strange nonchaotic attractors in quasiperiodically forced period-doubling systems
Post on 12-Jan-2016
37 Views
Preview:
DESCRIPTION
TRANSCRIPT
1
Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems
Quasiperiodically Forced Systems
1). (mod ),,( 11 nnnnn xFx
: Irrational No.
Typical Appearance of Strange Nonchaotic Attractors (SNAs)
Property of SNAs: 1. No Sensitivity to Initial Condition (<0) 2. Fractal Phase Space Structure
Smooth Torus SNA(Intermediate State)
Chaotic Attractor
Sang-Yoon Kim (KWNU, UMD)
2
Typical Dynamical Transitions in Quasiperiodically Forced Period-Doubling Systems
Quasiperiodically Forced Logistic Map
Phase Diagram
Main Interesting Feature
“Tongue,” where Rich Dynamical Transitions Occur: • Route a Intermittency • Route b or c Interior Crisis of SNA or CA • Route d or e Boundary Crisis of Torus or SNA
(All These Dynamical Transitions May Occur through Collision with a New Type of “Ring-Shaped Unstable Set.”)
Smooth Torus (Light Gray): T and 2TCA (Black), SNA (Gray and Dark Gray)
1), (mod
),1()2cos(:
1
1
nn
nnnn xxaxM .
2
15
•
3
Phase Sensitivity Exponent to Characterize Strangeness of an Attractor
Phase Sensitivity with Respect to the Phase of Quasiperiodic Forcing:
Measured by Calculating a Derivative x/ along a Trajectory and Finding its Maximum Value:
n
NnN
x0max
Phase Sensitivity Function: Nx
N ),( 00
min
• Smooth Torus (a=3.38, =0.584 7) N: Bounded No Phase Sensitivity
• SNA (a=3.38, =0.584 75) N ~ N: Unbounded [( 19.5): PSE] Phase Sensitivity Strange Geometry
(Taking the minimum value of N(x0,0) with respect to an ensemble of randomly chosen initial conditions)
~_
4
Typical Phase Diagrams in Quasiperiodically Forced Period-Doubling Systems
Quasiperiodically Forced Hénon Map
1). (mod
,
),2cos(
1
1
21
nn
nn
nnnn
bxy
yxax
Quasiperiodically Forced Ring Map
1). (mod
),2sin()2/(
,1) (mod )2cos(
)sin()2/(
1
1
1
nn
nnn
n
nnnn
xabyy
byxaxx
=0 and b=0.01b=0.05
(a: Intermittency, b & c: Interior Crisis, d & e: Boundary Crisis)
Tongues (near the Terminal Points of the Torus Doubling Bifurcation Lines)
5
Intermittent Route to SNAs
Absorbing Area (AA) in the Quasiperiodically Forced Logistic Map M
M: Noninvertible [ detDM=0 along the Critical Curve L0={x=0.5}]
Images of the Critical Curve x=0.5 [i.e., Lk=Mk(L0): Critical Curve of Rank k]:Used to Define a Bounded Trapping Region inside the Basin of Attraction.
The AA determines the Global Structure of a Newly-Born Intermittent SNA.
Smooth Torus inside an AAfor a=3.38 and =0.584 7(x -0.059)
Intermittent SNA filling the AAfor a=3.38 and =0.584 75(x -0.012, 19.5)
*=0.584 726 781
~_ ~_ ~_
6
Global Structure of an Intermittent SNAThe Global Structure of the SNA may be Determined by the Critical Curves Lk.
7
Rational Approximations
Rational Approximation (RA)• Investigation of the Intermittent Transition in a Sequence of Periodically Forced Systems with Rational Driving Frequencies k, Corresponding to the RA to the Quasiperiodic Forcing ( ) :
• Properties of the Quasiperiodically Forced Systems Obtained by Taking the Quasiperiodic Limit k .
Unstable OrbitsThe Intermittent Transition is Expected to Occur through Collision with an Unstable Orbit:
• Smooth Unstable Torus x=0 (developed from the unstable fixed point for the unforced case): Outside the AA No Interaction with the Smooth Attracting Torus
• Ring-Shaped Unstable Set (without correspondence for the unforced case) Using the RA, a New Type of Ring-Shaped Unstable Set that Interacts with the Smooth Torus is found inside the AA.
1 and 0,;/ 10111 FFFFFFF kkkkkk
2/)15(
8
Metamorphoses of the Ring-Shaped Unstable Set The kth RA to a Smooth Torus e.g. k=6 RA: Composed of Stable Orbits with Period F6 (=8) inside the AA.
a=3.246, =0.446, k=6
Birth of a Ring-Shaped Unstable Set (RUS) via a Phase-Dependent Saddle-Node Bifurcation
• RUS of Level k=6: Composed of 8 Small Rings
Each Ring: Composed of Stable (Black) and Unstable (Gray) Orbits with Period F6 (=8)
(Unstable Part: Toward the Smooth Torus They may Interact.)
a=3.26, =0.46, k=6
Evolution of the Rings
• Appearance of Chaotic Attractor (CA) via Period-Doubling Bifurcations (PDBs) and Its Disappearance via a Boundary Crisis
(Upper Gray Line: Period-F6 (=8) Orbits Destabilized via PDBs)
9
a=3.326, =0.526, k=6
a=3.326, =0.526, k=8
Change in the Shape and Size in the Rings
Quasiperiodic Limit
No. of Rings (=336): Significantly IncreasedUnstable Part (Gray): DominantAttracting Part (Black): Negligibly Small
Each Ring: Composed of the Large Unstable Part (Gray) and a Small Attracting Part (Black)
Expectation: In the Quasiperiodic Limit, the RUS forms a Complicated Unstable Set Composed of Only Unstable Orbits
10
Mechanism for the Intermittency Smooth Torus (Black) and RUS (Gray) in the RA of Level k=8 (F8=21)
a=3.38, =0.5864, k=8 a=3.38, =0.5864, k=8
a=3.38, =0.586, k=8 a=3.38, =0.586, k=8
Phase-Dependent Saddle-Node Bifurcation for 8=0.586 366 Appearance of Gaps, Filled by Intermittent Chaotic Attractors RA to the Whole Attractor: Composed of Periodic and Chaotic (in 21 Gaps) Components Average Lyapunov Exponent < 0 (<x> -0.09)~_
11
• Algebraic Convergence of the Phase-Dependent SNB Values k (up to k=15) to its Limit Value *(=0.584 726 781) of Level k.
Quasiperiodic Limit
* ,|~| kkkk F
• A Dense Set of Gaps (Filled with Intermittent CAs)
Using This RA, One Can Explain the Intermittent Route to SNA.
a=3.38, =0.584 75
12
Transition from SNA to CA
Average Lyapunov Exponents (in the x-direction) in the RAs
<x>= p + c;
p(c): “Weighted” Lyapunov Exponents of the Periodic (Chaotic) Component
p(c) = <x>p(c) Mp(c);
Mp(c): Lebesgue Measure in and <x>p(c): Average Lyapunov Exponent of the Periodic (Chaotic) Component
Chaotic Transition (c=0.5848)
Solid Line: Quasiperiodic LimitSolid Circles: RA of Level k=15
13
Other Dynamical Transitions in the Tongues Interior Crisis
Attractor-Widening Interior Crisis Occurs through Collision with the RUS:
Route b: SNA (Born via Gradual Fractalization) Intermittent SNA
Route c: CA Intermittent CA
a=3.4441=0.55
x -0.018 1.4
~_
a=3.4443=0.55
x -0.005 8.0
~_
a=3.44=0.48
x 0.124~_
a=3.43=0.48
x 0.069~_
~_~_
14
Boundary Crisis
Boundary Crisis of Type I (Heavy Solid Line) through Collision with the RUS • Route d: Smooth Torus Divergence • Route e: SNA (Born via Gradual Fractalization) Divergence
Boundary Crisis of Type II (Heavy Dashed Line) through Collision with the Smooth Unstable Torus • Route (): CA (SNA) Divergence
Double Crises near the Vertex Points
T: Torus, S: SNA, C: CA, D: Divergence,Dashed Line: Birth of SNA via Gradual Fractalization, Solid Line: Transition to Chaos, Dash-Dotted Line: Basin Boundary Metamorphosis Line, Dotted Line: Interior Crisis Line
•
•
•
15
Appearance of Higher-Order Tongues Appearance of Similar Higher-Order Tongues
Band-Merging Transitions near the Higher-Order Tongues
• Hard Band-Merging Transition (Heavy Solid Line) via Collision with the RUS• Soft Band-Merging Transition (Heavy Dashed Line) via Collision with an Unstable Parent Torus• Double Crises near the Vertex Points
2T: Doubled Torus, S & 2S: SNA, C & 2C: CA,Dashed Line: Birth of SNA via Gradual Fractalization, Solid Line: Transition to Chaos, Dash-Dotted Line: Basin Boundary Metamorphosis Line, Dotted Line: Interior Crisis Line
Torus (Light Gray)SNA (Gray)CA (Black)
(a*=3.569 945 ...)
16
Summary Using the RA , the Quasiperiodically Forced Logistic Map has been Investigated: Appearance of a New Type of Ring-Shaped Unstable Sets via Phase-Dependent Saddle-Node Bifurcations near the Tongues Occurrence of Rich Dynamical Transitions such as the Intermittency, Interior and Boundary Crises, and Band-Merging Transitions through Interaction with the RUS Such Dynamical Transitions: “Universal,” in the Sense that They Occur Typically in a Large Class of Quasiperiodically Forced Period-Doubling Systems
• Phase Diagram of the Quasiperiodically Forced Toda Oscillator.2/)15()/( ,coscos1 1221 ttaexx x
=0.8 and 1=2
S.-Y. Kim, W. Lim, and E. Ott, “Mechanism for the Intermittent Route to Strange Nonchaotic Attractors,” nlin.CD/0208028 (2002).
17
Basin Boundary Metamorphoses Main Tongue
When the Critical Curve L1 of Rank 1 Passes the Upper Basin Boundary (x=1),“Holes,” Leading to Divergent Trajectories, Appears inside the Basin.
2nd-Order Tongue
In the Twice Iterated Map, when the Critical Curve L1 of Rank 1 Passes the UpperBasin Boundary, “Holes,” Leading to an Another Attractor, Appears inside the Basin.
a=3.4 =0.58
a=3.43 =0.58
a=3.44 =0.14
a=3.45 =0.14
18
Dynamical Transitions in Quasiperiodically Forced Systems
Rich dynamical transitions in the quasiperiodically forced systems have been reported:
1. Transitions from a Smooth Torus to a Strange Nonchaotic Attractor (SNA) 1.1 Gradual Fractalization [T. Nishikawa and K. Kaneko, Phys. Rev. E 54, 6114 (1996)]
1.2 Torus Collision [J.F. Heagy and S.M. Hammel, Physica D 70, 140 (1994)]
1.3 Intermittent Transition [A. Prasad, V. Mehra, and R. Ramaswamy, Phys. Rev. Lett. 79, 4127 (1997), A. Witt, U. Feudel, and A. Pikovsky, Physica D 109, 180 (1997)]
1.4 Blowout Transition [C. Grebogi, E. Ott, S. Pelikan, and J. A. Yorke, Physica D 13, 261 (1984), T. Yalcinkaya and Y.-C. Lai, Phys. Rev. Lett. 77, 5039 (1996)]
2. Crises for the SNA and Chaotic Attractor (CA) 2.1 Band-Merging Crisis [O. Sosnovtseva, U. Feudel, J. Kurths, and A. Pikovsky, Phys. Lett. A 218, 255 (1996)]
2.2 Interior Crisis [A. Witt, U. Feudel, and A. Pikovsky, Physica D 109, 180 (1997)]
2.3 Boundary Crisis [H.M. Osinga and U. Feudel, Physica D 141, 54 (2000)]
However, the Mechanisms for such Transitions are Much Less Understood than those in the Unforced or Periodically Forced systems. Illumination of the Mechanisms for the Dynamical Transitions: Necessary
19
Band-Merging Transition Band-Merging Transition of Type I through Collision with the RUS
2T 1 SNA
2CA 1CA
a=3.431=0.16
a=3.43=0.165
x -0.014 9.7
~_
a=3.32=0.202
x 0.033~_
a=3.32=0.198
x 0.014~_
~_
Band-Merging Transition of Type II through Collision with the Unstable Parent Torus
top related