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
Comments on 'Non-existence of Shilnikov chaos in continuous-time systems' are given.An error has been found in the proof of Theorem 1 in the paper by Elhadj and Sprott(Elhadj,Z.and Sprott,J.Non-existence of Sh...Comments on 'Non-existence of Shilnikov chaos in continuous-time systems' are given.An error has been found in the proof of Theorem 1 in the paper by Elhadj and Sprott(Elhadj,Z.and Sprott,J.Non-existence of Shilnikov chaos in continuous-time systems.Applied Mathematics and Mechanics(English Edition),33(3),1-4(2012)).It makes the main conclusion of the paper incorrect,that is to say,the non-existence of Shilnikov chaos in the continuous-time systems considered cannot be ensured.Furthermore,a counter-example shows that Theorem 1 in the paper is incorrect.展开更多
The Shilnikov sense Smale horseshoe chaos in a simple 3D nonlinear system is studied. The proportional integral derivative （PID） controller is improved by introducing the quadratic and cubic nonlinearities into the ...The Shilnikov sense Smale horseshoe chaos in a simple 3D nonlinear system is studied. The proportional integral derivative （PID） controller is improved by introducing the quadratic and cubic nonlinearities into the governing equa- tions. For the discussion of chaos, the bifurcate parameter value is selected in a reasonable regime at the requirement of the Shilnikov theorem. The analytic expression of the Shilnikov type homoclinic orbit is accomplished. It depends on the series form of the manifolds surrounding the saddle-focus equilibrium. Then the methodology is extended to research the dynamical behaviours of the simplified solar-wind-driven-magnetosphere-ionosphere system. As is illustrated, the Lyapunov characteristic exponent spectra of the two systems indicate the existence of chaotic attractor under some specific parameter conditions.展开更多
In this paper, the Pad6 approximant and analytic solution in the neighborhood of the initial value are introduced into the process of constructing the Shilnikov type homoclinic trajectories in three-dimensional nonlin...In this paper, the Pad6 approximant and analytic solution in the neighborhood of the initial value are introduced into the process of constructing the Shilnikov type homoclinic trajectories in three-dimensional nonlinear dynamical systems. The P1D controller system with quadratic and cubic nonlinearities, the simplified solar-wind-driven-magnetosphere-ionosphere system, and the human DNA sequence system are considered. With the aid of presenting a new condition, the solutions of solving the boundary-value problems which are formulated for the trajectory and evaluating the initial amplitude values become available. At the same time, the value of the bifurcation parameter is obtained directly, which is almost consistent with the numerical result.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Project supported by the National Natural Science Foundation of China(No.11102156)the Northwestern Polytechnical University Foundation for Fundamental Research
文摘Comments on 'Non-existence of Shilnikov chaos in continuous-time systems' are given.An error has been found in the proof of Theorem 1 in the paper by Elhadj and Sprott(Elhadj,Z.and Sprott,J.Non-existence of Shilnikov chaos in continuous-time systems.Applied Mathematics and Mechanics(English Edition),33(3),1-4(2012)).It makes the main conclusion of the paper incorrect,that is to say,the non-existence of Shilnikov chaos in the continuous-time systems considered cannot be ensured.Furthermore,a counter-example shows that Theorem 1 in the paper is incorrect.
基金Project supported by the National Natural Science Foundation of China (Grant No. 10872141)
文摘The Shilnikov sense Smale horseshoe chaos in a simple 3D nonlinear system is studied. The proportional integral derivative （PID） controller is improved by introducing the quadratic and cubic nonlinearities into the governing equa- tions. For the discussion of chaos, the bifurcate parameter value is selected in a reasonable regime at the requirement of the Shilnikov theorem. The analytic expression of the Shilnikov type homoclinic orbit is accomplished. It depends on the series form of the manifolds surrounding the saddle-focus equilibrium. Then the methodology is extended to research the dynamical behaviours of the simplified solar-wind-driven-magnetosphere-ionosphere system. As is illustrated, the Lyapunov characteristic exponent spectra of the two systems indicate the existence of chaotic attractor under some specific parameter conditions.
基金supported by the National Natural Science Foundation of China (Grant Nos. 11072168 and 11102127)the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20100032120006)the Research Program of Application Foundation and Advanced Technology of Tianjin, China (Grant Nos. 12JCYBJC12500 and 11JCYBJC05800)
文摘In this paper, the Pad6 approximant and analytic solution in the neighborhood of the initial value are introduced into the process of constructing the Shilnikov type homoclinic trajectories in three-dimensional nonlinear dynamical systems. The P1D controller system with quadratic and cubic nonlinearities, the simplified solar-wind-driven-magnetosphere-ionosphere system, and the human DNA sequence system are considered. With the aid of presenting a new condition, the solutions of solving the boundary-value problems which are formulated for the trajectory and evaluating the initial amplitude values become available. At the same time, the value of the bifurcation parameter is obtained directly, which is almost consistent with the numerical result.
