Showing papers on "Van der Pol oscillator published in 2015"
TL;DR: In this article, the authors studied the dynamics of the single and coupled van der Pol-Duffing oscillators and performed a detailed bifurcation analysis of these attractors and investigated how this plethora of states influences the network of coupled oscillators.
Abstract: We study the dynamics of the single and coupled van der Pol-Duffing oscillators Each oscillator is characterized by the multistability (the coexistence of attractors) Some of the coexisting attractors have very small basins of attraction (the rare ones) and some of them do not contain equilibria in their basin of attraction (the hidden ones) We perform the detailed bifurcation analysis of these attractors and investigate how this plethora of states influences the dynamics of the network of coupled oscillators We have observed the cluster synchronization on different attractors as well as different types of chimera states
97 citations
TL;DR: Varying the bifurcation parameter of the Van der Pol system can interpolate between regular sinusoidal and strongly nonlinear relaxation oscillations and it is demonstrated that more pronounced nonlinearity induces multi-chimera states with multiple incoherent domains.
Abstract: Chimera states are complex spatio-temporal patterns in which domains of synchronous and asynchronous dynamics coexist in coupled systems of oscillators. We examine how the character of the individual elements influences chimera states by studying networks of nonlocally coupled Van der Pol oscillators. Varying the bifurcation parameter of the Van der Pol system, we can interpolate between regular sinusoidal and strongly nonlinear relaxation oscillations and demonstrate that more pronounced nonlinearity induces multi-chimera states with multiple incoherent domains. We show that the stability regimes for multi-chimera states and the mean phase velocity profiles of the oscillators change significantly as the nonlinearity becomes stronger. Furthermore, we reveal the influence of time delay on chimera patterns.
85 citations
TL;DR: In this paper, the synchronization of two dissipatively coupled Van der Pol oscillators in the quantum regime is studied, where strict frequency locking is absent and is replaced by a crossover from weak to strong frequency entrainment.
Abstract: Synchronization of two dissipatively coupled Van der Pol oscillators in the quantum regime is studied. Due to quantum noise strict frequency locking is absent and is replaced by a crossover from weak to strong frequency entrainment. The differences to the behavior of one quantum Van der Pol oscillator subject to an external drive are discussed. Moreover, a possible experimental realization of two coupled quantum Van der Pol oscillators in an optomechanical setting is described.
76 citations
TL;DR: This study shows that, by introducing aslow variable and finding the relation between the slow variable and the slow excitations, PEESTLEFs can be transformed into a fast-slow form with a singleSlow variable and therefore MMOs observed in PEESTleFs can been understood by the classical machinery of fast subsystem analysis of the transformed fast- slow system.
Abstract: We present a general method for analyzing mixed-mode oscillations (MMOs) in parametrically and externally excited systems with two low excitation frequencies (PEESTLEFs) for the case of arbitrary m:n relation between the slow frequencies of excitations. The validity of the approach has been demonstrated using the equations of Duffing and van der Pol, separately. Our study shows that, by introducing a slow variable and finding the relation between the slow variable and the slow excitations, PEESTLEFs can be transformed into a fast-slow form with a single slow variable and therefore MMOs observed in PEESTLEFs can be understood by the classical machinery of fast subsystem analysis of the transformed fast-slow system.
75 citations
01 Jan 2015
TL;DR: In this paper, the qualitative properties of the forced Van der Pol chaotic oscillator have been discussed, and new results for the global chaos synchronization of the identical forced van der pol chaotic oscillators via adaptive control method have been derived.
Abstract: Chaos theory has a lot of applications in science and engineering. This paper first details the qualitative properties of the forced Van der Pol chaotic oscillator, which has important applications. Since its introduction in the 1920's, the Van der Pol equation has been a prototype model for systems with self-excited limit cycle oscillations. The Van der Pol equation has been studied over wide parameter regimes, from perturbations of harmonic motion to relaxation oscillations. It has been used by scientists to model a variety of physical and biological phenomena. Next, we derive new results for the global chaos synchronization of the identical forced Van der Pol chaotic oscillators via adaptive control method. MATLAB plots have been shown to illustrate the phase portraits of the forced Van der Pol chaotic oscillator and the adaptive synchronization of the forced Van der Pol chaotic oscillator.
72 citations
Book•
10 Jan 2015
TL;DR: The BvP model as mentioned in this paper is a generalization of the van der Pol oscillator for the description of oscillatory phenomena, which can be seen as an extension of the Hodgkin-Huxley model.
Abstract: its variables can still be interpreted in terms of physiological variables, since they have been derived from the physiological variables of the Hodgkin–Huxley model. The other way in which Fitzhugh derived these equations was by introducing one additional parameter in the original van der Pol oscillator equations which modifies them so that they have the qualitative features which Bonhoeffer had postulated, namely showing excitable and oscillatory behavior. The original van der Pol oscillator equations are symmetric in the sense that they are unaffected by a reversal of the sign of the variables. Since this symmetry is broken in the BvP model by the introduction of an additional parameter, the BvP model represents an important generalization of the van der Pol oscillator for the description of oscillatory phenomena. By studying the BvP system, we can thus learn something about physiological neuronal systems, and study a very important modification and generalization of the van der Pol oscillator dynamics. In the Bonhoeffer–van der Pol model, the dynamics of a single neuron is described by a system of two coupled differential equations ẋ1 = F1(x1, x2) = c · (x1 − x1/3 + x2 + z) ẋ2 = F2(x1, x2) = −1c · (x1 + bx2 − a). (8.1) According to Fitzhugh’s derivation of the BvP equations [?], x1 represents the negative transmembrane voltage and x2 is closely related to the potassium conductivity. The dynamical character of the solutions of this system of equations is determined by the parameter z, which represents the excitation of a neuron. In the absence of noise, z determines whether the system is an oscillator which periodically changes its voltage, or an excitable element which rests at a fixed voltage. The phase portrait for these two dynamical modes is shown in Figs. 8.1 and 8.2. For an analysis of the BvP system, we first want to determine the stationary point of the system. It can be obtained as the intersection point of the nullclines. The nullclines for the dynamical system April 23, 2000 Preliminary version
68 citations
01 Jan 2015
TL;DR: In this article, the qualitative properties of the forced Van der Pol chaotic oscillator have been discussed and new results for output regulation of the chaotic oscillators via adaptive control method have been obtained.
Abstract: Chaos theory has a lot of applications in science and engineering. This paper first details the qualitative properties of the forced Van der Pol chaotic oscillator, which has important applications. Since its introduction in the 1920's, the Van der Pol equation has been a prototype model for systems with self-excited limit cycle oscillations. The Van der Pol equation has been studied over wide parameter regimes, from perturbations of harmonic motion to relaxation oscillations. It has been used by scientists to model a variety of physical and biological phenomena. Next, new results are obtained for the output regulation of the forced Van der Pol chaotic oscillator via adaptive control method. MATLAB plots have been shown to illustrate the phase portraits of the forced Van der Pol chaotic oscillator and the output regulation of the forced Van der Pol chaotic oscillator.
67 citations
TL;DR: In this paper, the authors examine how the character of individual elements influences chimera states by studying networks of non-locally coupled Van der Pol oscillators and reveal the influence of time delay on chimera patterns.
Abstract: Chimera states are complex spatio-temporal patterns in which domains of synchronous and asynchronous dynamics coexist in coupled systems of oscillators. We examine how the character of the individual elements influences chimera states by studying networks of nonlocally coupled Van der Pol oscillators. Varying the bifurcation parameter of the Van der Pol system, we can interpolate between regular sinusoidal and strongly nonlinear relaxation oscillations, and demonstrate that more pronounced nonlinearity induces multi-chimera states with multiple incoherent domains. We show that the stability regimes for multi-chimera states and the mean phase velocity profiles of the oscillators change significantly as the nonlinearity becomes stronger. Furthermore, we reveal the influence of time delay on chimera patterns.
60 citations
TL;DR: In this paper, the authors present a method for explicit leapfrog integration of inseparable Hamiltonian systems by means of an extended phase space, which can be numerically integrated by standard symplectic leapfrog (splitting) methods.
Abstract: We present a method for explicit leapfrog integration of inseparable Hamiltonian systems by means of an extended phase space. A suitably defined new Hamiltonian on the extended phase space leads to equations of motion that can be numerically integrated by standard symplectic leapfrog (splitting) methods. When the leapfrog is combined with coordinate mixing transformations, the resulting algorithm shows good long term stability and error behaviour. We extend the method to non-Hamiltonian problems as well, and investigate optimal methods of projecting the extended phase space back to original dimension. Finally, we apply the methods to a Hamiltonian problem of geodesics in a curved space, and a non-Hamiltonian problem of a forced non-linear oscillator. We compare the performance of the methods to a general purpose differential equation solver LSODE, and the implicit midpoint method, a symplectic one-step method. We find the extended phase space methods to compare favorably to both for the Hamiltonian problem, and to the implicit midpoint method in the case of the non-linear oscillator.
52 citations
TL;DR: In this paper, the authors proposed a weak signal detection method based on the van der Pol-Duffing oscillator, which is more robust against the different frequency signal compared to the Duffing oscillators.
51 citations
TL;DR: The chaotic behavior of Van der Pol oscillator is studied and a method to generate Substitution Box (S-box) from it is proposed that have some good statistical properties such as PSNR, MSE, correlation, energy, Homogeneity, entropy and contrast.
Abstract: Chaos is the impromptu behavior exhibited by some nonlinear dynamical systems and has been applied extensively in secure communication over the last decade In this paper, the chaotic behavior of Van der Pol oscillator is studied and proposed a method to generate Substitution Box (S-box) from it The generated S-box have some good statistical properties such as PSNR, MSE, correlation, energy, Homogeneity, entropy and contrast The performance of proposed S-box is compared with other S-boxes like AES, gray, APA, Lui J and S8 to show the strength of anticipated technique
TL;DR: Methods for proving upper and lower bounds on infinite-time averages in deterministic dynamical systems and on stationary expectations in stochastic systems are described, using sum-of-squares polynomials to formulate sufficient conditions that can be checked by semidefinite programming.
Abstract: We describe methods for proving upper and lower bounds on infinite-time averages in deterministic dynamical systems and on stationary expectations in stochastic systems. The dynamics and the quantities to be bounded are assumed to be polynomial functions of the state variables. The methods are computer-assisted, using sum-of-squares polynomials to formulate sufficient conditions that can be checked by semidefinite programming. In the deterministic case, we seek tight bounds that apply to particular local attractors. An obstacle to proving such bounds is that they do not hold globally; they are generally violated by trajectories starting outside the local basin of attraction. We describe two closely related ways past this obstacle: one that requires knowing a subset of the basin of attraction, and another that considers the zero-noise limit of the corresponding stochastic system. The bounding methods are illustrated using the van der Pol oscillator. We bound deterministic averages on the attracting limit cycle above and below to within 1%, which requires a lower bound that does not hold for the unstable fixed point at the origin. We obtain similarly tight upper and lower bounds on stochastic expectations for a range of noise amplitudes. Limitations of our methods for certain types of deterministic systems are discussed, along with prospects for improvement.
TL;DR: In this article, a hierarchical terminal sliding surface is proposed and its finite time convergence to the origin is shown based on the fractional Lyapunov stability theorem and SMC theory.
Abstract: This study presents a novel fractional hierarchical terminal sliding mode control (SMC) scheme for finite-time stabilisation of non-autonomous fractional-order dynamical systems. It is assumed that the fractional-order system is disturbed by some model uncertainties and external noises. A novel fractional hierarchical terminal sliding surface is proposed and its finite time convergence to the origin is shown. Based on the fractional Lyapunov stability theorem and SMC theory, a robust sliding mode switching control law is derived to ensure the existence of the sliding motion in finite time. It is mathematically proved that the states of the error can reach the proposed hierarchical terminal sliding surface in finite time. The introduced method is applied for synchronisation of the fractional-order chaotic Arneodo and Genesio systems to show the usefulness of the method. Furthermore, two non-autonomous fractional-order systems, namely Van der Pol equation and gyro system, are successfully stabilised using the proposed strategy to confirm the theoretical results of this study.
TL;DR: In this paper, the stochastic response of a class of self-excited systems with Caputo-type fractional derivative driven by Gaussian white noise is considered, where the generalized harmonic function technique is applied to the fractional selfexcited system.
Abstract: The stochastic response of a class of self-excited systems with Caputo-type fractional derivative driven by Gaussian white noise is considered. Firstly, the generalized harmonic function technique is applied to the fractional self-excited systems. Based on this approach, the original fractional self-excited systems are reduced to equivalent stochastic systems without fractional derivative. Then, the analytical solutions of the equivalent stochastic systems are obtained by using the stochastic averaging method. Finally, in order to verify the theoretical results, the two most typical self-excited systems with fractional derivative, namely the fractional van der Pol oscillator and fractional Rayleigh oscillator, are discussed in detail. Comparing the analytical and numerical results, a very satisfactory agreement can be found. Meanwhile, the effects of the fractional order, the fractional coefficient, and the intensity of Gaussian white noise on the self-excited fractional systems are also discussed in detail.
TL;DR: In this article, the influence of time delay on noise-induced oscillations was investigated using the model of a generalized Van der Pol oscillator in the regime of subcritical Hopf bifurcation.
Abstract: Using the model of a generalized Van der Pol oscillator in the regime of subcritical Hopf bifurcation, we investigate the influence of time delay on noise-induced oscillations. It is shown that for appropriate choices of time delay, either suppression or enhancement of coherence resonance can be achieved. Analytical calculations are combined with numerical simulations and experiments on an electronic circuit.
TL;DR: The objective of this letter is to convey two essential principles of biological computing—synchronization and memory—in an electronic circuit with two van der Pol oscillators coupled via a memristive device, and the experimental results are underlined by theoretically considering a system of two coupled vdP equations.
Abstract: The objective of this paper is to explore the possibility to couple two van der Pol (vdP) oscillators via a resistance-capacitance (RC) network comprising a Ag-TiOx-Al memristive device. The coupling was mediated by connecting the gate terminals of two programmable unijunction transistors (PUTs) through the network. In the high resistance state (HRS) the memresistance was in the order of MOhm leading to two independent selfsustained oscillators characterized by the different frequencies f1 and f2 and no phase relation between the oscillations. After a few cycles and in dependency of the mediated pulse amplitude the memristive device switched to the low resistance state (LRS) and a frequency adaptation and phase locking was observed. The experimental results are underlined by theoretically considering a system of two coupled vdP equations. The presented neuromorphic circuitry conveys two essentials principle of interacting neuronal ensembles: synchronization and memory. The experiment may path the way to larger neuromorphic networks in which the coupling parameters can vary in time and strength and are realized by memristive devices.
TL;DR: In this paper, a reduced-order method was introduced to study the parametric excitations and lock-in of flexible hydrofoils caused by unsteady two-phase (cavitating) flow.
TL;DR: In this paper, a dynamic state feedback controller is applied to control Hopf bifurcations arising from a fractional-order Van Der Pol oscillator, and the system possesses the stability in a larger parameter range.
Abstract: In this paper, a dynamic state feedback is applied to control Hopf bifurcations arising from a fractional-order Van Der Pol oscillator. The degree parameter indicating the strength of the nonlinear damping is chosen as the bifurcation parameter. It is shown that in the absences of the dynamic state feedback controller, the fractional-order Van Der Pol oscillator loses the stability via the Hopf bifurcation early, and can maintain the stability only in a certain domain of the degree parameter. When applying the state feedback controller to the fractional-order Van Der Pol oscillator, the onset of the undesirable Hopf bifurcation is postponed. Thus, the stability domain is extended, and the system possesses the stability in a larger parameter range. Numerical simulations are given to justify the validity of the dynamic state feedback controller in bifurcation controls.
TL;DR: A stochastic dynamics of systems with hard excitement of auto-oscillations possessing a bistability mode with coexistence of the stable equilibrium and limit cycle and a detailed parametric description of the response of this model on the additive and multiplicative noise and corresponding Stochastic bifurcations are presented and discussed.
Abstract: We study a stochastic dynamics of systems with hard excitement of auto-oscillations possessing a bistability mode with coexistence of the stable equilibrium and limit cycle. A principal difference in the results of the impact of additive and parametric random disturbances is shown. For the stochastic van der Pol oscillator with increasing parametric noise, qualitative transformations of the probability density function form "crater"-"peak+crater"-"peak" are demonstrated by numerical simulation. An analytical investigation of such P bifurcations is carried out for the stochastic Hopf-like model with hard excitement of self-oscillations. A detailed parametric description of the response of this model on the additive and multiplicative noise and corresponding stochastic bifurcations are presented and discussed.
TL;DR: A modified Lindstedt–Poincare method (modified L–P method) is adopted to derive approximate periodic solutions for the forced and damped system and its frequency-response curves are obtained through numerical simulation.
TL;DR: This study experimentally and numerically verify that an extremely simple dynamical circuit captures the essence of the underlying mechanism causing MMOIBs, and it is observed that MMO IBs and chaos with distinctive waveforms in real circuit experiments are observed.
Abstract: Bifurcations of complex mixed-mode oscillations denoted as mixed-mode oscillation-incrementing bifurcations (MMOIBs) have frequently been observed in chemical experiments. In a previous study [K. Shimizu et al., Physica D 241, 1518 (2012)], we discovered an extremely simple dynamical circuit that exhibits MMOIBs. Our model was represented by a slow/fast Bonhoeffer-van der Pol circuit under weak periodic perturbation near a subcritical Andronov-Hopf bifurcation point. In this study, we experimentally and numerically verify that our dynamical circuit captures the essence of the underlying mechanism causing MMOIBs, and we observe MMOIBs and chaos with distinctive waveforms in real circuit experiments.
TL;DR: In this article, the Bonhoeffer-van der Pol oscillator with non-autonomous periodic perturbation was considered and the presence of mixed mode oscillations reported in that paper can be explained using geometric singular perturbations theory.
Abstract: Following the paper of Shimizu et al. (Phys Lett A 375:1566, 2011), we consider the Bonhoeffer-van der Pol oscillator with non-autonomous periodic perturbation. We show that the presence of mixed mode oscillations reported in that paper can be explained using the geometric singular perturbation theory. The considered model can be re-written as a four-dimensional (locally three-dimensional) autonomous system, which under certain conditions has a folded saddle-node singularity and additionally can be treated as a three time scale one.
TL;DR: The results establish that an appropriately tuned stochastic coupled nonlinear oscillator network such as the Duffing-van der Pol system could provide a useful framework for modeling and analysis of the EEG signal.
TL;DR: In this paper, the nonlinear dynamics of two coupled glow discharge plasma sources were investigated and a variety of nonlinear phenomena including frequency synchronization and frequency pulling were observed as the coupling strength is varied.
Abstract: Experimental results on the nonlinear dynamics of two coupled glow discharge plasma sources are presented. A variety of nonlinear phenomena including frequency synchronization and frequency pulling are observed as the coupling strength is varied. Numerical solutions of a model representation of the experiment consisting of two coupled asymmetric Van der Pol type equations are found to be in good agreement with the observed results.
TL;DR: In this paper, the authors considered a van der Pol equation with state-dependent delayed feedback and constructed solutions near equilibria using perturbation methods to determine the sub/supercriticality of the bifurcation and hence their stability.
Abstract: In this paper, we consider a classical van der Pol equation with state-dependent delayed feedback. Firstly, solutions near equilibria are constructed using perturbation methods to determine the sub/supercriticality of the bifurcation and hence their stability. Then, we choose a few examples of state-dependant delay to test our analytical results by comparing them to numerical continuation.
TL;DR: A neural identifier is proposed in order to obtain a mathematical model for the unknown discrete-time nonlinear systems, then a novel second order sliding mode controller is proposed and both, the neural identifier and the controller are optimized using bacterial foraging algorithm.
TL;DR: The main finding of the work is that statistically significant unique models represent the EC and EO conditions for both CTL and AD subjects and it is shown that the inclusion of sample entropy in the optimization process, to match the complexity of the EEG signal, enhances the stochastic non-linear oscillator model performance.
Abstract: In this article, the Electroencephalography (EEG) signal of the human brain is modeled as the output of stochastic nonlinear coupled oscillator networks. It is shown that EEG signals recorded under different brain states in healthy as well as Alzheimer's disease (AD) patients may be understood as distinct, statistically significant realizations of the model. EEG signals recorded during resting eyes-open (EO) and eyes-closed (EC) resting conditions in a pilot study with AD patients and age-matched healthy control subjects (CTL) are employed. An optimization scheme is then utilized to match the output of the stochastic Duffing - van der Pol double oscillator network with EEG signals recorded during each condition for AD and CTL subjects by selecting the model physical parameters and noise intensity. The selected signal characteristics are power spectral densities in major brain frequency bands Shannon and sample entropies. These measure allow matching of linear time varying frequency content as well as nonlinear signal information content and complexity. The main findings of the work is that statistically significant unique models represent the EC and EO conditions for both CTL and AD subjects. However, it is also shown that the inclusion of sample entropy in the optimization process, to match the complexity of the EEG signal, enhances the stochastic nonlinear oscillator model performance.
TL;DR: In this article, the existence of chimera states in an assembly of identical nonlinear oscillators that are globally linked to each other in a simple planar cross-coupled form was reported.
Abstract: We report the existence of chimera states in an assembly of identical nonlinear oscillators that are globally linked to each other in a simple planar cross-coupled form. The rotational symmetry breaking of the coupling term appears to be responsible for the emergence of these collective states that display a characteristic coexistence of coherent and incoherent behaviour. The finding, observed in both a collection of van der Pol oscillators and chaotic Rossler oscillators, further simplifies the existence criterion for chimeras, thereby broadens the range of their applicability to real-world situations.
TL;DR: Various diagnostics reveal a close parallelism between classical regular as well as chaotic dynamics and that obtained from the Bohmian mechanics.
Abstract: The orbital free density functional theory and the single density equation approach are formally equivalent. An orbital free density based quantum dynamical strategy is used to study the quantum-classical correspondence in both weakly and strongly coupled van der Pol and Duffing oscillators in the presence of an external electric field in one dimension. The resulting quantum hydrodynamic equations of motion are solved through an implicit Euler type real space method involving a moving weighted least square technique. The Lagrangian framework used here allows the numerical grid points to follow the wave packet trajectory. The associated classical equations of motion are solved using a sixth order Runge–Kutta method and the Ehrenfest dynamics is followed through the solution of the time dependent Schrodinger equation using a time dependent Fourier Grid Hamiltonian technique. Various diagnostics reveal a close parallelism between classical regular as well as chaotic dynamics and that obtained from the Bohmian mechanics.