1

B



X

2

C



X

2

B

1

d A



A d t f

11 

f

22  

f



f

< 0  dissipative

X

1

C



X

1

B



X

2

C



X

2

B



f

22 

X

1

B

,

X

2

B

 1

dV V d t



j N

  1

f j j

 

f

  

t

0

X i



f i

Jacobian Matrix at Fixed Point

X i



X i

0 

x i

→

x i



j N



 1

f x i j j x i



Ce



t

→ 

x j



j N



 1

f x i j j

→

J

x

 

x

J

    

x f i j

    

f

11

f N

1

f

1

N

 

f NN

Jacobian matrix

f

11  

f N

1

f

1

N f NN

   0

Tr f



i N

  1

f i i



i N

  1 

i

det

f



i N

  1 

i

2-D State Space    1 2  

f

11 

f

22  

f

11 

f

22  2  4 

f f

11 22 

f f

12 21     1 2 

Tr J

 

Tr J

 2  4

det J

   1 2 

Tr J

  Δ < 0 Δ > 0 det J > 0 Δ > 0 det J det J < 0 Tr J < 0 Both λ complex, Real part negative Spiral node Both λ real & negative node Both λ real & of of opposite signs, Saddle point Tr J > 0 Both λ complex, Real part positive Spiral repellor Both λ real & positive repellor Both λ real & of of opposite signs, Saddle point

Example: The Brusselator

X Y



BX

 

B

 1 

X

 

X

0 ,

Y

0 

A

,

B A J

   

B

 

B

1

A

2 

A

2  

Tr J B

det

J



A

2

A

2    1 2  

B A

2 



B

Set A = 1 & let B be control parameter :

A

2



2  4

A

2      1 2  

B

• B < 2, spiral node • 2 < B < 4, spiral repellor (converge to another limit cycle) • B > 4, do exercise 3.14-2.



B

 2  2  4  

Limit Cycles

Limit cycle : closed loop in state space to (from) which nearby trajectories are attracted (repelled).  vortex Invariant set : region in state space where a trajectory starting in it will remain there forever.

Poincare-Bendixson theorem : Let R be a finite invariant set in a 2-D state space, then any trajectory in it must, as t → ∞, approach a 1.

2.

fixed point , or limit cycle.

Implications : • no chaos in 2-D systems.

• limit cycle in Brusselator.

Delayed DE

u

 :    DoF =  : IC for t  [-T,0] needed Topology : Poincare index theorem 

Poincare Sections

Poincare section in n-D state space: An (n-1)-D hyper-surface that cuts through the trajectory of a n-D continuous flow and reduces it to a (n-1)-D discrete map.

Example : Limit cycle in 2-D state space

Poincare Map

P n

 1  Exercise 3.16-1 Let Fixed point of F : Near P*:

d n



P n



P

*

P

* 

P

2 

P

*  → 

d

2 

d F d P P

*

d

1 

Md

1

d F d P P

* 

P

1 

P

*  

M



d F d P P

* = (characteristic / Floquet / Lyapunov) multiplier

d n



Md n

 1 →

d n



M n

 1

d

1 Characteristic exponent   ln

M

M < 1 M > 1 M = 1 Attracting Repelling Saddle (rare in 2-D )

Bifurcation Theory

Study of changes in the character of fixed points.

( limit cycles are fixed points in Poincare sections ) Appendix B 2 types of bifurcation diagrams: • control parameter vs location of fixed point.

• control parameter vs characteristic value.

x

    1 

x



a



Bifurcations in 1-D

y

 

y

Normal form δ> 0 repellor δ< 0 node Bifurcation at δ= 0

x x

2

d f d x

  2

x

No Fixed points if μ< 0.

2 fixed points for μ> 0

x

*   node

x

*    repellor

d x

2   2 -4 -2 -2 -4 -6 -8 -10 4 2 • For μ= 0, x* = 0 is a saddle point. • For μ> 0, x* = ±  μ form repellor-node pair.

• μ= 0 is repellor-node bifurcation point.

• Other names: saddle-node / tangent / fold bifurcation Note that for μ= 0, x* = 0 is structurally unstable.

2 4

Lifted (Suspended) State Space

Flow along extra dimension X 2 always towards original axis X 1 .

x

1

x

1 2

x

2  

x

2 Repellor ↓ Saddle point Node ↓ Node No fixed point

Bifurcation in 2-D The Brusselator    1 2  

B



B

 2  2  4   B > 4, λ ± real (Transcritical) bifurcation at B = 2.

Node → Repellor + limit cycle 4  B  0, λ ± complex

Normal Form Equations

Fixed point at

x

=

0

. Bifurcation at μ = 0.

Saddle – node bifurcation :

x x

2 1  

x

2

x

1 2 μ> 0 : node at saddle at μ< 0 : no fixed point 

x x

1 , 2    

x x

1 , 2    ,0  ,0  μ= 0 : bifurcation

Transcritical bifurcation :

x



x

 

y

 

y



x

 2 fixed points switching types of stability Pitchfork bifurcation :

x y



x

   

y



x

2  1 → 3 fixed points

Limit Cycle Bifurcations

Spiral-in-spiral-out bifurcation at Re( λ) = 0 Hopf bifurcation : birth of stable limit cycle Poincare section of limit cycle in 2 D → 1-D dynamics Normal form:

x

1    2

x

1

x

2

x

2     

x

1 2 

x

1 2  

x

2 2

x

2 2   Polar coordinates:

r



r

  

r

2     1

r



x

1 2 

x

2 2   tan  1

x

2

x

1    0 

t



r

  

r

2 

f

 3

r

2 Fixed points: for μ< 0 r* = 0 spiral node for μ> 0 r* = 0 spiral repellor

r

*   limit cycle, period = 2π Hopf bifurcation at μ= 0 Limit cycle: asymptotic time-dependent behavior of dissipative system.