A generic, hard transition to chaos

Abstract

A hard-in-amplitude transition to chaos in a class of dissipative flows of broad applicability is presented. For positive values of a parameter F, no matter how small, a fully developed chaotic attractor exists within some domain of additional parameters, whereas no chaotic behavior exists for F < 0. As F is made positive, an unstable fixed point reaches an invariant plane to enter a phase half-space of physical solutions; the ghosts of a line of fixed points and a rich heteroclinic structure existing at F = 0 make the limits t --* +oc, F ~ +0 non-commuting, and allow an exact description of the chaotic flow. The formal structure of flows that exhibit the transition is determined. A subclass of such flows (coupled oscillators in near-resonance at any 2 : q frequency ratio, with F representing linear excitation of the first oscillator) is fully analysed.

Sudden transition to chaos in plasma wave interactions

“…͑5a͒-͑5c͒ and ͑11͒, in the limit ␥ 3 Ϫ␥ 2 →0, recover known results for the special case ␥ 3 ϭ␥ 2 . 12 In Eqs. ͑5a͒-͑5c͒ the entire line ⌳ collapses into the plane rϭ1 and, using ͑5c͒ in ͑5b͒, becomes a 2 ϭ0, rϭ1, 2a 1 cos ␤ϩ ϭ0.…”

“…The limits t→ϩϱ and ⌫→0ϩ do not commute because times on the attractor diverge as ⌫→0ϩ, which also makes an exact description of 3-D chaotic flows possible. 12 Both the hard transition, and the effects of noise on it, have been experimentally verified by using two coupled electronic oscillators. 13 The spatio-temporal evolution of fully conservative wave packets can be determined by means of an inverse scattering transform; this system has been found to support soliton solutions.…”

“…21 In this paper we reconsider the hard transition to chaos for the general case of different dampings of the two stable waves (␥ 3 ␥ 2 , 0, ⌫→0ϩ), thus keeping the interaction fully three-wave and the flow four dimensional. We want to ascertain whether, and how, the sudden setup of chaos, which seems to have a certain universal character, 12 is retained in passing from 3-D to four dimensons ͑4-D͒. In Sec.II we recall the 3WRI model.…”

Hard transition to chaotic dynamics in Alfvén wave fronts

“…13 Actually, the system exhibits a hard transition to complex phase-space dynamics: no matter how small ⌫Ͼ0 there exists a fully developed attractor that is absent at ⌫р0 and is chaotic for some parametric domain; this is an example of a broad scenario for chaos also present in the resonant coupling of two oscillators at frequency ratio 2:q, q integer, with the first oscillator unstable. 14 For quadratic coupling ͑corresponding to qϭ1 in the two-oscillator case͒, the hard transition and the effects of noise have been experimentally verified using electronic oscillators; 15 also, the hard transition was found to persist when the daughter waves had unequal dampings, the flow then being 4D rather than 3D. 16,17 Cubic interaction, corresponding to qϭ2 ͑or 1:1 frequency ratio͒ in the two-oscillator case, allows a variety of coupling structures.…”

“…A reduced 3-wave truncation of the nonlinear Schrödinger equation showed chaotic behavior at finite ⌫; 18 a hard transition was encountered in a two-oscillator model of a spherical swing. 14,19 In the present paper we explore weakly nonlinear dynamics in a truncation of the DNLS equation that shows more complex cubic coupling, in both reduced and full 3-wave models. We want, in particular, to ascertain whether gross features in dynamical behavior, like fully developed phase-space attractors at ⌫ϭ0 ϩ , are structurally stable, as suggested by their appearance for quadratic coupling in both 3D and 4D flows, and for the particular cubic coupling of Refs.…”