Blowout Bifurcation Route to Strange Nonchaotic Attractors

Abstract

Strange nonchaotic attractors are attractors that are geometrically strange, but have nonpositive Lyapunov exponents. We show that for dynamical systems with an invariant subspace in which there is a quasiperiodic torus, the loss of the transverse stability of the torus can lead to the birth of a strange nonchaotic attractor. A physical phenomenon accompanying this route to strange nonchaotic attractors is an extreme type of intermittency. [S0031-9007(96)01861-3]

Dynamical Mechanism for Coexistence of Dispersing Species

“…intermittency, the &&on'' state, or the burst, occurs in time intervals that are typically much shorter than those for the &&o!'' state (Spiegel, 1980;Fujisaka & Yamada, 1985, 1986Yu et al, 1991;Platt et al, 1993;Heagy et al, 1994;Lai & Grebogi, 1995;Lai, 1996a, b;Yalcinkaya & Lai, 1996;Venkataramani et al, 1995Venkataramani et al, , 1996Marthaler et al, 2001), because dynamically bursting occurs when the trajectory is su$ciently near a transversely unstable set, such as a transversely unstable periodic orbit, and is therefore exponentially fast (in contrast, the &&o!'' state corresponds to the trajectory's wandering through many transversely stable sets, which tends to keep the trajectory in the &&o!''…”

“…intermittency, a dynamical behavior that has received extensive recent attention (Spiegel, 1980;Fujisaka & Yamada, 1985, 1986Yu et al, 1991;Platt et al, 1993;Heagy et al, 1994;Lai & Grebogi, 1995;Lai, 1996a, b;Yalcinkaya & Lai, 1996;Venkataramani et al, 1995Venkataramani et al, , 1996Marthaler et al, 2001). In ecology, Ferriere & Cazelles (1999) arises because of a local storage e!ect (Chesson, 1986).…”

A plethora of strange nonchaotic attractors

“…Thus, it follows that the attractor has a dense set of points on the line x = 0, θ ∈ [0, 1] (since ω is irrational), but the entire line itself cannot be the attractor for α > 1, since the dynamics is unstable on that line. There will, therefore at some α, be a "blowout bifurcation" [8,9] transition to strange nonchaotic dynamics.…”

“…Several studies, both theoretical [3][4][5][6][7][8][9] and experimental [10][11][12][13][14][15], have, over the years, elucidated the principal features of such attractors which appear to be generic in quasiperiodically driven nonlinear dynamical systems. These are geometrically strange sets (fractals) on which all Lyapunov exponents are either zero or negative.…”

Dynamics of a Parametrically Excited System with Two Forcing Terms

“…Another route to chaos includes the scenario called intermittency [53][54][55]. Here, one observes long periods of periodic motion with bursts of chaos.…”