Chaos in a generalized van der Pol system and in its fractional order system

In this paper, chaos of a generalized van der Pol system with fractional orders is studied. Both nonautonomous and autonomous systems are considered in detail. Chaos in the nonautonomous generalized van der Pol system excited by a sinusoidal time function with fractional orders is studied. Next, chaos in the autonomous generalized van der Pol system with fractional orders is considered. By numerical analyses, such as phase portraits, Poincaré maps and bifurcation diagrams, periodic, and chaotic motions are observed. Finally, it is found that chaos exists in the fractional order system with the order both less than and more than the number of the states of the integer order generalized van der Pol system.     

Haar wavelet solutions of nonlinear oscillator equations

In this paper, we present a numerical scheme using uniform Haar wavelet approximation and quasilinearization process for solving some nonlinear oscillator equations. In our proposed work, quasilinearization technique is first applied through Haar wavelets to convert a nonlinear differential equation into a set of linear algebraic equations. Finally, to demonstrate the validity of the proposed method, it has been applied on three type of nonlinear oscillators namely Duffing, Van der Pol, and Duffing–van der Pol. The obtained responses are presented graphically and compared with available numerical and analytical solutions found in the literature. The main advantage of uniform Haar wavelet series with quasilinearization process is that it captures the behavior of the nonlinear oscillators without any iteration. The numerical problems are considered with force and without force to check the efficiency and simple applicability of method on nonlinear oscillator problems.     

A new modification of the variational iteration method for van der Pol equations

In this paper, we provide a new modification of the variational iteration method (MVIM) for solving van der Pol equations. The modification couples the classical variational iteration method with He’s polynomials, where the He’s polynomials are applied to the approximate solution and the initial condition to eliminate secular terms. For the large ?, the numerical results demonstrate that the modification method get an accurate approximate period than the other presented methods.     

Solutions to Painleve III and other nonlinear equations by a generalized Cole–Hopf transformation

In this paper, we introduce a generalized form of Cole‐Hopf transformation and apply it to find new closed‐form (analytic) solutions to Painleve III equation. The same transformation is used then to find analytic solutions for the van der Pol and other nonlinear convective equations. These solutions provide analytic insights to some practical problems and might be used also to test the accuracy of numerical solutions. Copyright © 2017 John Wiley & Sons, Ltd.     

Nonintegrability of Parametrically Forced Nonlinear Oscillators

We discuss nonintegrability of parametrically forced nonlinear oscillators which are represented by second-order homogeneous differential equations with trigonometric coefficients and contain the Duffing and van der Pol oscillators as special cases. Specifically, we give sufficient conditions for their rational nonintegrability in the meaning of Bogoyavlenskij, using the Kovacic algorithm as well as an extension of the Morales–Ramis theory due to Ayoul and Zung. In application of the extended Morales–Ramis theory, for the associated variational equations, the identity components of their differential Galois groups are shown to be not commutative even if the differential Galois groups are triangularizable, i. e., they can be solved by quadratures. The obtained results are very general and reveal their rational nonintegrability for the wide class of parametrically forced nonlinear oscillators. We also give two examples for the van der Pol and Duffing oscillators to demonstrate our results.     

Iterative solution of some nonlinear differential equations

We propose an iterative method to solve some non-linear ordinary differential equations. Comparing on the Mathieu, van der Pol and Hill equation of fourth order, we see that this method is much more efficient than the well known methods by Lyapunov or Picard.     

Numerical Scaling Invariance Applied to the van der Pol Model

In this study we use the van der Pol model to explain a novel numerical application of scaling invariance. The model in point is not invariant to a scaling group of transformations, but by introducing an embedding parameter we are able to recover it from an extended model which is invariant to an extended scaling group. As well known, within a similarity analysis we can define a family of solutions from a computed one, so that the solution of a target problem can be obtained by rescaling the solution of a reference problem. The main idea is to use scaling invariance and numerical analysis to find a reference problem easier to solve, from a numerical viewpoint, than the target problem. This allows us to save human efforts and computational resources every time we have to solve a challenging problem. We test our approach using three stiff solvers available within the most recent releases of MATLAB. Independently from the solver used, by employing the described scaling invariance we are able to significantly reduce the computational cost of the numerical solution of the van der Pol model.      

Generalizations of the image conjecture and the Mathieu conjecture

We first propose a generalization of the image conjecture Zhao (submitted for publication) [31] for the commuting differential operators related with classical orthogonal polynomials. We then show that the non-trivial case of this generalized image conjecture is equivalent to a variation of the Mathieu conjecture Mathieu (1997) [21] from integrals of G-finite functions over reductive Lie groups G to integrals of polynomials over open subsets of Rⁿ with any positive measures. Via this equivalence, the generalized image conjecture can also be viewed as a natural variation of the Duistermaat and van der Kallen theorem Duistermaat and van der Kallen (1998) [14] on Laurent polynomials with no constant terms. To put all the conjectures above in a common setting, we introduce what we call the Mathieu subspaces of associative algebras. We also discuss some examples of Mathieu subspaces from other sources and derive some general results on this newly introduced notion.     

Existence of limit cycles for coupled van der Pol equations

In this paper, we consider the existence of limit cycles of coupled van der Pol equations by using S¹-degree theory due to Dylawerski et al. (see Ann. Polon. Math. 62 (1991) 243).     

The Order Completion Method for Systems of Nonlinear PDEs Revisited

In this paper we presents further developments regarding the enrichment of the basic Theory of Order Completion as presented in Oberguggenberger and Rosinger (Solution of continuous nonlinear PDEs through order completion, North-Holland, Amsterdam, 1994). In particular, spaces of generalized functions are constructed that contain generalized solutions to a large class of systems of continuous, nonlinear PDEs. In terms of the existence and uniqueness results previously obtained for such systems of equations (van der Walt, Acta Appl. Math. 103:1–17, 2008), one may interpret the existence of generalized solutions presented here as a regularity result. Furthermore, it is indicated how the methods developed in this paper may be adapted to solve initial and/or boundary value problems. In particular, we consider the Navier-Stokes equations in three spacial dimensions, subject to an initial condition on the velocity. In this regard, we obtain the existence of a generalized solution to a large class of such initial value problems.      

Applicability of the Interval Taylor Model to the Computational Proof of Existence of Periodic Trajectories in Systems of Ordinary Differential Equations

We consider the construction of the interval Taylor model used to prove the existence of periodic trajectories in systems of ordinary differential equations. Our model differs from the ones available in the literature in the method for describing the algorithms for the computation of arithmetic operations over Taylor models. In the framework of the current model, this permits reducing the computational expenditures for obtaining interval estimates on computers. We prove an assertion that permits establishing the existence of a periodic solution of a system of ordinary differential equations by verifying the convergence of the Picard iterations in the sense of embedding of the proposed Taylor models. An example illustrating how the resulting assertion can be used to prove the existence of a closed trajectory in the van der Pol system is given.     

Numerical solution to generalized Lyapunov/Stein and rational Riccati equations in stochastic control

We consider the numerical solution of the generalized Lyapunov and Stein equations in \(\mathbb {R}^{n}\), arising respectively from stochastic optimal control in continuous- and discrete-time. Generalizing the Smith method, our algorithms converge quadratically and have an O(n³) computational complexity per iteration and an O(n²) memory requirement. For large-scale problems, when the relevant matrix operators are “sparse”, our algorithm for generalized Stein (or Lyapunov) equations may achieve the complexity and memory requirement of O(n) (or similar to that of the solution of the linear systems associated with the sparse matrix operators). These efficient algorithms can be applied to Newton’s method for the solution of the rational Riccati equations. This contrasts favourably with the naive Newton algorithms of O(n⁶) complexity or the slower modified Newton’s methods of O(n³) complexity. The convergence and error analysis will be considered and numerical examples provided.     

Hamiltonian formulation of energy conservative variational equations by wavelet expansion

Hamiltonian formulation of various energy conservative evolution equations is given by means of wavelet expansion of solutions on the whole real axis R. The KdV equation, wave equations and Schrödinger equations are treated in a unified similar manner. A matrix representation of operators with respect to a nice wavelet base plays an important role in the formulation. Since the procedure is very concrete, our results can be used to efficiently compute numerical solutions of partial differential equations described in the text. In fact, we may also use symplectic schemes to solve derived Hamiltonian systems.     

Continuous block implicit hybrid one-step methods for ordinary and delay differential equations

A class of high order continuous block implicit hybrid one-step methods has been proposed to solve numerically initial value problems for ordinary and delay differential equations. The convergence and Aω-stability of the continuous block implicit hybrid methods for ordinary differential equations are studied. Alternative form of continuous extension is constructed such that the block implicit hybrid one-step methods can be used to solve delay differential equations and have same convergence order as for ordinary differential equations. Some numerical experiments are conducted to illustrate the efficiency of the continuous methods.     

Dynamics of three coupled van der Pol oscillators with application to circadian rhythms

In this work we study a system of three van der Pol oscillators. Two of the oscillators are identical, and are not directly coupled to each other, but rather are coupled via the third oscillator. We investigate the existence of the in-phase mode in which the two identical oscillators have the same behavior. To this end we use the two variable expansion perturbation method (also known as multiple scales) to obtain a slow flow, which we then analyze using the computer algebra system MACSYMA and the numerical bifurcation software AUTO.Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We model the circadian oscillator in each eye as a van der Pol oscillator. Although there is no direct connection between the two eyes, they are both connected to the brain, especially to the pineal gland, which is here represented by a third van der Pol oscillator.     

Stochastic optimal control of first-passage failure for coupled Duffing–van der Pol system under Gaussian white noise excitations

In this paper, the stochastic averaging method of quasi-non-integrable-Hamiltonian systems is applied to Duffing–van der Pol system to obtain partially averaged Ito stochastic differential equations. On the basis of the stochastic dynamical programming principle and the partially averaged Ito equation, dynamical programming equations for the reliability function and the mean first-passage time of controlled system are established. Then a non-linear stochastic optimal control strategy for coupled Duffing–van der Pol system subject to Gaussian white noise excitation is taken for investigating feedback minimization of first-passage failure. By averaging the terms involving control forces and replacing control forces by the optimal ones, the fully averaged Ito equation is derived. Thus, the feedback minimization for first-passage failure of controlled system can be obtained by solving the final dynamical programming equations. Numerical results for first-passage reliability function and mean first-passage time of the controlled and uncontrolled systems are compared in illustrative figures to show effectiveness and efficiency of the proposed method.     

Duffing's equation and nonlinear resonance

The phenomenon of nonlinear resonance (sometimes called the ‘jump phenomenon’) is examined and second-order van der Pol plane analysis is employed to indicate that this phenomenon is not a feature of the equation, but rather the result of accumulated round-off error, truncation error and algorithm error that distorts the true bounded solution onto an unbounded one. This is a common occurrence when numerically solving differential equations with initial values very close to a separatrix that distinguishes between stable (bounded) solutions and unstable (unbounded) solutions. This numerical phenomenon is not discussed in most texts and it is the purpose of this article to describe the effect is such a way as to make it suitable for beginning students to understand why things happen the way they do. Given the modern trend for computer laboratory projects in beginning differential equations courses, it is important for students to be aware of one of the common failings of numerical solutions.     

Global existence of solutions for a class of nonlinear differential equations

By making use of a special Lyapunov-type function and applying the comparison method due to Conti, we prove global existence of solutions for a general class of nonlinear second-order differential equations that includes, in particular, van der Pol, Rayleigh, and Liénard equations, widely encountered in applications. Relevant examples are discussed.     

Laguerre approximation with negative integer and its application for the delay differential equation

In this paper, two kinds of novel algorithms based on generalized Laguerre approximation with negative integer are presented to solve the delay differential equations. The algorithms differ from the spectral collocation method by the high sparsity of the matrices. Moreover, the use of generalized Laguerre polynomials leads to much simplified analysis and more precise error estimates. The numerical results indicate the high accuracy and the stability of long-time calculation of suggested algorithm.