Van Der Pol System Research Articles

An improved nonlinear proportional-integral-differential controller combined with fractional operator and symbolic adaptation algorithm

Parameter adjustment is usually applied for designing the proportional-integral-differential (PID) controllers. However, the ability to improve control performance by adjusting parameters is limited. Hence, with the goal to achieve ideal closed-loop response, this paper takes advantage of a structural optimization method for modifying the controller model. A symbolic adaptation algorithm for fractional order PID (FOPID) controller is employed to obtain precise nonlinear controller model. Firstly, a modeling comparison for nonlinear duffing system is carried out to highlight the efficiency of the symbolic adaptation algorithm. The case study indicates the proposed algorithm can establish compact dynamic models by amending the shortcomings of symbolic regression. Secondly, the proposed controller is restructured with the linear FOPID controller and its nonlinearity is increased by adjusting controllers’ components in symbolic form. The proposed controllers are simulated in an unstable second-order system, a time-delay system and a nonlinear VanderPol system. Compared with the IOPID and the FOPID controller, the symbolic adaptation algorithm improves the structural flexibility of these linear controllers. Meanwhile, the system response can better approximate the desired response and the structural integrity of the nonlinear controller model is guaranteed simultaneously. Finally, the nonlinear FOPD controllers for trajectory tracking experiments are carried out on a rotary inverted pendulum control system.

Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

Bifurcation and stability analysis of commensurate fractional-order van der Pol oscillator with time-delayed feedback

The stability and existence conditions of Hopf bifurcation of a commensurate fractional-order van der Pol oscillator with time-delayed feedback are studied. Firstly, the necessary and sufficient conditions for the asymptotic stability of the equilibrium point of fractional-order van der Pol oscillator with linear displacement feedback are obtained, and it is found that the conditions are not only related to the feedback gain, but also to the fractional order. Secondly, regarding time delay as a bifurcation parameter, the stability of the commensurate fractional-order van der Pol system with time-delayed feedback is investigated based on the characteristic equation. Under some conditions, the critical value of time delay is calculated. The equilibrium point is stable when the parameter is less than the critical value and will be unstable if the parameter is greater than it. Moreover, the conditions for the occurrence of Hopf bifurcation are obtained. Finally, choosing four typical system parameters, some numerical simulations are carried out to verify the correctness of the obtained theoretical results.

Active Learning of Dynamics for Data-Driven Control Using Koopman Operators

This paper presents an active learning strategy for robotic systems that takes into account task information, enables fast learning, and allows control to be readily synthesized by taking advantage of the Koopman operator representation. We first motivate the use of representing nonlinear systems as linear Koopman operator systems by illustrating the improved model-based control performance with an actuated Van der Pol system. Information-theoretic methods are then applied to the Koopman operator formulation of dynamical systems where we derive a controller for active learning of robot dynamics. The active learning controller is shown to increase the rate of information about the Koopman operator. In addition, our active learning controller can readily incorporate policies built on the Koopman dynamics, enabling the benefits of fast active learning and improved control. Results using a quadcopter illustrate single-execution active learning and stabilization capabilities during free fall. The results for active learning are extended for automating Koopman observables and we implement our method on real robotic systems.

Stochastic P-bifurcation in a generalized Van der Pol oscillator with fractional delayed feedback excited by combined Gaussian white noise excitations

This paper investigates the stochastic P-bifurcation behavior of bistability in a generalized Van der Pol system with fractional time-delay feedback under additive and multiplicative Gaussian white noise excitations. First, using the minimal mean square error principle, the fractional derivative is found to be equivalent to a linear combination of damping and restoring forces, and the original system is transformed into an equivalent integer-order system. Second, the stationary probability density function of system amplitude is obtained by stochastic averaging, and according to singularity theory, the critical parameters for stochastic P-bifurcation of the system are found. Finally, the different types of stationary probability density function curves of the system amplitude are qualitatively analyzed by choosing the corresponding parameters in each region divided by the transition set curves. The consistency between the analytical solutions and the Monte Carlo simulation results verifies the theoretical analysis in this paper. The method used in this paper can directly guide the design of the fractional-order controller to adjust the response of the system.

Super and sub-harmonic synchronization in generalized van der Pol oscillator

Effects of quasiperiodic beatings and sub- and super-harmonic synchronization reveal in plasma physics, optics, acoustics, neurophysiology, etc., but also very often in aero-elasticity of large slender engineering structures, e.g., bridge decks, towers, masts, high rise buildings, ropes, where these phenomena emerge in relation with vortex shedding effects. This applies mainly to quasiperiodic beatings that can be encountered especially in the lock-in regimes, when the vortex frequency becomes close to the structure eigenfrequency ω0 with a small positive or negative detuning. This phenomenon has been thoroughly investigated for the van der Pol system in the previous study by the authors of this paper. However, experimental testing in a wind tunnel showed also other phenomena that can occur in the system resonance zone due to excitations with a multiple or rational fraction frequency leading to sub- or super-harmonic synchronization. All these effects are very dangerous from the viewpoint of reliability and safety of respective systems and hence, a robust theoretical background for design of adequate countermeasures are worthwhile to be developed. In this subsequent study the original sub- and super-synchronization effects are identified and quantified including assessment of their dynamic stability.

Transient motion and chaotic dynamics in a pair of van der Pol oscillators

The transient chaos and the stable chaotic dynamics of coupled autonomous van der Pol (VdP) oscillators with cubic term are investigated. Transient chaos is a common phenomenon in externally driven van der Pol oscillators. Nevertheless, in coupled autonomous VdP oscillators the occurrence of transient chaos, even stable chaos, is a rare scenario. To the best of our knowledge, transient chaos has not often been observed in coupled autonomous van der Pol systems. We demonstrate that the nonlinear restoring forces in a pair of van der Pol oscillators can induce a transient chaotic route to deterministic chaos. The symmetric coupling has been considered and provided by perturbing the amplitude of one oscillator by a fraction another oscillator's amplitude. The coupled systems undergo a crisis when the coupling parameter passes through a certain threshold. The crisis occurs when the chaotic attractor behaves as a chaotic repeller for a transient time and transient chaos emerges. After transient motion, the system's dynamics is attracted either to a periodic motion or stationary state. The Lyapunov spectrum, bifurcation diagram, phase space trajectories and Poincare section were used to study the chaotic motion. The effects of nonlinear restoring force have been investigated through bifurcation diagram.

Cooperative robust output regulation for a class of nonlinear multi-agent systems subject to a nonlinear leader system

In this paper, we study the cooperative robust output regulation problem for a class of nonlinear multi-agent systems subject to a nonlinear leader system. We first establish a result on the existence of a distributed observer for a class of nonlinear leader systems. Then, we convert the cooperative robust output regulation problem into a robust stabilization problem of an augmented system composed of the given multi-agent system and the distributed internal model. Finally, we stabilize this augmented system by a decentralized output feedback control law, which together with the distributed observer and the distributed internal model, constitutes an overall distributed output feedback control law to solve the cooperative robust output regulation problem of the given multi-agent system. An example with the well-known Van der Pol system as the leader system is used to illustrate our design.

Generalised energy conservation law for chaotic phenomena

Chaotic phenomena are increasingly being observed in all fields of nature, where investigations reveal that a natural phenomenon exhibits nonlinearities and attempts to reveal their deep underlying mechanisms. Chaos is normally understood as “a state of disorder”, for which there is as yet no universally accepted mathematical definition. A commonly used concept states that, for a dynamical system to be classified as chaotic, it must have the following properties: be sensitive to initial conditions, show topological transitivity, have densely periodical orbits etc. Revealing the rules that govern chaotic motion is thus an important unsolved task for exploring nature. We present herein a generalised energy conservation law governing chaotic phenomena. Based on two scalar variables, viz. generalised potential and kinetic energies defined in the phase space describing nonlinear dynamical systems, we find that chaotic motion is periodic motion with infinite time period whose time-averaged generalised potential and kinetic energies are conserved over its time period. This implies that, as the averaging time is increased, the time-averaged generalised potential and kinetic energies tend to constants while the time-averaged energy flows, i.e., their rates of change with time, tend to zero. Numerical simulations on reported chaotic motions, such as the forced van der Pol system, forced Duffing system, forced smooth and discontinuous oscillator, Lorenz’s system, and Rössler’s system, show the above conclusions to be correct according to the results presented herein. This discovery may indicate that chaotic phenomena in nature could be controlled because, even though their instantaneous states are disordered, their long-time averages can be predicted.

An efficient optimized adaptive step-size hybrid block method for integrating differential systems

This paper deals with the development, analysis and implementation of an optimized hybrid block method having different features, for integrating numerically initial value ordinary differential systems. The hybrid nature of the proposed one-step scheme allows us to bypass the first Dahlquist’s barrier on linear multi-step methods. The theory of interpolation and collocation has been used in the development of the method. We assume an appropriate polynomial representation of the theoretical solution of the problem and consider three off-step points in a one-step block. One of these three off-step points is fixed and the other two off-step points are optimized in order to minimize the local truncation errors of the main method and other additional formula. The resulting scheme is of order five having the property of A-stability. An embedded-type approach is used in order to formulate the proposed method in adaptive form, showing a high efficiency. The adaptive method is tested on well-known differential systems viz. the Robertson’s system, a Gear’s system, a system related with Jacobi elliptic functions, the Brusselator system, and the Van der Pol system, and compared with some well-known numerical codes in the scientific literature.

Lévy noise induced transitions and enhanced stability in a birhythmic van der Pol system

This work describes the effects of Lévy noise on a birhythmic van der Pol like oscillator. The two periodic attractors are characterized by different periods, and the stability in the presence of Gaussian noise can be described by an effective, or quasi-potential. Numerical simulations demonstrate that in the presence of Lévy noise the induced escapes from an attractor to another are similar to the escapes between stable points in an ordinary potential. Assuming that the attractors are almost separated by a barrier of a quasi (or pseudo) potential, the theory for Lévy noise escapes captures the qualitative features of the escapes across the quasi-potential. The differences to the Gaussian case are more pronounced for large values of the Lévy index. We found that for the symmetric quasi-potential, the relative stability of the two attractors are similar, while in the asymmetric case the properties of the two attractors differ for increasing a. The global stability is also characterized by means of the residence times, that give indications for future theoretical analysis.

Applying higher order stochastic averaging method to the Van der Pol system with timedelay under random excitation

The paper investigated the Van der Pol system with time-delay under random excitation by the higher stochastic averaging method. The original system was expressed in terms without time-delay under the assumption that the state variabled of the system were slowly varying processed. Then the higher stochastic averaging method was applied on the approximation system. By this technique, the analytical expression of the stationary probability density function for the Van der Pol system with time-delay under random excitation was showed in higher order approximation for the first time. Effects of the parameter time-delay on the system’s response were investigated. The analytical results were suited well to numerical ones obtained by Monte-Carlo method. It was also showed that the higher order averaging solution was better than the one obtained by the traditional stochastic averaging method.

Stochastic P-bifurcation in a tri-stable Van der Pol system with fractional derivative under Gaussian white noise

In this paper, we study the tri-stable stochastic P-bifurcation problem of a generalized Van der Pol system with fractional derivative under Gaussian white noise excitation. Firstly, using the principle for minimal mean square error, we show that the fractional derivative term is equivalent to a linear combination of the damping force and restoring force, so that the original system can be transformed into an equivalent integer order system. Secondly, we obtain the stationary Probability Density Function (PDF) of the system’s amplitude by the stochastic averaging, and using the singularity theory, we find the critical parametric conditions for stochastic P-bifurcation of amplitude of the system, which can make the system switch among the three steady states. Finally, we analyze different types of the stationary PDF curves of the system amplitude qualitatively by choosing parameters corresponding to each region divided by the transition set curves, and the system response can be maintained at the small amplitude near the equilibrium by selecting the appropriate unfolding parameters. We verify the theoretical analysis and calculation of the transition set by showing the consistency of the numerical results obtained by Monte Carlo simulation with the analytical results. The method used in this paper directly guides the design of the fractional order controller to adjust the response of the system.

A Distributed Observer for a Class of Nonlinear Systems and Its Application to a Leader-Following Consensus Problem

In this note, we first present a framework for designing a distributed control law to solve a cooperative control problem of nonlinear multiagent systems with a nonlinear leader system. The distributed control law is composed of a purely decentralized control law and a distributed observer for the leader system, based on the certainty equivalence principle. Then, we establish a sufficient condition to guarantee the existence of a distributed observer for a class of nonlinear autonomous systems. The framework is then applied to solve the leader-following consensus problem for a class of high-order nonlinear heterogeneous multiagent systems subject to external disturbances. Our design is illustrated by an example with the Van der Pol system as the leader system.

Parabolic bursting, spike-adding, dips and slices in a minimal model

A minimal system for parabolic bursting, whose associated slow flow is integrable, is presented and studied both from the viewpoint of bifurcation theory of slow-fast systems, of the qualitative analysis of its phase portrait and of numerical simulations. We focus the analysis on the spike-adding phenomenon. After a reduction to a periodically forced one-dimensional system, we uncover the link with the dips and slices first discussed by J.E. Littlewood in his famous articles on the periodically forced van der Pol system.

Frequency-truncation fast-slow analysis for parametrically and externally excited systems with two slow incommensurate excitation frequencies

This paper aims to report an approximation method, the frequency-truncation fast-slow analysis, for analyzing fast-slow dynamics in parametrically and externally excited systems with two slow incommensurate excitation frequencies (PEESTSIEFs). We obtain truncated, commensurate excitation frequencies, which are approximations of the incommensurate excitation frequencies. Then, we show numerically that bursting behavior in PEESTSIEFs can be approximated in the same systems but with truncated, commensurate excitation frequencies, and therefore bursting dynamics in PEESTSIEFs can be understood by analyzing the same systems with truncated, commensurate excitation frequencies. Based on this, the approximation method for analyzing bursting dynamics in PEESTSIEFs is proposed. The validity of the approach is demonstrated by the Duffing and van der Pol systems, respectively.

Stochastic P-bifurcation of fractional derivative Van der Pol system excited by Gaussian white noise

This paper aimed to investigate the stochastic P-bifurcation of Van der Pol oscillator with a fractional derivative damping term driven by Gaussian white noise excitation. Firstly, based on the method of stochastic averaging method and Stratonovich–Khasminskii theorem, the corresponding Fokker–Plank–Kolmogorov (FPK) equation is deduced. To describe the P-bifurcation of system, the stationary probability densities of amplitude can be obtained by solving the FPK equation. Then, the effects of the fractional order, the fractional coefficient, and the intensity of Gaussian white noise on the fractional systems are discussed in detail. The results show that increasing order α will change obviously the number and the height of peaks under certain parameter conditions. Finally, comparing the analytical and numerical results, a very satisfactory agreement can be found.

Explosive death of conjugate coupled Van der Pol oscillators on networks.

Explosive death phenomenon has been gradually gaining attention of researchers due to the research boom of explosive synchronization, and it has been observed recently for the identical or nonidentical coupled systems in all-to-all network. In this work, we investigate the emergence of explosive death in networked Van der Pol (VdP) oscillators with conjugate variables coupling. It is demonstrated that the network structures play a crucial role in identifying the types of explosive death behaviors. We also observe that the damping coefficient of the VdP system not only can determine whether the explosive death state is generated but also can adjust the forward transition point. We further show that the backward transition point is independent of the network topologies and the damping coefficient, which is well confirmed by theoretical analysis. Our results reveal the generality of explosive death phenomenon in different network topologies and are propitious to promote a better comprehension for the oscillation quenching behaviors.

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.

Novel algebraic aspects of Liouvillian integrability for two-dimensional polynomial dynamical systems

The general structure of irreducible invariant algebraic curves for a polynomial dynamical system in C2 is found. Necessary conditions for existence of exponential factors related to an invariant algebraic curve are derived. As a consequence, all the cases when the classical force-free Duffing and Duffing–van der Pol oscillators possess Liouvillian first integrals are obtained. New exact solutions for the force-free Duffing–van der Pol system are constructed.