In this paper, a non-existence condition for homoclinic and heteroclinic orbits in n-dimensional, continuous-time, and smooth systems is obtained, Based on this result and an elementary example, it can be conjectured ...In this paper, a non-existence condition for homoclinic and heteroclinic orbits in n-dimensional, continuous-time, and smooth systems is obtained, Based on this result and an elementary example, it can be conjectured that there is a fourth kind of chaos in polynomial ordinary differential equation （ODE） systems characterized by the nonexistence of homoclinic and heteroclinic orbits.展开更多
A paper, ＂Non-existence of Shilnikov chaos in continuous-time systems＂ was published in the journal Applied Mathematics and Mechanics （English Edition）. The authors gave sufficient conditions for the non-existence...A paper, ＂Non-existence of Shilnikov chaos in continuous-time systems＂ was published in the journal Applied Mathematics and Mechanics （English Edition）. The authors gave sufficient conditions for the non-existence of homoclinic and heteroclinic orbits in an nth-order autonomous system. Unfortunately, we show in this comment that the proof presented is erroneous and the result is invalid. We also provide two counterexamples of the wrong criterion stated by the authors.展开更多
This paper is concerned with the pattern dynamics of the generalized nonlinear Schrodinger equations(NSEs)relatedwith various nonlinear physical problems in plasmas.Our theoretical and numerical results show that the ...This paper is concerned with the pattern dynamics of the generalized nonlinear Schrodinger equations(NSEs)relatedwith various nonlinear physical problems in plasmas.Our theoretical and numerical results show that the higher-order nonlinear effects,acting as a Hamiltonian perturbation,break down the NSE integrability and lead to chaotic behaviors.Correspondingly,coherent structures are destroyed and replaced by complex patterns.Homoclinic orbit crossings in the phase space and stochastic partition of energy in Fourier modes show typical characteristics of the stochastic motion.Our investigations show that nonlinear phenomena,such as wave turbulence and laser filamentation,are associated with the homoclinic chaos.In particular,we found that the unstable manifolds W(u)possessing the hyperbolic fixed point correspond to an initial phase θ=45° and 225° ,and the stable manifolds W(s)correspond toθ=135° and 315° .展开更多
文摘In this paper, a non-existence condition for homoclinic and heteroclinic orbits in n-dimensional, continuous-time, and smooth systems is obtained, Based on this result and an elementary example, it can be conjectured that there is a fourth kind of chaos in polynomial ordinary differential equation （ODE） systems characterized by the nonexistence of homoclinic and heteroclinic orbits.
基金supported by the Ministerio de Educacion y Ciencia,Plan Nacional I+D+I co-financed with FEDER Funds(No.MTM2010-20907-C02)the Consejeria de Educacion y Ciencia de la Juntade Andalucia(Nos.FQM-276,TIC-0130,and P08-FQM-03770)
文摘A paper, ＂Non-existence of Shilnikov chaos in continuous-time systems＂ was published in the journal Applied Mathematics and Mechanics （English Edition）. The authors gave sufficient conditions for the non-existence of homoclinic and heteroclinic orbits in an nth-order autonomous system. Unfortunately, we show in this comment that the proof presented is erroneous and the result is invalid. We also provide two counterexamples of the wrong criterion stated by the authors.
基金This work is also supported by the National Natural Science Foundation of China grant Nos.10575013 and 10576007partially by the National Basic Research Program of China(973)(2007CB814802 and 2007CB815101).
文摘This paper is concerned with the pattern dynamics of the generalized nonlinear Schrodinger equations(NSEs)relatedwith various nonlinear physical problems in plasmas.Our theoretical and numerical results show that the higher-order nonlinear effects,acting as a Hamiltonian perturbation,break down the NSE integrability and lead to chaotic behaviors.Correspondingly,coherent structures are destroyed and replaced by complex patterns.Homoclinic orbit crossings in the phase space and stochastic partition of energy in Fourier modes show typical characteristics of the stochastic motion.Our investigations show that nonlinear phenomena,such as wave turbulence and laser filamentation,are associated with the homoclinic chaos.In particular,we found that the unstable manifolds W(u)possessing the hyperbolic fixed point correspond to an initial phase θ=45° and 225° ,and the stable manifolds W(s)correspond toθ=135° and 315° .
