L'Ordre dans le Choas and Deterministic Chaos: An Introduction

Length distribution of laminar phases for type-I intermittency in the presence of noise

“…To develop the theory of type-I intermittency in the presence of noise, we consider the same quadratic map (1) with the addition of a stochastic term ξ n…”

“…The standard model that is used to study the type-I intermittency [1] is the one-parameter quadratic map…”

“…Its arousal and main statistical properties have been studied and characterized already since long time ago, and different types of intermittency have been classified as types I-III [1,2], on-off intermittency [3,4,5,6], eyelet intermittency [7,8,9] and ring intermittency [10].…”

Bifurcation to spatially induced chaos in a reaction-diffusion system

“…Steady state behavior was followed by periodicity, quasiperiodicity, frequency-locking of the quasiperiodicity, and finally period-doubling from the frequency-locked behavior into low-dimensional chaos. This is just the expected bifurcation scenario for a single, periodically forced, nonlinear oscillator [36]. We will now describe how such a bifurcation sequence occurs in our model system.…”

The Century of Turbulence Theory: The Main Achievements and Unsolved Problems

“…Note also studied in [28,29] subcritical-flow transition scenarios which don't include any cascade of successive instabilities. During the last 15 years many hundreds of papers and many dozens of books and lengthy surveys appeared were these and some other scenarios of transition of dynamic systems to chaotic regimes are discussed (the books [43][44][45][46][47] and the papers [48][49][50] discussing the applicability of the concept of chaos to turbulence are only a few examples). Let us mention in this respect also a few laboratory and numerical studies [51][52][53][54] where there were described some flow-transition phenomena having features close to those of some of the proposed transition scenarios.…”