Devaney's Definition Research Articles

Maximally chaotic homeomorphisms of sigma-compact manifolds

Let μ be a locally positive Borel measure on a σ-compact n-manifold X,n≥2. We show that there is always a μ-preserving homeomorphism of X which is maximally chaotic in that it satisfies Devaney's definition of chaos, with the sensitivity constant chosen maximally. Furthermore, maximally chaotic homeomorphisms are compact-open topology dense in the space of all μ-preserving homeomorphisms of X if and only if (X,μ) has at most one end of infinite measure. (For example, for Lebesgue measure λ on X= R 2 , but not for λ on the strip X= R×[0,1] .) This work extends that of Aarts and Daalderop, Daalderop and Fokkink, Kato et al., and Alpern, regarding chaotic phenomena on compact manifolds, and that of Besicovitch and Prasad for other dynamical properties on noncompact manifolds.

Additive one-dimensional cellular automata are chaotic according to Devaney's definition of chaos

We study the chaotic behavior of a particular class of dynamical systems: cellular automata. We specialize the definition of chaos given by Devaney for general dynamical systems to the case of cellular automata. A dynamical system ( X, F) is chaotic according to Devaney's definition of chaos if its transition map F is sensitive to the initial conditions, topologically transitive, and has dense periodic orbits on X. Our main result is the proof that all the additive one-dimensional cellular automata defined on a finite alphabet of prime cardinality are chaotic in the sense of Devaney.

The Role of Transitivity in Devaney's Definition of Chaos

The Role of Transitivity in Devaney's Definition of Chaos

The Role of Transitivity in Devaney's Definition of Chaos

The Role of Transitivity in Devaney's Definition of Chaos

Dynamics of a harmonically excited impact damper: Bifurcations and chaotic motion

The dynamic behavior of a vibro-impact damper model with two degrees of freedom is investigated. The motions of this model include not only periodic motions but also irregular motions. Instead of employing classical methods for analyzing the dynamic characteristics of this two-degree-of-freedom impact-clearance problem, the concept of using the Poincaré map for observing the bifurcation phenomena is utilized in studying the existence and stability of subharmonic motions. Global bifurcations are discussed by using numerical simulation, and the evolution from the period-doubling sequence to chaotic motions is illustrated. In addition, the occurrence of chaotic motions is examined numerically, on the basis of Devaney's definition for the existence of chaos.

On Devaney's Definition of Chaos

(1992). On Devaney's Definition of Chaos. The American Mathematical Monthly: Vol. 99, No. 4, pp. 332-334.

On Devaney's Definition of Chaos

On Devaney's Definition of Chaos