The Necessary Condition for the Existence of Periodic Solutions of Certain Three Dimensional Nonlinear Dynamical Systems

In this paper, we investigate a class of three dimensional nonlinear dynamical systems whose unperturbed systems have a family of periodic orbits. Firstly, we establish the moving Frenet Frame on these closed orbits. Secondly, the successor functions are defined by the orbits which go through the normal plane. Finally, by judging the existence of solutions of the equations obtained from the Successor functions, we obtain the necessary condition for the existence of periodic solutions of these three dimensional nonlinear dynamical systems. The result has important significance for the basic research of applied mechanics.

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The Sufficient Condition for the Existence of Periodic Solution of Certain Three Dimensional Nonlinear Dynamical Systems

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.249-250.147 ◽

2012 ◽

Vol 249-250 ◽

pp. 147-152

Author(s):

Jing Li ◽

Ting Ting Quan ◽

Xiao Dong Jin

Keyword(s):

Periodic Solution ◽

Dynamical Systems ◽

Nonlinear Dynamical Systems ◽

Closed Orbit ◽

Three Dimensional ◽

Existence Of Solution ◽

Sufficient Condition ◽

Frenet Frame ◽

Nonlinear Dynamical ◽

Important Significance

In this paper, the sufficient condition for the existence of periodic solution of a class of three dimensional nonlinear dynamical systems is investigated. The moving Frenet frame is established on the closed orbit and the successor functions near the closed orbit are defined. According to the study of the existence of solution of the equation which obtained from the successor functions with the implicit function theorem, the sufficient condition for the existence of periodic solution of these systems is obtained. The results in this paper have important significance to decide the existence of periodic solution of three dimensional nonlinear dynamical systems.

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ON THE ONSET OF QUASI-PERIODIC SOLUTIONS IN THIRD-ORDER NONLINEAR DYNAMICAL SYSTEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127405014064 ◽

2005 ◽

Vol 15 (10) ◽

pp. 3165-3180 ◽

Cited By ~ 4

Author(s):

R. GENESIO ◽

C. GHILARDI

Keyword(s):

Dynamical Systems ◽

Periodic Solutions ◽

Harmonic Balance ◽

Nonlinear Dynamical Systems ◽

General Relation ◽

Three Dimensional ◽

Third Order ◽

Van Der Pol ◽

Nonlinear Dynamical ◽

Classical Systems

The paper considers the existence of quasi-periodic solutions in three-dimensional systems. Since these solutions commonly arise as a consequence of a Neimark–Sacker bifurcation of a limit cycle, a fairly general relation connected to this phenomenon is pointed out as the main result of the paper. Then, the application of harmonic balance techniques makes possible to exploit such a relation. In particular, a simplified condition denoting the quasi-periodicity onset can be derived, in making evident the main elements for this transition in terms of structure and parameters, and hence some remarks on the features of the interested systems. Several examples show the application of the above condition to detect "tori" in the state space in a qualitative (not simply numerical) way. They consider classical systems — Rössler, where such behavior seems to be unknown, Chua, forced Van der Pol — and new quadratic systems.

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Prediction for the Rub-Impact Phenomena in Rotor Systems

Volume 6C: 18th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc2001/vib-21633 ◽

2001 ◽

Author(s):

Zhiqiang Wu ◽

Yushu Chen

Keyword(s):

Dynamical Systems ◽

Periodic Solutions ◽

Nonlinear Dynamical Systems ◽

Nonlinear Dynamical ◽

Rotor Systems ◽

Different Types ◽

Impact Phenomena ◽

Direct Guidance

Abstract By the method (Wu, 2001) developed by authors for singularity analyzing of the bifurcation of the periodic solutions in nonlinear dynamical systems with clearance, the bifurcation patterns of non-impact-rub response and a method for predicting rub-impact are given. It is shown that there are much more types of bifurcation patterns when the clearance constraint is take into account. Given their physical meanings of the parameters in practical rotor systems, the resonant periodic solutions of rotor systems consist of 11 different types of bifurcation patterns among of which the following four types are more likely to appear, (1) patterns without impact and jump, (2) jump patterns without impact, (3) impact pattern without jump and (4) patterns with impact and jump. Based on these results, parameter conditions for rub-impact phenomena are derived. These conditions can give more direct guidance to the design of rotor systems. The method proposed here can be used to predict rub-impact phenomena in more complicated rotor systems.

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