From Limit Cycles to Strange Attractors

@article{Ott2010FromLC,
  title={From Limit Cycles to Strange Attractors},
  author={William Ott and Mikko Stenlund},
  journal={Communications in Mathematical Physics},
  year={2010},
  volume={296},
  pages={215-249}
}
  • W. Ott, Mikko Stenlund
  • Published 31 March 2010
  • Mathematics, Physics
  • Communications in Mathematical Physics
We define a quantitative notion of shear for limit cycles of flows. We prove that strange attractors and SRB measures emerge when systems exhibiting limit cycles with sufficient shear are subjected to periodic pulsatile drives. The strange attractors possess a number of precisely-defined dynamical properties that together imply chaos that is both sustained in time and physically observable. 
Bifurcation Analysis of a Stochastically Driven Limit Cycle
AbstractWe establish the existence of a bifurcation from an attractive random equilibrium to shear-induced chaos for a stochastically driven limit cycle, indicated by a change of sign of the first
Dissipative homoclinic loops of two‐dimensional maps and strange attractors with one direction of instability
We prove that when subjected to periodic forcing of the form p�;�;! .t/ D�.�h.x;y/ C sin.!t//; certain two-dimensional vector fields with dissipative homoclinic loops generate strange attractors with
Homoclinic Loops, Heteroclinic Cycles, and Rank One Dynamics
We prove that genuine nonuniformly hyperbolic dynamics emerge when flows in ${R}^{N}$ with homoclinic loops or heteroclinic cycles are subjected to certain time-periodic forcing. In particular, we
RANK ONE DYNAMICS NEAR HETEROCLINIC CYCLES AND CONDITIONAL MEMORY LOSS FOR NONEQUILIBRIUM DYNAMICAL SYSTEMS
There are two parts in this dissertation. In the first part we prove that genuine nonuniformly hyperbolic dynamics emerge when flows in R with homoclinic loops or heteroclinic cycles are subjected to
"Large" strange attractors in the unfolding of a heteroclinic attractor
We present a mechanism for the emergence of strange attractors in a one-parameter family of differential equations defined on a 3-dimensional sphere. When the parameter is zero, its flow exhibits an
Strange attractors for asymptotically zero maps
Abstract A discrete dynamical system in Euclidean m-space generated by the iterates of an asymptotically zero map f, satisfying f ( x ) → 0 as x → ∞ , must have a compact global attracting set A. The
New Strange Attractors for Discrete Dynamical Systems
A discrete dynamical system in Euclidean m-space generated by the iterates of an asymptotically zero map f , satisfying |f(x)| → 0 as |x| → ∞, must have a compact global attracting set A. The
Generalized Attracting Horseshoes and Chaotic Strange Attractors
A generalized attracting horseshoe is introduced as a new paradigm for describing chaotic strange attractors (of arbitrary finite rank) for smooth and piecewise smooth maps f from Q to Q, where Q is
Positive Lyapunov exponent by a random perturbation
We study the effect of a random perturbation on a one-parameter family of dynamical systems whose behaviour in the absence of perturbation is ill-understood. We provide conditions under which the
Dynamics of periodically kicked oscillators
We review some recent results surrounding a general mechanism for producing chaotic behavior in periodically kicked oscillators. The key geometric ideas are illustrated via a simple linear shear
...
1
2
3
...

References

SHOWING 1-10 OF 36 REFERENCES
From Invariant Curves to Strange Attractors
Abstract: We prove that simple mechanical systems, when subjected to external periodic forcing, can exhibit a surprisingly rich array of dynamical behaviors as parameters are varied. In particular,
Strange Attractors in Periodically-Kicked Limit Cycles and Hopf Bifurcations
We prove the emergence of chaotic behavior in the form of horseshoes and strange attractors with SRB measures when certain simple dynamical systems are kicked at periodic time intervals. The settings
The simplest case of a strange attractor
Abstract It is shown that stochastic motion of strange attractor type may arise in a system with stable limit cycle if the perturbation of the system is periodical. Analytical and numerical analyses
Shear-Induced Chaos
Guided by a geometric understanding developed in earlier works of Wang and Young, we carry out numerical studies of shear-induced chaos in several parallel but different situations. The settings
Strange Attractors with One Direction of Instability
Abstract: We give simple conditions that guarantee, for strongly dissipative maps, the existence of strange attractors with a single direction of instability and certain controlled behaviors. Only
Recurrence times and rates of mixing
The setting of this paper consists of a map making “nice” returns to a reference set. Criteria for the existence of equilibria, speed of convergence to equilibria and for the central limit theorem
Two distinct mechanisms of coherence in randomly perturbed dynamical systems.
TLDR
It is shown that coherence resonance arises only at the onset of bifurcation and is rather insensitive against variations in the noise amplitude and the time scale separation ratio, and self-induced stochastic resonance may arise away from b ifurcations and the properties of the limit cycle it induces are controlled by both the noise Amplitude and time scale separates ratio.
Application of the 0-1 Test for Chaos to Experimental Data
TLDR
A new method of detecting chaos which applies directly to the time series data and does not require phase space reconstruction is proposed and to illustrate the effectiveness of the method, data from a bipolar motor is analyzed.
Toward a theory of rank one attractors
Introduction 1 Statement of results PART I PREPARATION 2 Relevant results from one dimension 3 Tools for analyzing rank one maps PART II PHASE-SPACE DYNAMICS 4 Critical structure and orbits 5
A new test for chaos in deterministic systems
  • G. Gottwald, I. Melbourne
  • Mathematics, Physics
    Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
  • 2004
We describe a new test for determining whether a given deterministic dynamical system is chaotic or non–chaotic. In contrast to the usual method of computing the maximal Lyapunov exponent, our method
...
1
2
3
4
...