Blowout Bifurcation Route to Strange Nonchaotic Attractors

Tolga Yalçınkaya, Ying-Cheng Lai
1996 Physical Review Letters  
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] PACS numbers: 05.45. + b
more » ... nonchaotic attractors are attractors that are geometrically complicated, but typical trajectories on these attractors exhibit no sensitive dependence on initial conditions asymptotically [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] . Here the word strange refers to the complicated geometry of the attractor: A strange attractor is not a finite set of points, and it is not piecewise differentiable. The word chaotic refers to a sensitive dependence on initial conditions: trajectories originating from nearby initial conditions diverge exponentially in time. Strange nonchaotic attractors occur in dissipative dynamical systems driven by several incommensurate frequencies (quasiperiodically driven systems) [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] . For example, it was demonstrated that in twofrequency quasiperiodically driven systems, there exist regions of finite Lebesgue measure in the parameter space for which there are strange nonchaotic attractors [3, 4] . More recent work demonstrates that a typical trajectory on a strange nonchaotic attractor actually possesses positive Lyapunov exponents in finite time intervals, although asymptotically the exponent is negative [7] . Strange nonchaotic attractors can arise in physically relevant situations such as quasiperiodically forced damped pendula and quantum particles in quasiperiodic potentials [2] , and in biological oscillators [4] . These exotic attractors have been observed in physical experiments [8, 9] . While the existence of strange nonchaotic attractors was firmly established, a question that remains interesting is how these attractors are created as a system parameter changes through a critical value, i.e., what the possible routes to strange nonchaotic attractors are. One route was investigated by Heagy and Hammel [5] who discovered that, in quasiperiodically driven maps, the transition from two-frequency quaisperiodicity to strange nonchaotic attractors occurs when a period-doubled torus collides with its unstable parent torus [5] . Near the collision, the period-doubled torus becomes extremely wrinkled and develops into a fractal set at the collision, although the Lyapunov exponent remains negative throughout the collision process. Recently, Feudel et al. found that the collision between a stable torus and an unstable one at a dense set of points leads to a strange nonchaotic attractor [10] . A renormalization-group analysis was also devised for the transition to strange nonchaotic attractors in a particular class of quasiperiodically driven maps [11] . In this paper, we present at route to strange nonchaotic attractors in dynamical systems with a symmetric lowdimensional invariant subspace S in the phase space. Since S is invariant, initial conditions in S result in trajectories which remain in S forever. We consider the case where there is a quasiperiodic torus in S [12] . Whether the torus attracts or repels initial conditions in the vicinity of S is determined by the sign of the largest transverse Lyapunov exponent L T computed for trajectories in S with respect to perturbations in the subspace T which is transverse to S. When L T is negative, S attracts trajectories transversely in the phase space, and the quasiperiodic torus in S is also an attractor of the full phase space. When L T is positive, trajectories in the vicinity of S are repelled away from it, and, consequently, the torus is transversely unstable, and it is hence not an attractor of the full phase space. Assume that a system parameter changes through a critical value a c ; L T passes through zero from the negative side. This bifurcation is referred to as the "blowout bifurcation" [13] [14] [15] . Our main result is that blowout bifurcation can lead to the birth of a strange nonchaotic attractor. A physical phenomenon associated with this route to strange nonchaotic attractors is that the dynamical variables in the transverse subspace T exhibit an extreme type of temporally intermittent bursting behavior: on-off intermittency [16] . Thus, our work also demonstrates that on-off intermittency can occur in quasiperiodically driven dynamical systems, whereas, to our knowledge, these intermittencies have been reported only for systems that are driven either randomly or chaotically. We consider the following class of N-dimensional dynamical systems, dx dt F͑x, z, p͒ ,
doi:10.1103/physrevlett.77.5039 pmid:10062699 fatcat:n7mzwd57ybbrrbibayok6czumu