On the concept of attractor

  title={On the concept of attractor},
  author={John W. Milnor},
  journal={Communications in Mathematical Physics},
  • J. Milnor
  • Published 1 June 1985
  • Mathematics
  • Communications in Mathematical Physics
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 has at least one attractor. 
Analysis of a Class of Strange Attractors
This work contains the results from a comprehensive study of a new class of attractors. The attractors in this class are characterized by strong local instability, but they are not uniformly
Attractors and characteristic exponents
A natural definition of an attractor as an invariant measure is given (based on the ergodic theory of axiom A diffeomorphisms) and some results are proved which support this definition. It is also
Practical Stability of Chaotic Attractors
The pullback attractor
The aim of this chapter is to introduce the ‘pullback attractor’, which seems to be the correct generalisation of this concept for use with non-autonomous processes.
New developments in the ergodic theory of nonlinear dynamical systems
  • M. Benedicks
  • Mathematics
    Philosophical 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
Generic properties of one-dimensional dynamical systems
Some generic properties of continuous maps of the interval or the circle are proved, concerning global and local attractors, Ljapunov stability and pseudo-orbit shadowing.
Dimensional characteristics of diffusion chaos
  • S. Glyzin
  • Physics
    Automatic Control and Computer Sciences
  • 2013
It is shown by an extended numerical experiment that the Lyapunov dimension of the attractor of a distributed evolutionary dynamic system increases when the diffusion coefficient tends to zero.
Computational intractability of attractors in the real quadratic family
We show that special perturbations of a particular holomorphic map on Pk give us examples of maps that possess chaotic non-algebraic attractors. Furthermore, we study the dynamics of the maps on the


Small random perturbations of dynamical systems and the definition of attractors
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
Strange attractors and asymptotic measures of discrete-time dissipative systems
We investigate asymptotic properties of certain discrete-time dynamical systems in two and three dimensions with solenoidal attractor. It is proved that the asymptotic measures, relevant for the
The Lyapunov dimension of a nowhere differentiable attracting torus
Abstract The fractal dimension of an attracting torus Tk in × Tk is shown to be almost always equal to the Lyapunov dimension as predicted by a previous conjecture. The cases studied here can have
Invariant measures for Markov maps of the interval
There is a theorem in ergodic theory which gives three conditions sufficient for a piecewise smooth mapping on the interval to admit a finite invariant ergodic measure equivalent to Lebesgue. When
Sensitive dependence to initial conditions for one dimensional maps
This paper studies the iteration of maps of the interval which have negative Schwarzian derivative and one critical point. The maps in this class are classified up to topological equivalence. The
A problem on topological transformations of the plane. II
In the paper under the above title I have given an example of a homoeomorphism of a plane under which the positive half-orbit† of a certain point is everywhere dense on the plane, and I have made a
Two dimensional mapping with strange attractor
A system of three first-order differential equations, whose solutions tend toward a “strange attractor”, is investigated, and it is shown that the same properties can be observed in a simple mapping of the plane defined by:xi+1=yi+1−axi2,yi-1=bxi.