Sil'nikov chaos of the Liu system

A new chaotic attractor is discovered for the Liu system. The homoclinic and heteroclinic orbits in the Liu system have been found by using the undetermined coefficient method. It analytically demonstrates that there exists one heteroclinic orbit of the Sil'nikov type that connects two nontrivial equilibrium points, and therefore Smale horseshoes and the horseshoe chaos occur for this system via the Sil'nikov criterion. In addition, there also exists one homoclinic orbit joined to the origin. The convergence of the series expansions of these two types of orbits is proved.     

Comment on "Long time evolution of phase oscillator systems" [Chaos 19, 023117 (2009)

In a recent paper by Ott and Antonsen [Chaos 19, 023117 (2009)], it was shown for the case of Lorentzian distributions of oscillator frequencies that the dynamics of a very general class of large systems of coupled phase oscillators time-asymptotes to a particular simplified form given by Ott and Antonsen [Chaos 18, 037113 (2008)]. This comment extends this previous result to a broad class of oscillator distribution functions.     

Comment on "Finding finite-time invariant manifolds in two-dimensional velocity fields" [Chaos 10, 99 (2000)

This note serves as a commentary of the paper of Haller [Chaos 10, 99 (2000)] on techniques for detecting invariant manifolds. Here we show that the criterion of Haller can be improved in two ways. First, by using the strain basis reference frame, a more efficient version of theorem 1 of Haller (2000) allows to better detect the manifolds. Second, we emphasize the need to nondimensionalize the estimate of hyperbolic persistence. These statements are illustrated by the example of the Kida ellipse. (c) 2001 American Institute of Physics.     

Response to "Comment on 'Finding finite-time invariant manifolds in two-dimensional velocity fields' " [Chaos 11, 427 (2001)

Lapeyre, Hua, and Legras have recently suggested that the detection of finite-time invariant manifolds in two-dimensional fluid flows, as described by Haller and Haller and Yuan, can be substantially improved. In particular, they suggested (a) a change of coordinates to strain basis before the application of Theorem 1 of Haller and (b) the use of a nondimensionalized time computed from Theorem 1. Here we discuss why these proposed steps will not result in a significant overall improvement. We verify our arguments in a more detailed computation of the example analyzed in Lapeyre, Hau, and Legras (the Kida ellipse), as well as in a two-dimensional barotropic turbulence simulation. While in both of these examples the techniques suggested by Lapeyre, Hau, and Legras reveal additional thin regions of hyperbolicity near vortex cores, they also lead to an overall loss of detail in the global computation of finite-time invariant manifolds. (c) 2001 American Institute of Physics.     

Comment on "Exponential stabilization of chaotic systems with delay by periodically intermittent control" [Chaos 17, 013103 (2007)

In the referenced paper, there is a technical carelessness in the second lemma, and it is highlighted here to avoid possible failure when the result is used to design the intermittent controller for nonlinear systems.     

Comment on "Identification of low order manifolds: validating the algorithm of Maas and Pope" [Chaos 9, 108-123 (1999)

It is claimed by Rhodes, Morari, and Wiggins [Chaos 9, 108-123 (1999)] that the projection algorithm of Maas and Pope [Combust. Flame 88, 239-264 (1992)] identifies the slow invariant manifold of a system of ordinary differential equations with time-scale separation. A transformation to Fenichel normal form serves as a tool to prove this statement. Furthermore, Rhodes, Morari, and Wiggins [Chaos 9, 108-123 (1999)] conjectured that away from a slow manifold, the criterion of Maas and Pope will never be fulfilled. We present two examples that refute the assertions of Rhodes, Morari, and Wiggins. In the first example, the algorithm of Maas and Pope leads to a manifold that is not invariant but close to a slow invariant manifold. The claim of Rhodes, Morari, and Wiggins that the Maas and Pope projection algorithm is invariant under a coordinate transformation to Fenichel normal form is shown to be not correct in this case. In the second example, the projection algorithm of Maas and Pope leads to a manifold that lies in a region where no slow manifold exists at all. This rejects the conjecture of Rhodes, Morari, and Wiggins mentioned above.     

Comment on "Adaptive Q-S (lag, anticipated, and complete) time-varying synchronization and parameters identification of uncertain delayed neural networks" [Chaos 16, 023119 (2006)

This paper comments on a recent paper by Yu and Cao [Chaos, 16, 023119 (2006)]. We find that the theorem in this paper is incorrect by numerical simulations. The consequence of the incorrectness is analyzed as well.     

Self-organizing dynamics of the human brain: Critical instabilities and Sil'nikov chaos

Using a sensorimotor coordination task in conjunction with an array of SQUIDs (Superconducting QUantum Interference Devices) we demonstrate critical instabilities in human brain activity patterns. Analysis of the dominant spatial pattern of the brain and its time-varying amplitude displays a task-dependent geometry characteristic of Sil'nikov-like chaos, which changes qualitatively at the transition. (c) 1995 American Institute of Physics.