Three-Frequency Motion and Chaos in the Ginzburg-Landau Equation

Abstract

The Ginzburg-Landau equation with periodic boundary conditions on the interval L0,2/r/ q] is integrated numerically for large times. As q is decreased, the motion in phase space exhibits a sequence of bifurcations from a limit cycle to a two-torus to a threetorus to a chaotic regime. The three-torus is observed for a finite range of q and transition to chaotic flow is preceded by frequency locking.

Dynamics on the global attractor of a gradient flow arising from the Ginzburg-Landau equation

“…For a definition of the continuation of a Morse decomposition we refer the reader to [6,21,10], however, we wish to point out that the Morse sets may undergo tremendous bifurcations and yet the Morse decomposition may continue. It should also be mentioned that there is considerable evidence that the Ginzburg-Landau equation exhibits chaotic dynamics [1,15,18,19]. Closest, in terms of the choice of parameters, to our question is the work of [19].…”

“…We remark that the transformation (9) implies that any non-trivial solution v* to (18) gives rise to a periodic solution of (8) with u : (eos(~t) sin(~t) "~ v*. \-sin(at) cos(at)] Let us consider the equilibrium solution v* = (cos 9, sin 0)Tr to (18).…”

“…\-sin(at) cos(at)] Let us consider the equilibrium solution v* = (cos 9, sin 0)Tr to (18). Observe that the linearized equation of (10) around v* is…”

Dynamical playground of a higher-order cubic Ginzburg-Landau equation: From orbital connections and limit cycles to invariant tori and the onset of chaos

“…Various prototypical partial differential equation models have demonstrated a PB-type behavior as an intermediate bifurcation stage in the route to spatiotemporal chaos. Examples include the Kuramoto-Sivashinsky [17] and complex GinzburgLandau (CGL) equations; regarding the CGL model, which is of primary interest in this work, we refer to the seminal works [18] for the spatiotemporal transition to chaos. In addition to the above autonomous systems, spatiotemporal chaos was also found in non-autonomous ones, due to the interplay between loss and external forces, such as the damped-driven NLS [19][20][21] (where the hyperbolic structure of the underlying integrable NLS is a prerequisite [22]) and the sine-Gordon [23] system.…”

Instabilities of the Ginzburg-Landau equation. II. Secondary bifurcation

“…Introduction. The Ginzburg-Landau (G-L) amplitude equation governing the modulation of quasi-monochromatic waves in fluid systems with supercritical dimensionless parameter (e.g., Reynolds number) has been the focus of many recent studies on transition to chaos [1][2][3][4][5][6][7][8][9]. The general form of this equation is A, ~(Xr + i^i)Axx = arA -(ft + iPi)\A\2A (1) where Ar, ar, fir, fit are real quantities and under a suitable renormalization it may be written as [8] iA, +(1 -ic0)Axx = ipA -(1 + ip)\A\2A (2) where 0 < c20 < q, p = co/cj.…”