II. Towards a Theory of Nonlinear Dynamics & Chaos
3.
Dynamics in State Space: 1- & 2- D
4.
3-D State Space & Chaos
5.
Iterated Maps
6.
Quasi-Periodicity & Chaos
7.
Intermittency & Crises
8.
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  0 cos  t 
x  0 

m x  k x
sin  t
Cycle: Closed periodic trajectory

k
m
Systems of 1st Order ODEs
u  t   f  u 
ui  t   fi  u 
i  1,
u  t   f  u, t 
ui  t   fi  u, t 
Autonomous
,n
DoF = n
NonAutonomous
Dimension of state space = number of 1st order autonomous ODEs.
N-DoF non-autonomous → (N+1)-DoF
autonomous
i  1, , n  1
ui  t   fi  u 
un 1  t   t
un 1  t   f n 1  u   1
Autonomous
Non-crossing theorem is applicable
only to autonomous systems
One nth order ODE ~ n 1st order ODEs
Mass spring:
x  t   2 x  t 
x  t   y  t 
y   t    2 x  t 
u  f
u  u1, u2   x, y
f   f1 , f 2    y ,   2 x  u2 ,   2u1
2nd order ODE
Two 1st order ODEs
u1  t   u2  t 
u2  t    2u1  t 
Given
u  f  u
u* is a fixed point if
f  u *  0
Caution: Autonomous version of a non-autonomous system
requires special treatment [ un+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.
1-D State Space
Evolution eq. :
Fixed point:
X  f X 
X  0  f X 
Types of fixed points in 1-D state spaces:
• Nodes / sinks / stable fixed points
• Repellors / sources / unstable fixed points
• Saddle points
Type Determination
X
Let X0 be a fixed point:
df

dX
For λ > 0
X X0
X  X0
= characteristic value ( eigenvalue ) of X0
X


0 for X


X0
X t   X 0  X t     X t   X 0
X t   X 0  X t     X t   X 0
For λ < 0
 0  f  X0 
X


0 for X


X0
X t   X 0  X 0  X t     X t 
X  t   X 0  X 0  X t     X t 
λ = 0
d2 f
dX2
d2 f
dX2
 0 for X  X 0
X  X0
 0 for X  X 0
 0 for X  X 0
X  X0
 0 for X  X 0
λ = 0
d2 f
dX2
0
X X0
X  0 for X
X 0  X t   X 0  X t   X t   
X t   X 0  X t   X t     X 0


X0
Repells
Attracts
Convex
Saddle points
λ = 0
d2 f
dX2
0
X  0 for X
X X0
X 0  X t   X 0  X t     X t 
X t   X 0  X t     X t   X 0


X0
Attracts
Repells
Concave
Structural Instability
Saddle point is structurally unstable
Taylor series Expansion
X0 = fixed point
df
f  X   f  X0    X  X0
dX
X    X  X0 
X X0
2
1
2 d f
  X  X0
2
dX2

df
dX

X X0
0
X  X0
Lyapunov
exponent
X  t   X 0   X  0   X 0  e t
x  t   x  0  e t
 > 0  X0 repellor
 < 0  X0 node
x t   X t   X 0
 = 0  X0 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.
dL d
  X B  X A  X B  X A  f  XB   f  X A
dt dt
dL
Ld t
df
dX
df
dX
 XB  X A 
X X A
System is dissipative near XA if df/dX < 0.
X X A
e.g., near a node.
Divergence theorem
2-D State Space
X 1  f1  X 1 , X 2 
0  f1  X 10 , X 20 
X 2  f2  X 1, X 2 
0  f 2  X 10 , X 20 
X1 
X2 
 f1
f
 X 1  X 10   1
 X1 0
X2
 f2
 f2
X

X

 1 10 
 X1 0
X2
 X 2  X 20  
0
 X 2  X 20  
0
x1  f1,1 x1  f1, 2 x2
x j  X j  X j0
x2  f 2 ,1 x1  f 2 , 2 x2
fi , j 
x  fx
 fi
xj
0
Fixed point
Special case
λ1 < 0 λ 2 < 0
f1,1  1
f1, 2  0
f 2 ,1  0
f2 , 2  2
λ1 > 0 λ2 < 0
x1  1 x1
x2   2 x2
λ1 > 0 λ2 > 0
λ1 < 0 λ2 > 0
g  x, y   cos x sin y
Saddle point at (x,y) = (-π/2 , 0 )
1
0
-1
1
0.5
0
-0.5
-1
-3
-2
-1
0
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
Brusselator
X  A   B  1 X  X 2Y
Y  BX  X Y
2
→
X Y  A X
Fixed points:
A X  0
BX  X 2Y  0
B

 X 0 , Y0    A, 
A

A,B > 0
General 2-D Case
x1  f11 x1  f12 x2
x1  f11 x1  f12 x2
x2  f 21 x1  f 22 x2
 f11x1  f12  f 21x1  f 22 x2 

x f x 
 f11 x1  f12  f 21 x1  f 22 1 11 1 
f12 

  f11  f 22  x1   f12 f 21  f11 f 22  x1
x1  Ce t
→
 2   f11  f 22     f11 f 22  f12 f 21   0
1
   f11  f 22 
 f11  f 22 
1
f11  f 22 

2
 f11  f 22 
2 

x1  C e t  Ce t
State variables can be any pair from
2
2
 4  f11 f 22  f12 f 21  

 4 f12 f 21 

x2  B e t  Be t
x1, x1, x2 , x2 
Ex 3.11
Complex Characteristic Values
1
   f11  f 22 
2 
 f11  f 22 
 4 f12 f 21 

2
1
R
f11  f 22  Re

2
  R  i 
  Im
 f11  f 22   4 f12 f 21
Note:
2
x1  e Rt C ei  t  C e i  t 
x2  e  B e
Rt
i t
 B e
i  t

 f11  f 22 
2
 4 f12 f 21 

 f11  f 22   4 f12 f 21
2
is either real or purely imaginary
Spirals, inward if R < 0 (focus)
outward if R > 0
Limit cycle if R = 0
A
C  C 
2
A
B   B 
2i
R < 0
x1  Ae Rt cos t
x2  Ae Rt sin t
R > 0
Dissipation & Divergence Theorem
2-D state space
Area :
A   X1C  X1B  X 2C  X 2 B 
dA
  f1  X 1C , X 2 B   f1  X 1B , X 2 B    X 2C  X 2 B    X 1C  X 1B   f 2  X 1B , X 2 C   f 2  X 1B , X 2 B  
dt 
f1  X 1C , X 2 B 
f 2  X 1B , X 2 C 
f1  X 1B , X 2 B    X 1C  X 1B 
 f1
 X1  X
f 2  X 1B , X 2 B    X 2 C  X 2 B 
 f2
 X2

1B , X 2 B


 X1 B , X 2 B 
dA
 f11  X , X   X 1C  X 1B  X 2C  X 2 B    X 1C  X 1B  X 2C  X 2 B  f 22  X , X 
1B
2B
1B
2B
dt
1 dA
 f11  f 22   f
A dt
 f < 0  dissipative
N
1 dV
  f j j  f
V dt
j 1

  j  0
t
Jacobian Matrix at Fixed Point
N
X i  X i 0  xi
X i  fi
→
xi   fi j x j
j 1
N
xi  Ce t
→
 x j   fi j x j
 f11
  f i 

J 
   fi j   
  x j 
f
 N1
f11  
f1N
0
f N1
f NN  
Jx  x
→
j 1
f1N 


f NN 
Jacobian matrix
N
N
i 1
i 1
Tr f   f ii    i
N
det f    i
i 1
2-D State Space
 

1
f11  f 22 
2 
1
Tr J 

2
 f11 
f 22   4  f11 f 22  f12 f 21  

2
Tr J   4 det J 

2

1
Tr J 
2
 
Tr J < 0
Tr J > 0
Δ<0
Both λ complex,
Real part negative
Spiral node
Both λ complex,
Real part positive
Spiral repellor
Δ>0
det J > 0
Both λ real & negative
node
Both λ real & positive
repellor
Δ > 0 det J
det J < 0
Both λ real &
of
of opposite signs,
Saddle point
Both λ real &
of
of opposite signs,
Saddle point
Example: The Brusselator
X  A   B  1 X  X 2Y
B

X
,
Y

A
,
 0 0 

A

Y  BX  X 2Y
TrJ  B  1  A2
 B 1 A 
J 
2

B

A


2
   B  1  A2 
2
1
det J  A2
2 2
B

1

A

  4 A2 
Set A = 1 & let B be control parameter :
1
   B  2 
2 
• B < 2, spiral node
• 2 < B < 4, spiral repellor (converge to another limit cycle)
• B > 4, do exercise 3.14-2.
 B  2  4 

2
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.
fixed point , or
2.
limit cycle.
Implications:
• no chaos in 2-D systems.
• limit cycle in Brusselator.
Delayed DE:
u  f t    g t  T 
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
Pn1  F  Pn 
Exercise 3.16-1
P*  F  P *
Fixed point of F :
Near P*:
Let d n  Pn  P *
M
dF
dP
P2  P*  F  P1   F  P *
d2 
→
dF
dP
dF
dP
 P1  P * 
P*
d1  Md1
P*
= (characteristic / Floquet / Lyapunov) multiplier
P*
d n  Md n1
→
d n  M n 1d1
Characteristic exponent
  ln M
M<1
Attracting
M>1
Repelling
M=1
Saddle (rare in 2-D )
Bifurcation Theory
Appendix
B
Study of changes in the character of fixed points.
( limit cycles are fixed points in Poincare sections )
2 types of bifurcation diagrams:
• control parameter vs location of fixed point.
• control parameter vs characteristic value.
Bifurcations in 1-D
x     1 x  a 
y  y
Normal form
δ> 0 repellor
δ< 0 node
Bifurcation at δ= 0
df
 2 x
dx
x    x2
d2 f
 2
2
dx
4
2
-4
-2
2
-2
-4
-6
No Fixed points if μ< 0.
-8
-10
2 fixed points for μ> 0
x*  
x*   
node
repellor
• 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.
4
Lifted (Suspended) State Space
Flow along extra dimension X2 always towards original axis X1.
x1    x12
Repellor
↓
Saddle point
x2   x2
Node
↓
Node
No fixed point
Bifurcation in 2-D
The Brusselator
1
   B  2 
2 
 B  2  4 

2
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:
x1    x12
x2   x2
μ> 0 :
node at
saddle at
μ< 0 : no fixed point
μ= 0 : bifurcation
 x1, x2     ,0
 x1, x2      ,0
Transcritical bifurcation:
x  x   x 
y  y
2 fixed points switching
types of stability
Pitchfork bifurcation:
x  x   x 2 
y  y
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:
Polar coordinates:
x1   x2  x1     x12  x22 
x2   x1  x2     x12  x22 
r x x
r  r    r2   f r 
 1
2
1
2
2
  tan 1
x2
x1
  t   0  t
f r  r    r2 
f   r     3r 2
Fixed points:
for μ< 0
r* = 0 spiral node
for μ> 0
r* = 0
r*  
spiral repellor
limit cycle, period = 2π
Hopf bifurcation at μ= 0
Limit cycle:
asymptotic time-dependent
behavior of dissipative system.