scholarly journals Geometric Numerical Integration of Liénard Systems via a Contact Hamiltonian Approach

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1960
Author(s):  
Federico Zadra ◽  
Alessandro Bravetti ◽  
Marcello Seri
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Van Der Pol Oscillator ◽  
Geometric Numerical Integration ◽  
Time Step ◽  
Van Der Pol ◽  
Hamiltonian Description ◽  
Nonlinear Dynamical ◽  
New Family ◽  
Geometric Integrators ◽  
Liénard Systems

Starting from a contact Hamiltonian description of Liénard systems, we introduce a new family of explicit geometric integrators for these nonlinear dynamical systems. Focusing on the paradigmatic example of the van der Pol oscillator, we demonstrate that these integrators are particularly stable and preserve the qualitative features of the dynamics, even for relatively large values of the time step and in the stiff regime.

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.

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.

The Generalized Shooting Arc-Length Method for Periodic Solutions and the Corresponding Periods of Nonlinear Dynamical Systems

2001 ◽  
Author(s):  
Dexin Li ◽  
Jianxue Xu
Keyword(s):  
Dynamical System ◽  
Periodic Solution ◽  
Periodic Orbit ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Optimization Method ◽  
Arc Length ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Nonlinear Dynamical

Abstract In this paper, a generalized shooting/arc-length method for determining periodic orbit and its period of nonlinear dynamical system is presented. At first, by changing the time scale the period value of periodic orbit of the nonlinear system is drawn into the governing equation of this system. Then, by using the period value as a parameter, the shooting/arc-length procedure is taken for seeking such a periodic solution and its period simultaneously. The value of increment changed in iteration procedure is selected by using optimization method. The procedure involves the detennining of periodic orbit and its period value of the system. Thereby, the periodic orbit and period value of the system can be sought out rapidly and precisely. At last, the validity of such method is verified by determining the periodic orbit and period value for van der pol equation and nonlinear rotor-bear system.

Periodic Solutions for Coupled Van Der Pol Oscillators of Two-Degree-of-Freedom Solved by Homotopy Analysis Method

2009 ◽  
Author(s):  
Wei Zhang ◽  
Youhua Qian ◽  
Qian Wang
Keyword(s):  
Dynamical Systems ◽  
Analytical Solutions ◽  
Nonlinear Dynamical Systems ◽  
Degree Of Freedom ◽  
Analysis Method ◽  
Van Der Pol ◽  
Nonlinear Dynamical ◽  
Homotopy Analysis ◽  
Van Der Pol Oscillators ◽  
Two Degree Of Freedom

Innumerable engineering problems can be described by multi-degree-of-freedom (MDOF) nonlinear dynamical systems. The theoretical modelling of such systems is often governed by a set of coupled second-order differential equations. Albeit that it is extremely difficult to find their exact solutions, the research efforts are mainly concentrated on the approximate analytical solutions. The homotopy analysis method (HAM) is a useful analytic technique for solving nonlinear dynamical systems and the method is independent on the presence of small parameters in the governing equations. More importantly, unlike classical perturbation technique, it provides a simple way to ensure the convergence of solution series by means of an auxiliary parameter ħ. In this paper, the HAM is presented to establish the analytical approximate periodic solutions for two-degree-of-freedom coupled van der Pol oscillators. In addition, comparisons are conducted between the results obtained by the HAM and the numerical integration (i.e. Runge-Kutta) method. It is shown that the higher-order analytical solutions of the HAM agree well with the numerical integration solutions, even if time t progresses to a certain large domain in the time history responses.

scholarly journals Controlling Stochastic Sensitivity by Feedback Regulators in Nonlinear Dynamical Systems with Incomplete Information

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3229
Author(s):  
Irina Bashkirtseva
Keyword(s):  
Dynamical Systems ◽  
Incomplete Information ◽  
Matrix Equation ◽  
Nonlinear Dynamical Systems ◽  
Van Der Pol Oscillator ◽  
Nonlinear Dynamical ◽  
Stochastic Sensitivity ◽  
Quadratic Matrix Equation ◽  
Quadratic Matrix ◽  
Stochastic Oscillators

The problem of synthesis of stochastic sensitivity for equilibrium modes in nonlinear randomly forced dynamical systems with incomplete information is considered. We construct a feedback regulator that uses noisy data on some system state coordinates. For parameters of the regulator providing assigned stochastic sensitivity, a quadratic matrix equation is derived. Attainability of the assigned stochastic sensitivity is reduced to the solvability of this equation. We suggest a constructive algorithm for solving this quadratic matrix equation. These general theoretical results are used to solve the problem of stabilizing equilibrium modes of nonlinear stochastic oscillators under conditions of incomplete information. Details of our approach are illustrated on the example of a van der Pol oscillator.

DEVELOPMENT OF THE NONLINEAR DYNAMICAL SYSTEMS THEORY FROM RADIO ENGINEERING TO ELECTRONICS

2009 ◽  
Vol 19 (07) ◽  
pp. 2131-2163 ◽  
Author(s):  
CHRISTOPHE LETELLIER ◽  
JEAN-MARC GINOUX
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Dynamical System Theory ◽  
Three Body Problem ◽  
Van Der Pol ◽  
Nonlinear Dynamical ◽  
Radio Engineering ◽  
Chua Circuit ◽  
Three Body ◽  
Initial Results

Although initial results that contributed to the emergence of the nonlinear dynamical system theory arose from astronomy (the three-body problem), many subsequent developments were related to radio engineering and electronics. The path between the van der Pol equation and the Chua circuit is thus reviewed through main historical contributions.

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.

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