# 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} }

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.

## 23 Citations

Bifurcation Analysis of a Stochastically Driven Limit Cycle

- MathematicsCommunications in Mathematical Physics
- 2019

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

- Mathematics
- 2011

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

- Mathematics, Computer ScienceSIAM J. Appl. Dyn. Syst.
- 2015

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

- Mathematics
- 2013

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

- MathematicsDiscrete & Continuous Dynamical Systems
- 2021

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

- Mathematics
- 2014

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

- 2013

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

- Mathematics
- 2016

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

- Mathematics, Physics
- 2012

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

- Mathematics, Physics
- 2010

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…

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