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 "Sil'nikov chaos of the Liu system" [Chaos 18, 013113 (2008)

In the referenced paper, the authors use the undetermined coefficient method to prove analytically the existence of homoclinic and heteroclinic orbits in a Lorenz-like system. If the proof was correct, the existence of horseshoe chaos would be guaranteed via the Sil'nikov criterion. However, we hereby show that their demonstration is incorrect for two reasons. On the one hand, they wrongly use a symmetry the Lorenz-like system exhibits. On the other hand, they try to find structurally unstable global bifurcations by means of a series that is uniformly convergent in an open set of the parameter space: this would imply that the dynamical object they have found is structurally stable.     

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.     

Comment on phase evolution in two-photon amplifier systems

The evolution of the pulse chirp is examined for the case when two distinct ultrashort pulses are simultaneously amplified in a two-photon amplifier system.     

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 "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.     

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 "phase transitions in systems of self-propelled agents and related network models"

A Comment on the Letter by M. Aldana et al., Phys. Rev. Lett. 98, 095702 (2007)10.1103/PhysRevLett.98.095702.