Showing papers on "Van der Pol oscillator published in 2023"
TL;DR: In this paper , a time-delayed Van der Pol (VDP)-Duffing oscillator was converted to an equivalent linear one and the convergence between the numerical outcomes and the analytic solution was achieved and gave a satisfying accuracy of the equivalent result.
Abstract: The impetus for the current investigation relates to the implementation of an efficient novel technique to obtain a time-delayed vibration control analytical solution. The current technique follows simple and easy-to-apply criteria. This technique is based on converting the nonlinear time-delayed Van der Pol (VDP)-Duffing oscillator to an equivalent linear one. Details of the conversion to an equivalent linear ordinary differential equation are mentioned. The convergence between the numerical outcomes and the analytic solution is achieved and gives a satisfying accuracy of the equivalent result.
5 citations
TL;DR: In this article , the authors considered the Van der Pol equation nonlinearity aspect related to a moving singular point and proved the existence of moving singular points and the uniqueness of their solution for complex domains.
Abstract: The article considers the Van der Pol equation nonlinearity aspect related to a moving singular point. The fact of the existence of moving singular points and the uniqueness of their solution for complex domains have been proved. An answer to the question about the existence of moving singular points in the real domain was obtained. The proof of existence and uniqueness is based on an author’s modification of the technology of the classical Cauchy theorem. A priori estimates of the analytical approximate solution in the vicinity of a moving singular point are obtained. Calculations of a numerical experiment are presented.
2 citations
TL;DR: In this article , a master-slave configuration to obtain synchronization between the double-well Duffing-van der Pol (master system) and the three-well Φ6 Duffing oscillators (slave system) is studied.
Abstract: Abstract In this work a master-slave configuration to obtain synchronization between the double-well Duffing-van der Pol (master system) and the three-well Φ6 Duffing oscillators (slave system) is studied. For this configuration, we analyze the system when the dissipative and one that combines the elastic and dissipative couplings are used. Whenever the dissipative coupling is used, we observed a vertical shift synchronization in the slave system. However, when the combination of the elastic and dissipative couplings is used, the vertical shift disappears obtaining complete synchronization. We resort to perturbation method to corroborate analytically these kind of synchronization, when the master system provides an harmonic function to the slave system. These synchronizations are numerically corroborated.
2 citations
2 citations
TL;DR: In this paper , an improved approximate Bayesian method is proposed to identify unknown parameters of chaotic VEH system by combining the particle swarm optimization and sequential Monte Carlo methods, which identifies the parameters of the Mathieu-van der Pol-Duffing chaotic energy harvester system.
2 citations
TL;DR: In this paper , the existence of solutions for the Van der Pol differential equation is proved and the approximate structure of such solutions in the analyticity domain is obtained, and the majorant method is applied not to the right side of the differential equation, as per usual, but to the solution to the nonlinear differential equation under consideration.
Abstract: In the article, the existence of solutions for the Van der Pol differential equation is proved, and the approximate structure of such solutions in the analyticity domain is obtained. In the proof, the majorant method was applied not to the right side of the differential equation, as per usual, but to the solution to the nonlinear differential equation under consideration. Results of the numerical study are presented.
2 citations
TL;DR: In this article , the authors studied a coupled autonomous dynamical system composed of a Duffing oscillator coupled to a van der Pol oscillator, and the most gratifying behavior reported in this article concerns the coexistence of four attractors consisting of a limit cycle of period n, a pair of chaotic attractors and a hyperchaotic attractor.
Abstract: Undoubtedly, multistability represents one of the most followed venues for researchers working in the field of nonlinear science. Multistability refers to the situation where a combination of two or more attractors occurs for the same rank of parameters. However, to the best of our knowledge, the situation encountered in the relevant literature is never one where periodicity, chaos and hyperchaos coexist. In this article, we study a fourth-order autonomous dynamical system composing of a Duffing oscillator coupled to a van der Pol oscillator. Coupling consists in disturbing the amplitude of one oscillator with a signal proportional to the amplitude of the other. We exploit analytical and numerical methods (bifurcation diagrams, phase portraits, basins of attraction) to shed light on the plethora of bifurcation modes exhibited by the coupled system. Several ranks of parameters are revealed where the coupled system exhibits two or more competing behaviors. In addition to the transient dynamics, the most gratifying behavior reported in this article concerns the coexistence of four attractors consisting of a limit cycle of period-n, a pair of chaotic attractors and a hyperchaotic attractor. The impact of a fractional-order derivative is also examined. A physical implementation of the coupled oscillator system is performed and the PSpice simulations confirm the predictions of the theoretical study conducted in this work.
1 citations
31 May 2023
TL;DR: In this paper , a design problem of the impulsive Goodwin's oscillator (IGO) to admit a desired periodic solution is introduced, where the design objective is to select the free parameters so that the IGO exhibits a stable 1-cycle with desired characteristics.
Abstract: The impulsive Goodwin’s oscillator (IGO) is a hybrid model composed of a third-order continuous linear part and a pulse-modulated feedback. This paper introduces a design problem of the IGO to admit a desired periodic solution. The dynamics of the continuous states represent the plant to be controlled, whereas the parameters of the impulsive feedback constitute design degrees of freedom. The design objective is to select the free parameters so that the IGO exhibits a stable 1-cycle with desired characteristics. The impulse-to-impulse map of the oscillator is demonstrated to always possess a positive fixed point that corresponds to the desired periodic solution; the closed-form expressions to evaluate this fixed point are provided. Necessary and sufficient conditions for orbital stability of the 1-cycle are presented in terms of the oscillator parameters and exhibit similarity to the problem of static output control. An IGO design procedure is proposed and validated by simulation. The nonlinear dynamics of the designed IGO are reviewed by means of bifurcation analysis. Applications of the design procedure to dosing problems in chemical industry and biomedicine are envisioned.
1 citations
1 citations
TL;DR: In this paper , the authors considered the case in which the diode included in a driven BVP oscillator is assumed to operate as an ideal switch, and they showed that un-nested, singly, and doubly nested MMO-incrementing bifurcations generated by the driven Bonhoeffer-van der Pol (BVP) oscillator coincide with one of the possible unnested and singly nested period-additive bifurbcations, respectively, generated with the Nagumo-Sato map.
1 citations
TL;DR: In this paper , the authors studied a network of coupled oscillators whose local dynamics are governed by the fractional-order versions of the paradigmatic van der Pol and Rayleigh oscillators and showed that the networks exhibit diverse amplitude chimeras and oscillation death patterns.
Abstract: We study networks of coupled oscillators whose local dynamics are governed by the fractional-order versions of the paradigmatic van der Pol and Rayleigh oscillators. We show that the networks exhibit diverse amplitude chimeras and oscillation death patterns. The occurrence of amplitude chimeras in a network of van der Pol oscillators is observed for the first time. A form of amplitude chimera, namely, "damped amplitude chimera" is observed and characterized, where the size of the incoherent region(s) increases continuously in the course of time, and the oscillations of drifting units are damped continuously until they are quenched to steady state. It is found that as the order of the fractional derivative decreases, the lifetime of classical amplitude chimeras increases, and there is a critical point at which there is a transition to damped amplitude chimeras. Overall, a decrease in the order of fractional derivatives reduces the propensity to synchronization and promotes oscillation death phenomena including solitary oscillation death and chimera death patterns that were unobserved in networks of integer-order oscillators. This effect of the fractional derivatives is verified by the stability analysis based on the properties of the master stability function of some collective dynamical states calculated from the block-diagonalized variational equations of the coupled systems. The present study generalizes the results of our recently studied network of fractional-order Stuart-Landau oscillators.
TL;DR: A priori estimates of the error of the analytical approximate solution are obtained, which allows the solving of the inverse problem of the theory of error: what should the structure of the analytic approximate solution be in order to obtain a result with a given accuracy? Thanks to a new approach for obtaining a priori evaluations of errors, based on elements of differential calculus, the domain, used to obtain an analytical approximation solution, was substantially expanded as discussed by the authors .
Abstract: This paper considers a theoretical substantiation of the influence of a perturbation of a moving singular point on the analytical approximate solution to the Van der Pol equation obtained earlier by the author. A priori estimates of the error of the analytical approximate solution are obtained, which allows the solving of the inverse problem of the theory of error: what should the structure of the analytical approximate solution be in order to obtain a result with a given accuracy? Thanks to a new approach for obtaining a priori evaluations of errors, based on elements of differential calculus, the domain, used to obtain an analytical approximate solution, was substantially expanded. A variant of optimizing a priori estimates using a posteriori estimates is illustrated. The results of a numerical experiment are also presented.
09 Mar 2023
TL;DR: In this paper , the authors describe and illustrate the application of a novel approximation technique for coupled, nonlinear dynamic systems, which is used to approximate particular terms in the fully-coupled nonlinear system in such a way that the target system is amenable to (approximate) analytical solution algorithms.
Abstract: Abstract In this paper, we describe and illustrate the application of a novel approximation technique for coupled, nonlinear dynamic systems. The technique begins by obtaining the analytical (or approximate analytical) solutions to the uncoupled system. Then, these solutions are used to approximate particular terms in the fully-coupled, nonlinear system in such a way that the target system is amenable to (approximate) analytical solution algorithms. This work forms part of a larger effort to develop robust control systems for large-scale industrial manipulators. To this end, the final example examined in this work considers the FutureForge manipulator: a state-of-the-art manipulator which forms part of a next-generation forging platform under development at the Advanced Forming Research Centre in Glasgow. To show the breadth of applications of our approach, we also apply it to more widely-recognised models like the Rayleigh and Van der Pol oscillators. In both of these cases, we consider a system of two oscillators each having dynamic behaviour described by Rayleigh/Van der Pol oscillators and coupled together through the resulting damping matrices.
01 Apr 2023
TL;DR: In this article , all the Darboux polynomials of a Mathieu-van der Pol-Duffing oscillator were characterized by transforming from the original system into a three dimensional system.
Abstract: In this paper, we characterize all the Darboux polynomials of a Mathieu-van der Pol-Duffing oscillator by transforming from the original system into a three dimensional system. We also provide a complete classification of the rational first integrals and of the Darboux first integrals through the analysis of its Darboux polynomials and its exponential factors.
31 May 2023
TL;DR: In this paper , the authors use Lyapunov-like functions and convex optimization to propagate uncertainty in the initial condition of nonlinear systems governed by ordinary differential equations, and obtain bounds on the expected future value of a quantity of interest.
Abstract: We use Lyapunov-like functions and convex optimization to propagate uncertainty in the initial condition of nonlinear systems governed by ordinary differential equations. We consider the full nonlinear dynamics without approximation, producing rigorous bounds on the expected future value of a quantity of interest even when only limited statistics of the initial condition (e.g., mean and variance) are known. For dynamical systems evolving in compact sets, the best upper (lower) bound coincides with the largest (smallest) expectation among all initial state distributions consistent with the known statistics. For systems governed by polynomial equations and polynomial quantities of interest, one-sided estimates on the optimal bounds can be computed using tools from polynomial optimization and semidefinite programming. Moreover, these numerical bounds provably converge to the optimal ones in the compact case. We illustrate the approach on a van der Pol oscillator and on the Lorenz system in the chaotic regime.
24 Apr 2023
TL;DR: In nonlinear dynamical systems, strong quasiperiodic beating effects appear due to combination of self-excited and forced vibration as mentioned in this paper , indicating an exchange of energy between individual degrees of freedom of the model or by multiple close dominant frequencies.
Abstract: In nonlinear dynamical systems, strong quasiperiodic beating effects appear due to combination of self-excited and forced vibration. The presence of symmetric or asymmetric beatings indicates an exchange of energy between individual degrees of freedom of the model or by multiple close dominant frequencies. This effect is illustrated by the case of the van der Pol equation in the vicinity of resonance. The approximate analysis of these nonlinear effects uses the harmonic balance method and the multiple scale method.
TL;DR: In this article , an orthogonal-function-based equivalent linearization method is proposed for nonlinear systems subjected to nonstationary random excitations. But it is not applicable to any general type of non-stationary non-excitation.
Abstract: Many mechanical systems manifest nonlinear behavior under nonstationary random excitations. Neglecting this nonlinearity in the modeling of a dynamic system would result in unacceptable results. However, it is challenging to find exact solutions to nonlinear problems. Therefore, equivalent linearization methods are often used to seek approximate solutions for this kind of problem. To overcome the limitations of the existing equivalent linearization methods, an orthogonal-function-based equivalent linearization method in the time domain is proposed for nonlinear systems subjected to nonstationary random excitations. The proposed method is first applied to a single-degree-of-freedom (SDOF) Duffing–Van der Pol oscillator subjected to stationary and nonstationary excitations to validate its accuracy. Then, its applicability to nonlinear MDOF systems is depicted by a 5DOF Duffing–Van der Pol system subjected to nonstationary excitation, with different levels of system nonlinearity strength considered in the analysis. Results show that the proposed method has the merit of predicting the nonlinear system response with high accuracy and computation efficiency. In addition, it is applicable to any general type of nonstationary random excitation.
TL;DR: In this paper , the influence of the coexisting steady states in high-dimensional systems on the dynamical evolution of the vector field when a slow-varying periodic excitation is introduced is studied.
Abstract: We study the influence of the coexisting steady states in high-dimensional systems on the dynamical evolution of the vector field when a slow-varying periodic excitation is introduced. The model under consideration is a coupled system of Bonhöffer–van der Pol (BVP) equations with a slow-varying periodic excitation. We apply the modified slow–fast analysis method to perform a detailed study on all the equilibrium branches and their bifurcations of the generalized autonomous system. According to different dynamical behaviors, we explore the dynamical evolution of existing attractors, which reveals the coexistence of a quasi-periodic attractor with diverse types of bursting attractors. Further investigation shows that the coexisting steady states may cause spiking oscillations to behave in combination of a 2D torus and a limit cycle. We also identify a period-2 cycle bursting attractor as well as a quasi-periodic attractor according to the period-2 limit cycle.
01 Jan 2023
TL;DR: In this paper , an analytical method (Homotopy perturbation method HPM) is used for solving the initial value problem represented by a neutrosophic nonlinear Van der Pol oscillator equation (N-VDP) arising in applied dynamics using the thick function.
Abstract: In this paper, an analytical method (Homotopy perturbation method HPM) is used for solving the initial value problem represented by a neutrosophic nonlinear Van der Pol oscillator equation (N-VDP) arising in applied dynamics using the thick function. We find the solutions of the (N-VDP) equation by HPM and then compare the numerical results with fourth order Runge-Kutta method (RK4). The results showed that the HPM lead to accurate and efficient results. Furthermore, these results of the HPM scheme and RK4 are implemented in Matlab.
31 May 2023
TL;DR: In this article , a data-driven approach for accurately modeling and estimating parameters of nonlinear dynamical systems using trust region optimization is presented, which can accurately identify the parameters of these nonlinear systems, providing a reliable characterization of their behavior.
Abstract: Nonlinear systems are prevalent in many fields of science and engineering, and understanding their behavior is essential for developing effective control and prediction strategies. In this paper, we present a novel data-driven approach for accurately modeling and estimating parameters of nonlinear systems using trust region optimization. Our method is applied to three classic systems: the Van der Pol oscillator, the Damped oscillator, and the Lorenz system, which have broad applications in various fields, including engineering, physics, and biology. Our results demonstrate that our approach can accurately identify the parameters of these nonlinear systems, providing a reliable characterization of their behavior. We show that the ability to capture the dynamics on the attractor is crucial for these systems, especially in chaotic systems like the Lorenz system. Overall, this article presents a robust data-driven approach for parameter estimation of nonlinear dynamical systems, with promising potential for real-world applications.
23 Feb 2023
TL;DR: In this article , a delay-coupling Van der Pol (VDP) oscillator was used to simulate the locomotion gait of a hexapod robot, where the structure topology of the CPG controller was scheduled as a unidirectional ring network consisting of six identical units.
Abstract: Abstract In this study, we construct a type of CPG (central pattern generator) controller by using the delay-coupling Van der Pol (VDP) oscillators and propose an analysis method of parameter modulation to illustrate the locomotion gait of a hexapod robot. The structure topology of the CPG controller is scheduled as a unidirectional ring network consisting of six identical units. Each unit has independentparameters to modulate amplitude and frequency of the period activity. Employing the Hopf bifurcation, we first propose parameter conditions to guarantee the existence of the period activity for the delayed CPG controller, where coupling delay can induce generation and extinction of the period activity. Due to the symmetry structure of the proposed CPG controller, the period activity induced by coupling delay presents multiple spatiotemporal patterns with a constant phase difference. Based on theoretical analysis of the equivariant Hopf bifurcation, we further pinpoint parameter regions for the corresponding spatiotemporal patterns. Then using the invariable phase difference between signal outputs of the VDP oscillators, we assign connection order of the CPG units to link the hexapod’s legs and produce the hexapodal locomotion gaits. The results show that the coupling delay in the delayed VDP-CPG controller is an effective method to obtain many types of the hexapodal locomotion gaits.
TL;DR: In this article , the authors identify the key geometric structures responsible for symmetry-breaking in coupled, identical, fast-slow systems, and show that many different types of symmetry breaking rhythms arise robustly.
Abstract: Symmetry-breaking in coupled, identical, fast-slow systems produces a rich, dramatic variety of dynamical behavior-such as amplitudes and frequencies differing by an order of magnitude or more and qualitatively different rhythms between oscillators, corresponding to different functional states. We present a novel method for analyzing these systems. It identifies the key geometric structures responsible for this new symmetry-breaking, and it shows that many different types of symmetry-breaking rhythms arise robustly. We find symmetry-breaking rhythms in which one oscillator exhibits small-amplitude oscillations, while the other exhibits phase-shifted small-amplitude oscillations, large-amplitude oscillations, mixed-mode oscillations, or even undergoes an explosion of limit cycle canards. Two prototypical fast-slow systems illustrate the method: the van der Pol equation that describes electrical circuits and the Lengyel-Epstein model of chemical oscillators.
TL;DR: In this article , a retarded van der Pol-Duffing oscillator with multiple delays is investigated and conditions for which Bogdanov-Takens bifurcation occurs around the trivial equilibrium of the proposed system.
Abstract: In this article, we will investigate a retarded van der Pol-Duffing oscillator with multiple delays. At first, we will find conditions for which Bogdanov-Takens (B-T) bifurcation occurs around the trivial equilibrium of the proposed system. The center manifold theory has been used to extract second order normal form of the B-T bifurcation. After that, we derived third order normal form. We also provide a few bifurcation diagrams, including those for the Hopf, double limit cycle, homoclinic, saddle-node, and Bogdanov-Takens bifurcation. In order to meet the theoretical requirements, extensive numerical simulations have been presented in the conclusion.
TL;DR: In this article , an analytical approximate method for Cauchy problems of the Van der Pol equation in the complex domain is presented, where the influence of variation in the initial data of the problem is used to control the computational process and improve the accuracy of the final results.
Abstract: In this paper, we substantiate the analytical approximate method for Cauchy problem of the Van der Pol equation in the complex domain. These approximate solutions allow analytical continuation for both real and complex cases. We follow the influence of variation in the initial data of the problem in order to control the computational process and improve the accuracy of the final results. Several simple applications of the method are given. A numerical study confirms the consistency of the developed method.
25 May 2023
05 Jul 2023
TL;DR: In this paper , a system of four coupled van der Pol oscillators with delay is studied, and conditions for the existence of multi-periodic solutions of the system are given.
Abstract: Abstract In this paper, a system of four coupled van der Pol oscillators with delay is studied. Firstly, the conditions for the existence of multi-periodic solutions of the system are given. Secondly, the multi-periodic solutions of spatiotemporal patterns of the system are obtained by using symmetric bifurcation theory. The normal form of the system on the central manifold and the direction of the bifurcation periodic solution are given. Finally, numerical simulations are used to support our theoretical results. MSC: 34K18, 35B32