# Sensitive dependence on initial conditions

@article{Glasner1993SensitiveDO, title={Sensitive dependence on initial conditions}, author={Eli Glasner and Benjamin Weiss}, journal={Nonlinearity}, year={1993}, volume={6}, pages={1067-1075} }

It is shown that the property of sensitive dependence on initial conditions in the sense of Guckenheimer follows from the other two more technical parts of one of the most common recent definitions of chaotic systems. It follows that this definition applies to a broad range of dynamical systems, many of which should not be considered chaotic. We investigate the implications of sensitive dependence on initial conditions and its relation to dynamical properties such as rigidity, ergodicity…

## 330 Citations

### On Sensitive Dependence on Initial Conditions and Existence of Physical Measure for 3-Flows

- Mathematics
- 2014

After reviewing known results on sensitiveness and also on robustness of attractors together with observations on their proofs, we show that for attractors of three-dimensional flows, robust chaotic…

### Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems

- Mathematics
- 2009

### When is a dynamical system mean sensitive?

- MathematicsErgodic Theory and Dynamical Systems
- 2017

This article is devoted to studying which conditions imply that a topological dynamical system is mean sensitive and which do not. Among other things, we show that every uniquely ergodic, mixing…

### Chaotic Properties of Mappings on a Probability Space

- Mathematics
- 2002

Sensitive dependence on initial conditions is widely understood as being the central idea of chaos. We first give sufficient conditions (both topological and ergodic) on an endomorphism to ensure the…

### Sensitivity and chaos of iterated function systems

- Mathematics
- 2016

The present work is concerned with sensitivity and chaos for iterated function systems (IFSs). First of all, we introduce the concept of S-transitivity for IFSs which is relevant with sensitive…

### How chaotic are strange non-chaotic attractors?

- Physics
- 2006

We show that the classic examples of quasiperiodically forced maps with strange non-chaotic attractors described by Grebogi et al and Herman in the mid-1980s have some chaotic properties. More…

### Two results on entropy, chaos and independence in symbolic dynamics

- Mathematics
- 2015

We survey the connections between entropy, chaos, and independence in topological dynamics.
We present extensions of two classical results placing the following notions in the context of symbolic…

### Chaotic dynamical systems

- Physics
- 2012

In this work, we look at the dynamics of four different spaces, the interval, the unit circle, subshifts of finite type and compact countable sets. We put our emphasis on chaotic dynamical system and…

### Chaotic iterated function systems

- MathematicsArchiv der Mathematik
- 2022

The present work is concerned with chaotic iterated function systems, in a more general case. In this regard, we consider a finite set of relations as the generators of our system. Then we study the…

## References

SHOWING 1-8 OF 8 REFERENCES

### 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…

### Rigidity in topological dynamics

- MathematicsErgodic Theory and Dynamical Systems
- 1989

Abstract By analogy with the ergodic theoretical notion, we introduce notions of rigidity for a minimal flow (X, T) according to the various ways a sequence Tni can tend to the identity…

### On Devaney's definition of chaos

- Mathematics
- 1992

Chaotic dynamical systems have received a great deal of attention in recent years (see for instance [2], [3]). Although there has been no universally accepted mathematical definition of chaos, the…

### Sensitive dependence to initial conditions for one dimensional maps

- Mathematics
- 1979

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…

### Ergodic Theory on Compact Spaces

- Mathematics
- 1976

Measure-theoretic dynamical systems.- Measures on compact metric spaces.- Invariant measures for continuous tranformations.- Time averages.- Ergodicity.- Mixing and transitivity.- Shifts and…

### Disjointness in ergodic theory

- minimal sets,and a problem in diophantine approximation, Math.System Th. 1
- 1967