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

2005 ◽  
Vol 15 (10) ◽  
pp. 3165-3180 ◽  
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.

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

2012 ◽  
Vol 252 ◽  
pp. 40-43
Author(s):  
Ting Ting Quan ◽  
Jing Li ◽  
Min Sun
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Nonlinear Dynamical Systems ◽  
Three Dimensional ◽  
Basic Research ◽  
Necessary Condition ◽  
Nonlinear Dynamical ◽  
Closed Orbits ◽  
Existence Of Periodic Solutions ◽  
Important Significance

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.

HIGH-DIMENSIONAL CHAOS IN DISSIPATIVE AND DRIVEN DYNAMICAL SYSTEMS

2009 ◽  
Vol 19 (09) ◽  
pp. 2823-2869 ◽  
Author(s):  
Z. E. MUSIELAK ◽  
D. E. MUSIELAK
Keyword(s):  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Nonlinear Dynamical Systems ◽  
High Dimensional ◽  
Van Der Pol ◽  
Nonlinear Dynamical ◽  
Onset Of Chaos ◽  
General Rules ◽  
Van Der Pol Oscillators ◽  
Control And Synchronization

Studies of nonlinear dynamical systems with many degrees of freedom show that the behavior of these systems is significantly different as compared with the behavior of systems with less than two degrees of freedom. These findings motivated us to carry out a survey of research focusing on the behavior of high-dimensional chaos, which include onset of chaos, routes to chaos and the persistence of chaos. This paper reports on various methods of generating and investigating nonlinear, dissipative and driven dynamical systems that exhibit high-dimensional chaos, and reviews recent results in this new field of research. We study high-dimensional Lorenz, Duffing, Rössler and Van der Pol oscillators, modified canonical Chua's circuits, and other dynamical systems and maps, and we formulate general rules of high-dimensional chaos. Basic techniques of chaos control and synchronization developed for high-dimensional dynamical systems are also reviewed.

Generalized Energy Functions for a Class of Third-Order Nonlinear Dynamical Systems

2020 ◽  
pp. 1-1
Author(s):  
Luis Fernando Costa Alberto ◽  
Daniel Siqueira ◽  
Newton Geraldo Bretas ◽  
Hsiao-Dong Chiang
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Third Order ◽  
Energy Functions ◽  
Nonlinear Dynamical

scholarly journals Analytic Matrix Method for Frequency Response Techniques Applied to Nonlinear Dynamical Systems I: Small and Medium Amplitude Oscillations

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3287
Author(s):  
Elena Hernandez ◽  
Octavio Manero ◽  
Fernando Bautista ◽  
Juan Paulo Garcia-Sandoval
Keyword(s):  
Dynamical Systems ◽  
Frequency Response ◽  
Harmonic Balance ◽  
Matrix Method ◽  
Series Solution ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Analytic Matrix ◽  
Solution Methods ◽  
Traditional Approaches

This is the first on a series of articles that deal with nonlinear dynamical systems under oscillatory input that may exhibit harmonic and non-harmonic frequencies and possibly complex behavior in the form of chaos. Frequency response techniques of nonlinear dynamical systems are usually analyzed with numerical methods because, most of the time, analytical solutions turn out to be difficult, if not impossible, since they are based on infinite series of trigonometric functions. The analytic matrix method reported here is a direct one that speeds up the solution processing compared to traditional series solution methods. In this method, we work with the invariant submanifold of the problem, and we propose a series solution that is equivalent to the harmonic balance series solution. However, the recursive relation obtained for the coefficients in our analytical method simplifies traditional approaches to obtain the solution with the harmonic balance series method. This method can be applied to nonlinear dynamic systems under oscillatory input to find the analog of a usual Bode plot where regions of small and medium amplitude oscillatory input are well described. We found that the identification of such regions requires both the amplitude as well as the frequency to be properly specified. In the second paper of the series, the method to solve problems in the field of large amplitudes will be addressed.

Prediction for the Rub-Impact Phenomena in Rotor Systems

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.

Lock-in and quasiperiodicity in a forced hydrodynamically self-excited jet

2013 ◽  
Vol 726 ◽  
pp. 624-655 ◽  
Author(s):  
Larry K. B. Li ◽  
Matthew P. Juniper
Keyword(s):  
Natural Frequency ◽  
Nonlinear Dynamical Systems ◽  
Three Dimensional ◽  
Van Der Pol Oscillator ◽  
Natural Mode ◽  
Van Der Pol ◽  
Nonlinear Dynamical ◽  
Frequency Pulling ◽  
Low Dimensional ◽  
Lock In

AbstractThe ability of hydrodynamically self-excited jets to lock into strong external forcing is well known. Their dynamics before lock-in and the specific bifurcations through which they lock in, however, are less well known. In this experimental study, we acoustically force a low-density jet around its natural global frequency. We examine its response leading up to lock-in and compare this to that of a forced van der Pol oscillator. We find that, when forced at increasing amplitudes, the jet undergoes a sequence of two nonlinear transitions: (i) from periodicity to ${ \mathbb{T} }^{2} $ quasiperiodicity via a torus-birth bifurcation; and then (ii) from ${ \mathbb{T} }^{2} $ quasiperiodicity to 1:1 lock-in via either a saddle-node bifurcation with frequency pulling, if the forcing and natural frequencies are close together, or a torus-death bifurcation without frequency pulling, but with a gradual suppression of the natural mode, if the two frequencies are far apart. We also find that the jet locks in most readily when forced close to its natural frequency, but that the details contain two asymmetries: the jet (i) locks in more readily and (ii) oscillates more strongly when it is forced below its natural frequency than when it is forced above it. Except for the second asymmetry, all of these transitions, bifurcations and dynamics are accurately reproduced by the forced van der Pol oscillator. This shows that this complex (infinite-dimensional) forced self-excited jet can be modelled reasonably well as a simple (three-dimensional) forced self-excited oscillator. This result adds to the growing evidence that open self-excited flows behave essentially like low-dimensional nonlinear dynamical systems. It also strengthens the universality of such flows, raising the possibility that more of them, including some industrially relevant flames, can be similarly modelled.

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