Van Der Pol System Research Articles

Nonlinear dynamics of the CAM circadian rhythm in response to environmental forcing.

Crassulacean acid metabolism (CAM) photosynthesis functions as an endogenous circadian rhythm coupled to external environmental forcings of energy and water availability. This paper explores the nonlinear dynamics of a new CAM photosynthesis model (Bartlett et al., 2014) and investigates the responses of CAM plant carbon assimilation to different combinations of environmental conditions. The CAM model (Bartlett et al., 2014) consists of a Calvin cycle typical of C3 plants coupled to an oscillator of the type employed in the Van der Pol and FitzHugh-Nagumo systems. This coupled system is a function of environmental variables including leaf temperature, leaf moisture potential, and irradiance. Here, we explore the qualitative response of the system and the expected carbon assimilation under constant and periodically forced environmental conditions. The model results show how the diurnal evolution of these variables entrains the CAM cycle with prevailing environmental conditions. While constant environmental conditions generate either steady-state or periodically oscillating responses in malic acid uptake and release, forcing the CAM system with periodic daily fluctuations in light exposure and leaf temperature results in quasi-periodicity and possible chaos for certain ranges of these variables. This analysis is a first step in quantifying changes in CAM plant productivity with variables such as the mean temperature, daily temperature range, irradiance, and leaf moisture potential. Results may also be used to inform model parametrization based on the observed fluctuating regime.

An exponential method for accurate and automatic integration of nonlinear (stiff and nonstiff) ODE systems

In this paper, an explicit Exponential Method (EM), which is an off-shoot of Jibunoh’s spectral decomposition is developed for the accurate and automatic integration of nonlinear (stiff and nonstiff) ODE systems. In particular, the Vanderpol system of equations is solved. The method is also applicable to linear systems, including linear oscillatory systems or systems with complex eigenvalues. It has simplicity of implementation by automatic computation using the QBASIC Codes and produces high accuracy or the exact theoretical solutions in any nonlinear or linear systems. The EM is, therefore, superior to many traditional methods which are less accurate and which integrate nonlinear systems with cumbersome procedures.

An enriched multiple scales method for harmonically forced nonlinear systems

This article explores enrichment to the method of Multiple Scales, in some cases extending its applicability to periodic solutions of harmonically forced, strongly nonlinear systems. The enrichment follows from an introduced homotopy parameter in the system governing equation, which transitions it from linear to nonlinear behavior as the value varies from zero to one. This same parameter serves as a perturbation quantity in both the asymptotic expansion and the multiple time scales assumed solution form. Two prototypical nonlinear systems are explored. The first considered is a classical forced Duffing oscillator for which periodic solutions near primary resonance are analyzed, and their stability is assessed, as the strengths of the cubic term, the forcing, and a system scaling factor are increased. The second is a classical forced van der Pol oscillator for which quasiperiodic and subharmonic solutions are analyzed. For both systems, comparisons are made between solutions generated using (a) the enriched Multiple Scales approach, (b) the conventional Multiple Scales approach, and (c) numerical simulations. For the Duffing system, important qualitative and quantitative differences are noted between solutions predicted by the enriched and conventional Multiple Scales. For the van der Pol system, increased solution flexibility is noted with the enriched Multiple Scales approach, including the ability to seek subharmonic (and superharmonic) solutions not necessarily close to the linear natural frequency. In both nonlinear systems, comparisons to numerical simulations show strong agreement with results from the enriched technique, and for the Duffing case in particular, even when the system is strongly nonlinear.

Canard-like phenomena in piecewise-smooth Van der Pol systems

We show that a nonlinear, piecewise-smooth, planar dynamical system can exhibit canard phenomena. Canard phenomena in nonlinear, piecewise-smooth systems can be qualitatively more similar to the phenomena in smooth systems than piecewise-linear systems, since the nonlinearity allows for canards to transition from small cycles to canards “with heads.” The canards are born of a bifurcation that occurs as the slow-nullcline coincides with the splitting manifold. However, there are conditions under which this bifurcation leads to a phenomenon called super-explosion, the instantaneous transition from a globally attracting periodic orbit to relaxations oscillations. Also, we demonstrate that the bifurcation—whether leading to canards or super-explosion—can be subcritical.

Numerical Analysis of the Coupled Modified van der Pol Equations in a Model of Heart Action

In this paper, a modified van der Pol equations are considered as a description of the heart action. Wide ranges of the model parameters yield interesting qualitative results, e.g. Hopf bifurcation, Bogdanov-Takens bifurcation, transcritical and pitchfork bifurcations but also some stable solutions can be found. The physiological model works in the narrowest range of parameters which allows to obtain a stable behaviour what is important in biological problem. When some kinds of pathologies appear in the heart, it is possible to obtain chaotic behaviour. My aim is to compare the influence of these two types of coupling (unidirectional and bidirectional) on the behaviour of the van der Pol system. The coupling takes place in a system with healthy conductivity, between two nodes: SA and AV, but in some circumstances, a pathological coupling may occur in the heart. The van der Pol oscillator is a type of relaxation oscillator which can be synchronized. In this paper, synchronization properties of such a system are studied as well. For the purpose of a numerical analysis of the system in question, a numerical model was created.

Numerical Analysis of the Coupled Modified van der Pol Equations in a Model of Heart Action

In this paper, a modified van der Pol equations are considered as a description of the heart action. Wide ranges of the model parameters yield interesting qualitative results, e.g. Hopf bifurcation, Bogdanov-Takens bifurcation, transcritical and pitchfork bifurcations but also some stable solutions can be found. The physiological model works in the narrowest range of parameters which allows to obtain a stable behaviour what is important in biological problem. When some kinds of pathologies appear in the heart, it is possible to obtain chaotic behaviour. My aim is to compare the influence of these two types of coupling (unidirectional and bidirectional) on the behaviour of the van der Pol system. The coupling takes place in a system with healthy conductivity, between two nodes: SA and AV, but in some circumstances, a pathological coupling may occur in the heart. The van der Pol oscillator is a type of relaxation oscillator which can be synchronized. In this paper, synchronization properties of such a system are studied as well. For the purpose of a numerical analysis of the system in question, a numerical model was created.

Study of mixed-mode oscillations in a parametrically excited van der Pol system

This paper investigates the effects of slowly varying parametric excitation on the dynamics of van der Pol system. Periodic bifurcation delay behaviors are exhibited when the parametric excitation slowly passes through Hopf bifurcation value of the controlled van der Pol system. The first bifurcation delay behavior relies on initial conditions, while the bifurcation delay behaviors that follow the first one are immune to initial conditions. These bifurcation delay behaviors result in a hysteresis loop between the spiking attractor and the rest state, which is responsible for the generation of mixed-mode oscillations. Then an approximate calculation for the number of spikes in each cluster of repetitive spiking of mixed-mode oscillations is explored based on bifurcation delay behaviors. Theoretical results agree well with numerical simulations.

Pragmatical adaptive synchronization – New fuzzy model of two different and complex chaotic systems by new adaptive control

In this paper, (1) a new fuzzy model is presented to simulate two different chaotic systems with different numbers of nonlinear terms and (2) a new adaptive approach and a new control Lyapunov function are proposed to synchronize these two different fuzzy chaotic systems and speed up the convergence of errors. By using this new model, the numbers of fuzzy rules of chaotic systems can be reduced from 2N to 2×N and only 2 subsystems are needed, where N is the number of nonlinear terms. The fuzzy systems become much simpler. In addition, through the new fuzzy model, the new fuzzy systems are much simpler than T–S fuzzy systems (when nonlinear systems are complicated) and can be used to any other kind of application in fuzzy logic control or fuzzy modeling. Mathieu–Van der Pol system (which is called M–V system in this paper) and Quantum cellular neural networks nanosystem (which is called Q-CNN system in this paper) are used for illustrations in numerical simulation results to show the effectiveness and feasibility of our new adaptive approach and new control Lyapunov function. The T–S fuzzy modeling and traditional adaptive control are also given in Appendices B and C for comparison.

Hopf Bifurcations of a Stochastic Fractional-Order Van der Pol System

The Hopf bifurcation of a fractional-order Van der Pol (VDP for short) system with a random parameter is investigated. Firstly, the Chebyshev polynomial approximation is applied to study the stochastic fractional-order system. Based on the method, the stochastic system is reduced to the equivalent deterministic one, and then the responses of the stochastic system can be obtained by numerical methods. Then, according to the existence conditions of Hopf bifurcation, the critical parameter value of the bifurcation is obtained by theoretical analysis. Then, numerical simulations are carried out to verify the theoretical results.

Canard-like phenomena in piecewise-smooth Van der Pol systems

Canard-like phenomena in piecewise-smooth Van der Pol systems

Limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations

This paper deals with local bifurcations occurring near singular points of planar slow-fast systems. In particular, it is concerned with the study of the slow-fast variant of the unfolding of a codimension 3 nilpotent singularity. The slow-fast variant of a codimension 1 Hopf bifurcation has been studied extensively before and its study has lead to the notion of canard cycles in the Van der Pol system. Similarly, codimension 2 slow-fast Bogdanov–Takens bifurcations have been characterized. Here, the singularity is of codimension 3 and we distinguish slow-fast elliptic and slow-fast saddle bifurcations. We focus our study on the appearance on small-amplitude limit cycles, and rely on techniques from geometric singular perturbation theory and blow-up.

Haar wavelet operational methods for the numerical solutions of fractional order nonlinear oscillatory Van der Pol system

In this paper, an efficient numerical method for the solution of nonlinear damped Van der Pol equation based on the Haar wavelets approach is proposed. The proposed scheme can be used for solving one of the widely studied and challenging equation in nonlinear dynamics like the damped Van der Pol equation. We have applied a numerical procedure involving haar series for the highest order of derivative. Moreover the numerical result shows that the proposed mechanism is quite reasonable when compare to exact solution.

Robust Stability and Stabilization of Fractional Order Systems Based on Uncertain Takagi-Sugeno Fuzzy Model With the Fractional Order 1≤v < 2

This paper addresses the problems of the robust stability and stabilization for fractional order systems based on the uncertain Takagi-Sugeno fuzzy model. A sufficient and necessary condition of asymptotical stability for fractional order uncertain T-S fuzzy model is given, and a parallel distributed compensate fuzzy controller is designed to asymptotically stabilize the model. The results are obtained in terms of linear matrix inequalities. Finally, a numerical example and fractional order Van der Pol system are given to show the effectiveness of our results.

The role of slow system dynamics in predicting the degeneracy of slow invariant manifolds: The case of vdP relaxation–oscillations

The development of dissipative time scales, which are much faster than the rest, force the solution of a system of 1st order ODEs to land and then evolve on a slow invariant manifold. The slow evolution on this manifold can be approximated by a reduced system, which is constructed with asymptotic analysis and is free of the fast scales. In this work, the dynamics of the reduced system are compared to the fast and slow dynamics of the original system. The comparison is based on the van der Pol system, which is typical of those exhibiting relaxation–oscillation behavior and is characterized by successive dissipative and explosive phases. Special attention is given to the dissipative phase, during which the solution evolves on a slow invariant manifold and the fast/slow time scale gap tends to contract. It is shown that one cannot predict from the dynamics of the van der Pol system the approaching explosive phase and the associated degeneracy of the manifold. Such a prediction is possible only through the dynamics of the reduced system, which involves an explosive component that is absent in the original system. This explosive component accelerates as higher order effects are incorporated in the reduced system, resulting in a smaller gap between the fast time scale of the original system and the fast time scale of the reduced one. This feature sets a limit on the accuracy singular perturbation techniques can deliver, even in the presence of a reasonably large fast/slow time scale gap in the dynamics of the original system. Given a fast/slow system, predicting forthcoming changes in the dimensions of the manifold is very important when it is desired to identify the underlying major physical mechanisms or to control the process.

Bifurcations in Van der Pol-Like Systems

Generalizations of the Van der Pol system with polynomial interactions involving additional parameters are studied in order to understand qualitative properties such as stability and additional bifurcations in these generalized Van der Pol systems. The generalizations include the one known as the Duffing-Van der Pol system which has properties similar to those of the Maxwell-Bloch system.

SOME EVOLUTION EQUATIONS FOR AN ORNSTEIN–UHLENBECK PROCESS-DRIVEN DYNAMICAL SYSTEM

The statistical properties of the Ornstein–Uhlenbeck (OU) process, a colored noise process, confirm the real noise statistics, since the real noise process has finite, nonzero correlation time. For this reason, it seems worthwhile to develop the estimation-theoretic scenarios of dynamical systems embedded in the colored noise environment as well. Importantly, the application of the Itô theory is not straightforward to the dynamical system in which the OU variable is a driving input. The augmented solution vector approach coupled with the Itô stochastic differential rule plays the pivotal role to develop the theory of the OU process-driven Duffing–van der Pol (DvdP) system of this paper. Notably, the noise analysis of the Duffing–van der Pol system, especially from the estimation-theoretic viewpoint, under the colored noise influence is not available yet in literature. Numerical experimentations with three different sets of data are demonstrated to examine the efficacy of analytical findings of this paper. The results of this paper will be of interest to noise scientists, especially research communities in systems and control, looking for the estimation-theoretic scenarios of the colored noise-driven "vector" stochastic differential system.

Horseshoes of periodically kicked van der Pol oscillators

This paper contains a numerical study of the periodically forced van der Pol system. Our aim is to determine the extent to which chaotic behavior occurs in this system as well as the nature of the chaos. Unlike previous studies, which used continuous forcing, we work with instantaneous kicks, for which the geometry is simpler. Our study covers a range of parameters describing nonlinearity, kick sizes, and kick periods. We show that horseshoes are abundant whenever the limit cycle is kicked to a specific region of the phase space and offer a geometric explanation for the stretch-and-fold behavior which ensues.

Effective Fokker-Planck equation for birhythmic modified van der Pol oscillator

We present an explicit solution based on the phase-amplitude approximation of the Fokker-Planck equation associated with the Langevin equation of the birhythmic modified van der Pol system. The solution enables us to derive probability distributions analytically as well as the activation energies associated with switching between the coexisting different attractors that characterize the birhythmic system. Comparing analytical and numerical results we find good agreement when the frequencies of both attractors are equal, while the predictions of the analytic estimates deteriorate when the two frequencies depart. Under the effect of noise, the two states that characterize the birhythmic system can merge, inasmuch as the parameter plane of the birhythmic solutions is found to shrink when the noise intensity increases. The solution of the Fokker-Planck equation shows that in the birhythmic region, the two attractors are characterized by very different probabilities of finding the system in such a state. The probability becomes comparable only for a narrow range of the control parameters, thus the two limit cycles have properties in close analogy with the thermodynamic phases.

Chaotic system synchronization with tridiagonal structure and its initial investigation in complex power systems

Recently, existence of dynamical bifurcations and chaos has been shown by some researchers in simple power system networks. In this paper, a novel approach is first proposed to design stabilizing controllers for synchronization of chaotic systems with a tridiagonal structure and its initial investigation in complex power systems. During the design procedure, an original nonlinear affine system is firstly transformed into a stable system with special tridiagonal structure. Secondly, controllers are developed to synchronize the transformed chaotic system. It is the first time that chaotic synchronization and its initial application have been studied in complex power systems with tridiagonal structure. Finally, the Lü system, the Chua circuit system and the Van der Pol system are used to demonstrate the effectiveness of the proposed approach by applying them to a standard power system.

Structural susceptibility and separation of time scales in the van der Pol oscillator

We use an extension of the van der Pol oscillator as an example of a system with multiple time scales to study the susceptibility of its trajectory to polynomial perturbations in the dynamics. A striking feature of many nonlinear, multiparameter models is an apparently inherent insensitivity to large-magnitude variations in certain linear combinations of parameters. This phenomenon of "sloppiness" is quantified by calculating the eigenvalues of the Hessian matrix of the least-squares cost function. These typically span many orders of magnitude. The van der Pol system is no exception: Perturbations in its dynamics show that most directions in parameter space weakly affect the limit cycle, whereas only a few directions are stiff. With this study, we show that separating the time scales in the van der Pol system leads to a further separation of eigenvalues. Parameter combinations which perturb the slow manifold are stiffer and those which solely affect the jumps in the dynamics are sloppier.