Mechanism for the Band-Merging Route to Strange Nonchaotic Attractors in Quasiperiodically Forced Systems (original) (raw)
Related papers
Dynamical mechanism for band-merging transitions in quasiperiodically forced systems
Physics Letters A, 2005
As a representative model for quasiperiodically forced period-doubling systems, we consider the quasiperiodically forced logistic map, and investigate the mechanism for the band-merging transition. When the smooth unstable torus loses its accessibility from the interior of the basin of an attractor, it cannot induce the "standard" band-merging transition. For this case, we use the rational approximation to the quasiperiodic forcing and show that a new type of band-merging transition occurs for a nonchaotic attractor (smooth torus or strange nonchaotic attractor) as well as a chaotic attractor through a collision with an invariant ring-shaped unstable set which has no counterpart in the unforced case. Particularly, a two-band smooth torus is found to transform into a single-band intermittent strange nonchaotic attractor via a new band-merging transition, which corresponds to a new mechanism for the appearance of strange nonchaotic attractors. Characterization of the intermittent strange nonchaotic attractor is made in terms of the average time between bursts and the local Lyapunov exponents.
Journal of Physics A: Mathematical and General, 2004
To examine the universality for the intermittent route to strange nonchaotic attractors (SNAs), we investigate the quasiperiodically forced Hénon map, ring map and Toda oscillator which are high-dimensional invertible systems. In these invertible systems, dynamical transition to an intermittent SNA occurs via a phase-dependent saddle-node bifurcation, when a smooth torus collides with a 'ring-shaped' unstable set. We note that this bifurcation mechanism for the appearance of intermittent SNAs is the same as that found in a simple system of the quasiperiodically forced noninvertible logistic map. Hence, the intermittent route to SNAs seems to be 'universal', in the sense that it occurs through the same mechanism in typical quasiperiodically forced systems of different nature.
Multiband strange nonchaotic attractors in quasiperiodically forced systems
Physics Letters A, 1996
We study the effect of quasiperiodic forcing on two-dimensional invertible maps. As basic models the H&on and the ring maps are considered. We verify the existence of strange nonchaotic attractors (SNA) in these systems by two methods which are generalized to higher dimensions: via bifurcation analysis of the rational approximations, and by calculating the phase sensitivity. Analyzing these systems we especially find a new phenomenon: the appearance of strange nonchaotic attractors which consist of 2" bands. Similar to the band-merging crisis in chaotic systems, such a 2" band SNA can merge to a 2"-' band SNA.
Mechanism for the intermittent route to strange nonchaotic attractors
Physical Review E, 2003
Intermittent strange nonchaotic attractors (SNAs) appear typically in quasiperiodically forced period-doubling systems. As a representative model, we consider the quasiperiodically forced logistic map and investigate the mechanism for the intermittent route to SNAs using rational approximations to the quasiperiodic forcing. It is thus found that a smooth torus is transformed into an intermittent SNA via a phase-dependent saddle-node bifurcation when it collides with a new type of "ring-shaped" unstable set. Besides this intermittent transition, other transitions such as the interior, boundary, and band-merging crises may also occur through collision with the ring-shaped unstable sets. Hence the ring-shaped unstable sets play a central role for such dynamical transitions. Furthermore, these kinds of dynamical transitions seem to be "universal," in the sense that they occur typically in a large class of quasiperiodically forced period-doubling systems.
The birth of strange nonchaotic attractors
Physica D: Nonlinear Phenomena, 1994
A mechanism is described for the development of strange nonchaotic attractors from two-frequency quasiperiodic attractors in quasiperiodically driven maps. This mechanism is intimately tied to the phenomenon of torus-doubling. The transition to strange nonchaotic behavior occurs when a period-doubled torus collides with its unstable parent torus. As the collision is approached, the period-doubled torus becomes extremely wrinkled, ultimately becoming fractal at the collision. The Lyapunov exponent remains negative through the collision. These collisions are shown to be a new type of attractor merging crisis; the new feature is the possibility of nonchaotic attractors taking part in the crisis. This mechanism is illustrated via numerical and analytical studies of a quasiperiodically driven logistic map.
Intermittency and strange nonchaotic attractors in quasi-periodically forced circle maps
Physics Letters A, 1998
A possible mechanism for the creation of strange nonchaotic attractors close to the boundary of mode-locked tongues in a family of maps of the torus is described. This mechanism is based on the numerical observation that there are parameter values on the boundary of the mode-locked tongues at which the saddlenode bifurcation of invariant curves is not smooth, and assumptions about the nature of intermittency just outside the mode-locked tongues. @ 1998 Elsevier Science B.V.
Attractors on an N-torus: Quasiperiodicity versus chaos
Physica D: Nonlinear Phenomena, 1985
The occurrence of quasiperiodic motions in nonconservative dynami cal systems is of great -fundamental importance. However, current understanding concerning the question of how prevalent such m otions should be is incomplete. With this in mind, the types of attractors which can exist for flows on an N·torus are studied numerically for N = 3 and 4. Specifically, nonlinear perturbations are applied to maps representing N·frequeney quasiperiodic attractors. These perturbations can cause the original N-frequency quasiperiodic attractors to bifurcate to other types of attractors. Our results show that for small and moderate nonlinearity the frequency of occurrence of quasiperiodic motions is as follows: N-frequency quasiperiodic attractors are the most common, followed by (N -Ij-Irequcncy quasiperiodic attractors , .. . , followed by periodic attractors. However, as the nonlinearity is further increased. N·frequency quasiperiodicity becomes less common, ceasing to occur when the map becomes noninvertible. Chaotic attractors are very rare for N = 3 for small to mod erate nonlinearity, but are somewhat more common for N = 4. Examination of the types of chaotic attractors that occur for N = 3 reveals a rich variety of structure and dynamics. In particular, we see that there are chaot ic attractors which apparently fill the ent ire N-torus (i.e., limit sets of orbits on these attractors are the entire torus); Iurthermore, these are the most. common types of chaotic attractors at moderate nonlinearit ies.
Route to chaos via strange non-chaotic attractors
Journal of Physics A-mathematical and General, 1990
The route to chaos in quasiperiodically forced systems is investigated. I t has been found that chaotic behaviour is obtained after breaking of three-frequency torus, but strange non-chaotic attractors are present before three-frequency quasiperiodic behaviour occurs.
Torus Doublings in Symmetrically Coupled Period-Doubling Systems
Journal of the Korean Physical Society, 2010
As a representative model for Poincaré maps of coupled period-doubling oscillators, we consider two symmetrically coupled Hénon maps. Each invertible Hénon map has a constant Jacobian b (0 < b < 1) controlling the "degree" of dissipation. For the singular case of infinite dissipation (b = 0), it reduces to the non-invertible logistic map. Instead of period-doubling bifurcations, antiphase periodic orbits (with a time shift of half a period) lose their stability via Hopf bifurcations, and then smooth tori, encircling the anti-phase mother orbits, appear. We study the fate of these tori by varying b. For large b, doubled tori are found to appear via torus doubling bifurcations. This is in contrast to the case of coupled logistic maps without torus doublings. With decreasing b, mechanisms for disappearance of torus doublings are investigated, and doubled tori are found to be replaced with simple tori, periodic attractors, or chaotic attractors for small b. These torus doublings are also observed in two symmetrically coupled pendula that individually display a period-doubling transition to chaos.
Route to chaos via strange non-chaotic attractors by reshaping periodic excitations
Europhysics Letters (EPL), 2002
We present theoretical and numerical evidence for a new route, strange nonchaotic behavior ←→ chaos, in two-period quasiperiodically driven dynamical systems by solely changing the excitation waveform. A characteristic signature of this route is the existence of a sequence of intervals over a wide range of waveforms for which only strange non-chaotic attractors or chaos appear, alternatingly. We also found that the largest nontrivial Lyapunov exponent passes through zero linearly near each transition point, which confirms and extends the scaling behavior previously reported for other control parameters.