# Small random perturbations of dynamical systems and the definition of attractors

@article{Ruelle1981SmallRP, title={Small random perturbations of dynamical systems and the definition of attractors}, author={David Ruelle}, journal={Communications in Mathematical Physics}, year={1981}, volume={82}, pages={137-151} }

The “strange attractors” plotted by computers and seen in physical experiments do not necessarily have an open basin of attraction. In view of this we study a new definition of attractors based on ideas of Conley. We argue that the attractors observed in the presence of small random perturbations correspond to this new definition.

## 166 Citations

On the definitions of attractors

- Mathematics
- 1985

We introduce the notion of attractor and present its historical evolution. Then we show that previous definitions are too stringent. We present two equivalent definitions of attractors, show that in…

New developments in the ergodic theory of nonlinear dynamical systems

- MathematicsPhilosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences
- 1994

The purpose of this paper is to give a survey of recent results on non-uniformly hyperbolic dynamical systems. The emphasis is on the existence of strange attractors and Sinai-Ruelle-Bowen measures…

On the Concept of Attractor in Community-Dynamical Processes

- Mathematics
- 2001

We introduce a notion of attractor to community-dynamical processes as they are studied in biological models and their computer simulations. This attractor concept is modeled after the Conley-Ruelle…

On the concept of attractor

- Mathematics
- 1985

This note proposes a definition for the concept of “attractor,” based on the probable asymptotic behavior of orbits. The definition is sufficiently broad so that every smooth compact dynamical system…

Wild pseudohyperbolic attractor in a four-dimensional Lorenz system

- Mathematics, PhysicsNonlinearity
- 2021

We present an example of a new strange attractor which, as we show, belongs to a class of wild pseudohyperbolic spiral attractors. We find this attractor in a four-dimensional system of differential…

Attractors via random perturbations

- Mathematics, Physics
- 1989

We discuss conditions which ensure that weak limits of invariant measures of small random perturbations of dynamical systems have their supports on attractors.

On three types of dynamics and the notion of attractor

- Physics
- 2017

We propose a theoretical framework for explaining the numerically discovered phenomenon of the attractor–repeller merger. We identify regimes observed in dynamical systems with attractors as defined…

## References

SHOWING 1-10 OF 41 REFERENCES

ON SMALL RANDOM PERTURBATIONS OF SOME SMOOTH DYNAMICAL SYSTEMS

- Mathematics
- 1974

This paper studies the problem of convergence of invariant measure which is a result of perturbation by a diffusion process with a small parameter (tending to zero) of certain smooth dynamical…

A MEASURE ASSOCIATED WITH AXIOM-A ATTRACTORS.

- Mathematics
- 1976

The future orbits of a diffeomorphism near an Axiom-A attrac- tor are investigated. It is found that their asymptotic behavior is in general described by a fixed probability measure yt carried by the…

Universal properties of maps on an interval

- Mathematics
- 1980

We consider itcrates of maps of an interval to itself and their stable periodic orbits. When these maps depend on a parameter, one can observe period doubling bifurcations as the parameter is varied.…

Deterministic nonperiodic flow

- Mathematics
- 1963

Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with…

Differentiable dynamical systems and the problem of turbulence

- Physics
- 1981

1. Conservative and dissipative dynamical systems. The mathematical study of differentiable dynamical systems has its origin in the desire to understand the time evolutions which occur in nature.…

Characteristic Exponents and Invariant Manifolds in Hilbert Space

- Mathematics
- 1982

The multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds (Pesin theory) are extended to differentiable dynamical systems on Hilbert manifolds under…

The transition to aperiodic behavior in turbulent systems

- Engineering
- 1980

Some systems achieve aperiodic temporal behavior through the production of successive half subharmonics. A recursive method is presented here that allows the explicit computation of this aperiodic…

Iterated maps on the interval as dynamical systems

- Mathematics
- 1980

Motivation and Interpretation.- One-Parameter Families of Maps.- Typical Behavior for One Map.- Parameter Dependence.- Systematics of the Stable Periods.- On the Relative Frequency of Periodic and…

Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms

- Mathematics
- 1975

Gibbs Measures.- General Thermodynamic Formalism.- Axiom a Diffeomorphisms.- Ergodic Theory of Axiom a Diffeomorphisms.