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3-D State Space & Chaos
• Fixed Points, Limit Cycles, Poincare Sections
• Routes to Chaos– Period Doubling– Quasi-Periodicity– Intermittency & Crises– Chaotic Transients & Homoclinic Orbits
• Homoclinic Tangles & Horseshoes• Lyapunov Exponents
Heuristics
No chaos in 1- & 2-D state space
Chaos: nearby trajectories diverge exponentially for short times
Restrictions:
• orbits bounded
• no intersection
• exponential divergence
Can’t be all satisfied in 1- or 2-D
Strange attractor
0 td t d e λ= Lyapunov exponent
Chaotic attractor
Chaos is interesting only in systems with attractors.Counter-example: ball perched on hill top.
Routes to Chaos
Surprises• Ubiquity of chaotic behavior• Universality of routes to chaos
Except for solitons, there are no general method for solving non-linear ODEs.
Asymptotic motion
Regular ( stationary / periodic ) Chaotic
Known Types of Transitions to Chaos
• Local bifurcations (involves 1 limit cycle)– Period doubling– Quasi-periodicity– Intermittency:
• Type I ( tangent bifurcation intermittency )• Type II ( Hopf bifurcation intermittency )• Type III ( period-doubling bifurcation intermittency )
• On-off intermittency
• Global bifurcations ( involves many f.p. or l.c. )– Chaotic transients– Crises
A system can possess many types of transitions to chaos.
3-D Dynamical Systems
1 2 3, ,i ix f x x x 1,2,3i autonomous
x f x
2-D system with external t-dependent force :
1 1 1 2
2 2 1 2
, ,
, ,
x f x x t
x f x x t
1 1 1 2 3
2 2 1 2 3
3
, ,
, ,
1
x f x x x
x f x x x
x
~
Ex 4.4-1. van der Pol eq.
21 cosx x x x F t
Fixed Points in 3-D
1 1 1
1 2 3
2 2 2
1 2 3
3 3 3
1 2 3
f f f
x x x
f f fJ
x x x
f f f
x x x
3 2 0p q r 3 0x ax b
2 3
2 3
b as
s = Discriminant
Index of a fixed point = # of Reλ > 0 = dim( out-set )
1
3x p 21
33
a q p 312 9 27
27b p q p r
→
1/ 3
2
bA s
1/ 3
2
bB s
1or 3
2x A B A B A B i
A+B (A-B)i Roots
s < 0
A = B* complex
real real 3 real
s = 0
A = B real real 0 3 real (2 equal)
s > 0
A B real real imaginary
1 real, 2 complex
index = 0
index = 0
index = 3
index = 3
S.P. Bif
Poincare Sections
Poincare sections:
• Autonomous n-D system: (n-1)-D transverse plane.
• Periodically driven n-D system: n-D transverse plane.
( stroboscopic portrait with period of driven force )
transverse plane
Non- transverse plane
Trajectory is on surface of torus in 3-D state space of equivalent autonomous system.
Phase : [0 , 2π)
Poincare section = surface of constant phase of force
Limit cycle (periodic) → single point in Poincare sectionSubharmonics of period T = N Tf → N points in Poincare section
Periodically driven 2-D system ( non-autonomous )
Approach to a limit cycle
Caution: Curve connecting points P0 , P1 , P2 etc, is not a trajectory.
Limit Cycles
Assume: uniqueness of solution to ODEs
→ existence of Poincare map function 1
1 , ,n n n nj j k jx F x x F x 1, ,j k
Fixed point ~ limit cycle:
1* *, , * *j j k jx F x x F x
Floquet matrix: *
i
j
FJM
x
x x
1 1
1
1 *
k
k k
k
F F
x x
F F
x x
x x
Characteristic values :: stabilityBut, F usually can’t be obtained from the original differential eqs.
Floquet multipliers = Eigenvalues of JM = Mj
Dissipative system: det 1jj
JM M
1*n n nj j j j jd y y M d
Mj < 0 alternation
Yj = coordinate along the
jth eigenvector of JM
( Not allowed in 2-D systems due to the non-crossing theorem )
fixed point Floquet Multiplier Cycle
Node |Mj| < 1 j Limit cycle
Repellor |Mj| > 1 j Repelling cycle
Saddle mixed Saddle cycle
3-D case:
Circle denotes |M| = 1
Ex 4.6-1
Quasi-Periodicity
T2 can be represented in 3-D state space :
1
2
3
sin cos
cos
sin sin
r R
r
r R
x R r t t
x r t
x R r t t
21
22
0
0
x x
y y
→ 4-D state space
2 2 21 1
2 2 22 2
x x E
y y E
→ trajectories on torus T2
System with 2 frequencies:
r
R
rational
r
R
irrational
• Commensurate
• Phase-locked
• Mode-locked
• Incommensurate
• quasi-periodic
• Conditionally periodic
• Almost periodic
Neither periodic, nor chaotic
Routes to Chaos I: Period-Doubling
Flip bifurcation:
all |M| < 1 (Limit cycle) → One M < -1 (period doubling)
( node ) ( 1 saddle + 2 nodes )
There’s no period-tripling, quadrupling, etc. See Chap 5.
Routes to Chaos II: Quasi-Periodicity
Hopf bifurcation:
spiral node → Limit cycle
Ruelle-Takens scenario :
2 incommensurate frequencies ( quasi-periodicity )
→ chaos
Landau turbulence: Infinite series of Hopf bifurcations
21
r r r
Details in Chap 6
Routes to Chaos III: Intermittency & Crises
Details in Chap 7
Intermittency:
periodic motion interspersed with irregular bursts of chaos
Crisis:
Sudden disappearance / appearance / change of the size of basin of chaotic attractor.
Cause: Interaction of attractor with unstable f.p. or l.c.
Routes to Chaos IV: Chaotic Transients & Homoclinc Orbits
Global bifurcation:
• Crises: Interaction between chaotic attractor & unstable f.p / l.c.
• sudden appearance / disappearance of attractor.
• Chaotic transients:
Interaction of trajectory with tangles near saddle cycle(s).
• not marked by changes in f.p. stability
→ difficult to analyse.
• most important for ODEs, e.g. Lorenz model.
• Involves homoclinic / heteroclinic orbits.
Homoclinic Connection
Saddle Cycle
Poincare section
Critical theorem:
The number of intersects between the in-sets & out-sets of a saddle point in the Poincare section is 0 or ∞.
See E.A.Jackson,
Perspectives of Nonlinear Dynamics
Poincare section
Non-Integrable systems: Homoclinic tangle
Integrable systems: Homoclinic connection
Heteroclinic Connection (Integrable systems)
Heteroclinic Tangle (Non-Integrable
systems)
Lorenz Eqs
Sil’nikov Chaos:
• 3-D: Saddle point with characteristic values
a, -b + i c, -b - i c a,b,c real, >0
• → 1-D outset, 2-D spiral in-set.
• If homoclinic orbit can form & a > b, then chaos occurs for parameters near homoclinic formation.
• Distinction: chaos occurs before formation of homoclinic connection.
Homoclinic Tangles & Horseshoes
• Stretching along WU.
• Compressing along WS.
• Fold-back
• → Horseshoe map
Smale-Birkoff theorem:
Homoclinic tangle ~ Horseshoe map
Details in Chap 5
Experiment: Fluid mixing.
2-D flow with periodic perturbation,
dye injection near hyperbolic point.
Lyapunov Exponents & Chaos
Quantify chaos:
• distinguish between noise & chaos.
• measure degree of chaoticity.
Chapters:
• 4: ODEs
• 5: iterated map
• 9,10: experiment
x f x 0
0 0
x
d ff x x x
d x
x(t), x0(t) = trajectories with nearby starting points x(0), x0(0).
0s x x = distance between the trajectories
For all x(t) near x0(t):
0 0s x x f x f x 0x
d fs
d x 00 x ts t s e
0
0
x
d fx
d x = Lyapunov exponent
at x0.
→
0x = average over x0 on same trajectory.
n-D system:
0 s x x 0i i is x x
x f x 0
0 0 x
f x x x f
i ix f x 0
0 0i
i j jj j
ff x x
x
x
x
0 0 s x x f x f x 0
x
s f
0 0i i i i is x x f f x x0
ij
j j
fs
x
x
Let ua be the eigenvector of J(x0) with eigenvalue λa(x0).
0i j jj
J s x
0 J x s
0a a au x u→ 00 a t
a at e xu u
Chaotic system: at least one positive average λa = <λa (x0) >.
0
0i
i jj
fJ
x
x
x
Behavior of a cluster of ICs
0 exp ii
V t V t 1
ii
dV
V dt
f
Dissipative system:
0ii
3-D ODE: • One <λ> must be 0 unless the attractor is a fixed point.
H.Haken, Phys.Lett.A 94,71-4 (83)• System dissipative → at least one <λ> must be negative.• System chaotic → one <λ> positive.
Hyperchaos: More than one positive <λ>.
Signs of λs Type of attractor
-, -, - Fixed point
0, -, - Limit cycle
0, 0, - Quasi-periodic torus
+, 0, - Chaotic
Spectra of Lyapunov exponents in 3-D state space
Cautionary Tale
Choatic → <λ> > 0 converse not necessarily true.
Pseudo-chaos:
On outsets of saddle point <λ> > 0 for short time
Example: pendulum
sing
uL
u
Saddle point: Θ=πg
L
0 03.00 0.
3.02 0.0724
u
same E