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Chaotic saddles in nonlinear modulational interactions in a plasma  [PDF]
Rodrigo A. Miranda,Erico L. Rempel,Abraham C. -L. Chian
Physics , 2012, DOI: 10.1063/1.4766472
Abstract: A nonlinear model of modulational processes in the subsonic regime involving a linearly unstable wave and two linearly damped waves with different damping rates in a plasma is studied numerically. We compute the maximum Lyapunov exponent as a function of the damping rates in a two-parameter space, and identify shrimp-shaped self-similar structures in the parameter space. By varying the damping rate of the low-frequency wave, we construct bifurcation diagrams and focus on a saddle-node bifurcation and an interior crisis associated with a periodic window. We detect chaotic saddles and their stable and unstable manifolds, and demonstrate how the connection between two chaotic saddles via coupling unstable periodic orbits can result in a crisis-induced intermittency. The relevance of this work for the understanding of modulational processes observed in plasmas and fluids is discussed.
Chaos in driven Alfvén systems: unstable periodic orbits and chaotic saddles
A. C.-L. Chian, W. M. Santana, E. L. Rempel, F. A. Borotto, T. Hada,Y. Kamide
Nonlinear Processes in Geophysics (NPG) , 2007,
Abstract: The chaotic dynamics of Alfvén waves in space plasmas governed by the derivative nonlinear Schr dinger equation, in the low-dimensional limit described by stationary spatial solutions, is studied. A bifurcation diagram is constructed, by varying the driver amplitude, to identify a number of nonlinear dynamical processes including saddle-node bifurcation, boundary crisis, and interior crisis. The roles played by unstable periodic orbits and chaotic saddles in these transitions are analyzed, and the conversion from a chaotic saddle to a chaotic attractor in these dynamical processes is demonstrated. In particular, the phenomenon of gap-filling in the chaotic transition from weak chaos to strong chaos via an interior crisis is investigated. A coupling unstable periodic orbit created by an explosion, within the gaps of the chaotic saddles embedded in a chaotic attractor following an interior crisis, is found numerically. The gap-filling unstable periodic orbits are responsible for coupling the banded chaotic saddle (BCS) to the surrounding chaotic saddle (SCS), leading to crisis-induced intermittency. The physical relevance of chaos for Alfvén intermittent turbulence observed in the solar wind is discussed.
Coherence resonance near blowout bifurcation in nonlinear dynamical systems  [PDF]
Bambi Hu,Changsong Zhou
Physics , 2000,
Abstract: Previous studies have shown that noise can induce coherence resonance in some nonlinear dynamical systems close to a bifurcation of a periodic motion, such as in excitable systems. We demonstrate that coherence resonance can be observed in systems close to a {\sl blowout bifurcation}. It is shown that for dynamical systems with an invariant subspace in which there is a phase-coherent chaotic attractor, the interplay among the oscillation of local transverse stability, noise and nonlinearity can lead to coherence resonance phenomenon. The mechanism of coherence resonance in this type of system is different from that in previously studied systems.
Multistability, noise and attractor-hopping: The crucial role of chaotic saddles  [PDF]
Suso Kraut,Ulrike Feudel
Physics , 2003,
Abstract: We investigate the hopping dynamics between different attractors in a multistable system under the influence of noise. Using symbolic dynamics we find a sudden increase of dynamical entropies, when a system parameter is varied. This effect is explained by a novel bifurcation involving two chaotic saddles. We also demonstrate that the transient lifetimes on the saddle obey a scaling law in analogy to crisis.
Periodic orbit analysis at the onset of the unstable dimension variability and at the blowout bifurcation  [PDF]
Rodrigo Frehse Pereira,Sandro Ely de Souza Pinto,Ricardo L. Viana,Sergio R. Lopes,Celso Grebogi
Physics , 2009, DOI: 10.1063/1.2748619
Abstract: Many chaotic dynamical systems of physical interest present a strong form of nonhyperbolicity called unstable dimension variability (UDV), for which the chaotic invariant set contains periodic orbits possessing different numbers of unstable eigendirections. The onset of UDV is usually related to the loss of transversal stability of an unstable fixed point embedded in the chaotic set. In this paper, we present a new mechanism for the onset of UDV, whereby the period of the unstable orbits losing transversal stability tends to infinity as we approach the onset of UDV. This mechanism is unveiled by means of a periodic orbit analysis of the invariant chaotic attractor for two model dynamical systems with phase spaces of low dimensionality, and seems to depend heavily on the chaotic dynamics in the invariant set. We also described, for these systems, the blowout bifurcation (for which the chaotic set as a whole loses transversal stability) and its relation with the situation where the effects of UDV are the most intense. For the latter point, we found that chaotic trajectories off, but very close to, the invariant set exhibit the same scaling characteristic of the so-called on-off intermittency.
Analysis of Chaotic Synchronization Stability and Its Application to a Two-dimensional Advertising Competing Model
混沌同步稳定性分析及在一个二维广告竞争模型中的应用

QI Jie,WANG Ding-Wei,
齐洁
,汪定伟

自动化学报 , 2005,
Abstract: A class of map in which chaotic synchronization can occur is defined. The transverse Lyapunov exponents are used to determine the stability of synchronized trajectories. Some complex phenomena closely related to chaotic synchronization, namely riddled basin, riddling bifurcation and blowout bifurcation are theoretically analyzed. Riddling bifurcation and blowout bifurcation may change the synchronization stability of the system. And two types of riddled basins, i.e., global riddled basin and local riddled basin, may come into being after riddling bifurcation. An advertising competing model based on Vidale-Wolfe model is proposed and analyzed by the above theories at the end of the paper.
Non-normal parameter blowout bifurcation: an example in a truncated mean field dynamo model  [PDF]
Eurico Covas,Peter Ashwin,Reza Tavakol
Physics , 1997, DOI: 10.1103/PhysRevE.56.6451
Abstract: We examine global dynamics and bifurcations occurring in a truncated model of a stellar mean field dynamo. This model has symmetry-forced invariant subspaces for the dynamics and we find examples of transient type I intermittency and blowout bifurcations to transient on-off intermittency, involving laminar phases in the invariant submanifold. In particular, our model provides examples of blowout bifurcations that occur on varying a non-normal parameter; that is, the parameter varies the dynamics within the invariant subspace at the same time as the dynamics normal to it. As a consequence of this we find that the Lyapunov exponents do not vary smoothly and the blowout bifurcation occurs over a range of parameter values rather than a point in the parameter space.
Equilibrium points and bifurcation control of a chaotic system
Liang Cui-Xiang,Tang Jia-Shi,
梁翠香
,唐驾时

中国物理 B , 2008,
Abstract: Based on the Routh--Hurwitz criterion, this paper investigates the stability of a new chaotic system. State feedback controllers are designed to control the chaotic system to the unsteady equilibrium points and limit cycle. Theoretical analyses give the range of value of control parameters to stabilize the unsteady equilibrium points of the chaotic system and its critical parameter for generating Hopf bifurcation. Certain nP periodic orbits can be stabilized by parameter adjustment. Numerical simulations indicate that the method can effectively guide the system trajectories to unsteady equilibrium points and periodic orbits.
A chaotic saddle in a wada fractal basin boundary
Wada分形吸引域边界上的混沌鞍

Hong Ling Xu Jianxue,
洪灵
,徐健学

力学学报 , 2002,
Abstract: In this paper, bifurcations near optimal escape for Thompson's escape equation are numerically studied by means of Generalized Cell Mapping Digraph (GCMD) method. We find a chaotic saddle embedded in a Wada fractal basin boundary. The chaotic saddle is an unstable (nonattracting) chaotic invariant set. The Wada fractal basin boundary has the Wada property that any point that is on the boundary of that basin is also simultaneously on the boundary of at least two other basins. The chaotic saddle in the Wada basin boundary plays an extremely important role in the bifurcations governing the escape. We demonstrate that the chaotic saddle in the Wada basin boundary leads to a local saddle-node fold bifurcation with globally indeterminate outcome. In such a case, the attractor (node) and the saddle of the saddle-node fold are merged into the chaotic saddle and the chaotic saddle also undergoes an abrupt enlargement in its size as a parameter passes through the bifurcation value, simultaneously the Wada basin boundary is also converted into the fractal basin boundary of two remaining attractors, in particular, the chaotic saddle after the saddle-node fold bifurcation is in the fractal basin boundary, this implies that the saddle-node fold bifurcation has indeterminate outcome, namely, after the system drifts through the bifurcation, which of the two remaining attractors the orbit goes to is indeterminate in that it is sensitively dependent on arbitrarily small effects such as how the parameter is changed and/or noise and/or computer roundoff, obviously, this presents an extreme form of indeterminacy in a dynamical system. We also investigate the origin and evolution of the chaotic saddle in the Wada basin boundary and demonstrate that the chaotic saddle in the Wada basin boundary is created by the collision between two chaotic saddles in different fractal basin boundaries. We demonstrate that a final escape bifurcation is the boundary crisis caused by the collision between a chaotic attractor and a chaotic saddle, and this implies that Grebogi's definition of the boundary crisis by the collision with a periodic saddle is generalized.
Equilibrium points and bifurcation control of a chaotic system

Liang Cui-Xiang,

物理学报 , 2008,
Abstract: Based on the Routh--Hurwitz criterion, this paper investigates the stability of a new chaotic system. State feedback controllers are designed to control the chaotic system to the unsteady equilibrium points and limit cycle. Theoretical analyses give the range of value of control parameters to stabilize the unsteady equilibrium points of the chaotic system and its critical parameter for generating Hopf bifurcation. Certain nP periodic orbits can be stabilized by parameter adjustment. Numerical simulations indicate that the method can effectively guide the system trajectories to unsteady equilibrium points and periodic orbits.
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