Modified Van Der Pol Oscillator Research Articles

SELF-ORGANIZATION IN NONADDITIVE SYSTEMS WITH EXTERNAL NOISE

A nonadditive generalization of Klimontovich's S-theorem [Bagci, 2008] has recently been obtained by employing Tsallis entropy. This general version allows one to study physical systems whose stationary distributions are of the inverse power law in contrast to the original S-theorem, which only allows exponential stationary distributions. The nonadditive S-theorem has been applied to the modified Van der Pol oscillator with inverse power law stationary distribution. By using nonadditive S-theorem, it is shown that the entropy decreases as the system is driven out of equilibrium, indicating self-organization in the system. The allowed values of the nonadditivity index q are found to be confined to the regime (0.5,1].

Concealed conduction effects in the atrium

A one-dimensional (1-D) model of the atrium together with the sinoatrial (SA) and atrioventricular (AV) nodes is presented in this article. The two nodes are each modeled by 15-element, diffusively coupled, modified van der Pol oscillator chains, while the atrium tissue is represented by a 90-element chain of diffusively coupled FitzHugh-Nagumo (FHN) equations. The modified van der Pol oscillators are able to reproduce physiologically important properties, such as the refraction period, phase-sensitivity properties, and modes of change of the action potential frequency. The activity of both branches of the autonomous nervous system may be introduced into the model in a simplified way. The model enables the study of the effect of the magnitude of the action potential conduction rate in the nodes on interspike intervals (ISIs; equivalent of RR intervals) and explains the occurrence of RR-interval alternans in certain patients. The effect of breathing modulation of heart rate and of a single deep breath can also be modeled. Finally, concealed conduction effects in the atrium are studied, yielding results comparable with recorded heart rate variability data.

Local Robustness of Hopf Bifurcation Stabilization

Local robust analysis via <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> gain method is presented for a class of Hopf bifurcation stabilizing controllers. In particular, we first construct a family of Lyapunov functions for the corresponding critical system, then derive a sufficient condition to compute the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> gain by solving the Hamilton-Jacobi-Bellman (HJB) inequalities. Local robust analysis can be conducted through computing the local <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> gain achieved by the stabilizing controllers at the critical situation. The theoretical results obtained in this brief provide useful guidance for selecting a robust controller from a given class of stabilizing controllers under Hopf bifurcation. As an example, application to a modified Van der Pol oscillator is discussed in details.

An analysis of heart rhythm dynamics using a three-coupled oscillator model

Rhythmic phenomena represent one of the most striking manifestations of the dynamic behavior in biological systems. Understanding the mechanisms responsible for biological rhythms is crucial for the comprehension of the dynamics of life. Natural rhythms could be either regular or irregular over time and space. Each kind of dynamical behavior may be related to both normal and pathological physiological functioning. The cardiac conducting system can be treated as a network of self-excitatory elements and, since these elements exhibit oscillatory behavior, they can be modeled as nonlinear oscillators. This paper proposes a mathematical model to describe heart rhythms considering three modified Van der Pol oscillators connected with time delay couplings. Therefore, the heart dynamics is represented by a system of differential difference equations. Numerical simulations are carried out presenting qualitative agreement with the general heart rhythm behavior. Normal and pathological rhythms represented by the ECG signals are reproduced. Pathological rhythms are generated by either the coupling alterations that represents communications aspects in the heart electric system or forcing excitation representing external pacemaker excitation.

Multisensor Input for CPG-Based Sensory---Motor Coordination

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper describes a method for providing in real time a reliable synchronization signal for cyclical motions such as steady-state walking. The approach consists in estimating online a phase variable on the basis of several implicit central pattern generator associated with a set of sensors. These sensors can be of any kind, provided their output strongly reflects the timed motion of a link. They can be, for example, spatial position or orientation sensors, or foot sole pressure sensors. The principle of the method is to use their outputs as inputs to nonlinear observers of modified Van der Pol oscillators that provide us with several independent estimations of the overall phase of the system. These estimations are then combined within a dynamical filter constituted of a Hopf oscillator. The resulting phase is a reliable indexing of the cyclic behavior of the system, which can finally be used as input to low-level controllers of a robot. Some results illustrate the efficiency of the approach, which can be used to control robots. </para>

The role of inter-generation memory in diel phytoplankton division patterns

Phytoplankton populations are strongly influenced by light and therefore exhibit diel rhythms in many ways, including biomass growth and division. Cells typically divide at about the same time of day, but when the average generation time is longer than 24 h, only a fraction of the population divides each day. This means that two given cells born at the same time and exposed to the same environment can have different generation times (e.g. 24, 48 h). Existing explanations attribute this variability to stochastic internal variability, and several cell cycle models have been proposed. This paper proposes that a deterministic inter-generation memory mechanism, specifically biomass inheritance, is the source for the generation time variability. To explore this hypothesis, the continuum model (an alternative to the cell cycle model that accounts for inter-generation memory) was modified for phytoplankton. Two models with the traditional stochastic and the new deterministic mechanisms were constructed and implemented in an agent-based simulation model. An autonomous endogenous circadian clock is included using a modified van der Pol oscillator. The two models were tested against cell density data from light/dark experiments with average generation times ≈24 h and >24 h from the literature. The applications demonstrate that the deterministic model can reproduce the data without introducing stochastic internal variability, and that it is applicable to a wider range of conditions. Under realistic field conditions, with a light intensity time series from a Lagrangian particle tracking model, the two approaches produce significantly different results. Additional model-data comparisons for cell densities from free-running experiments, cell biomass, and DNA content from direct measurement and flow cytometric analysis are presented. The results suggest that the inter-generation memory mechanism is responsible for the generation time variability underlying diel phytoplankton division patterns and that the difference is of practical significance.

Periodically-forced finite networks of heterogeneous globally-coupled oscillators: A low-dimensional approach

We study a network of 500 globally-coupled modified van der Pol oscillators. The value of a parameter associated with each oscillator is drawn from a normal distribution, giving a heterogeneous network. For strong enough coupling the oscillators all have the same period, and we consider periodic forcing of the network when it is in this state. By exploiting the correlations that quickly develop between the state of an oscillator and the value of its parameter we obtain an approximate low-dimensional description of the system in terms of the first few coefficients in a polynomial chaos expansion. Standard bifurcation analysis can then be performed on the low-dimensional system which results from this computational coarse-graining, and the results obtained from this predict very well the behaviour of the high-dimensional system for any set of realisations of the random parameter. Situations in which the method begins to fail are also discussed.

Second‐order sliding mode control of underactuated mechanical systems II: Orbital stabilization of an inverted pendulum with application to swing up/balancing control

AbstractOrbital stabilization of an underactuated cart–pendulum system is under study. The quasihomogeneous control synthesis is utilized to design a second‐order sliding mode controller that drives the actuated cart to a periodic reference orbit in finite time, while the non‐actuated pendulum produces bounded oscillations. A modified Van der Pol oscillator is introduced into the synthesis as an asymptotic generator of the periodic motion. The resulting closed‐loop system is capable of moving from one orbit to another by simply changing the parameters of the Van der Pol modification. Performance issues of the proposed synthesis are illustrated in numerical and experimental studies of the swing up/balancing control problem of moving a pendulum, located on an actuated cart, from its stable downward position to the unstable inverted position and stabilizing it about the vertical. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Golden mean relevance for chaos inhibition in a system of two coupled modified van der Pol oscillators

In this work, we present a novel evidence of the importance of the golden mean criticality of a system of oscillators in agreement with El Naschie’s E-infinity theory. We focus on chaos inhibition in a system of two coupled modified van der Pol oscillators. Depending on the coupling between the two oscillators, the system shows chaotic behavior for different ranges of the coupling parameter. Chaos suppression, as a transition from irregular behavior to a periodical one, is induced by perturbing the system with a harmonic signal with amplitude considerably lower than the value which causes entrainment. The frequency of the perturbation is related to the main frequencies in the spectrum of the freely running system (without perturbation) by the golden mean. We demonstrate that this effect is also obtained for a perturbation with frequency such that the ratio of half the frequency of the first main component in the freely running chaotic spectrum over the frequency of the perturbation is very close (five digits coincidence) to the golden mean. This result is shown to hold for arbitrary values of the coupling parameter in the various ranges of chaotic dynamics of the free running system.

Self-excited systems: Analytical determination of limit cycles

Abstract The determination of amplitude and period of limit cycles is a crucial question in non-linear mechanics. Recently, a van der Pol oscillator containing a periodic potential has been considered as a suitable self-excited system in tribological studies. Analytical expressions for the amplitude and the period of limit cycles for such modified van der Pol oscillator are calculated making use of the He’s variational method and the Krylov–Bogoliubov–Mitropolsky (KBM) method. The analytical results are then compared to numerical correspondent results, showing the goodness of the two methods in the range of application.

Quasihomogeneity Approach to the Pendubot Stabilization around Periodic Orbits

Orbital stabilization of a simple underactuted manipulator, namely, two-link PENDUlum roBOT is under study. Since underactuated systems can not be locally stabilized by means of smooth feedback, the solution to the stabilization problem is sought within switched control methods. The quasihomogeneity-based synthesis is applied to design a switched controller that drives the PENDUBOT to its zero dynamics in finite time and maintains it there in sliding mode. The constructed controller is such that the pendubot zero dynamics is generated by a modified Van der Pol oscillator, being viewed as the reference model. The proposed Van der Pol modification aims to enlarge the region of attraction of the zero dynamics while also exhibiting a desired stable periodic behavior. Performance issues of the controller constructed are illustrated in a simulation study.

Nonlinear instabilities and nonstationarity in human heart-rate variability

The human cardiovascular system exhibits complex and interesting dynamics, including heart-rate fluctuations. The heart rate fluctuates irregularly - a phenomenon called heart-rate variability. Here, we present a short review of our experience with the analysis of heart-rate variability. Initially, we focused on the assessment of the risk of sudden cardiac death due to cardiac arrest. During this research, we became aware of intermittency in heart-rate variability. We identify the intermittency as type I and describe how simple models helped us understand the differences between our results and textbook properties of this phenomenon. We also mention the use of modified van der Pol oscillators for the modeling of heart-rate variability. Finally, we review our latest research on the modeling of the heart tissue's properties using the true shape of the ventricles reconstructed from clinical electro-physiological measurement data.

Modeling cardiac pacemakers with relaxation oscillators

A modified van der Pol oscillator model was designed in order to reproduce the time series of the action potential generated by a natural pacemaker of the heart (i.e., the SA or the AV node). The main motivation was that the models published up to now were not altogether adequate for research on the heart. Based on either the classical van der Pol oscillator or other nonlinear oscillators, these models were interesting rather because of the physical phenomena that could be obtained (chaos and synchronization). However, they were unable to simulate many important physiological features of true physiological action potentials. We based our research on the experience of other groups which modeled neuronal oscillators. There complex nonlinear oscillators were used whose most important feature was a certain topology of the phase space. In our case, we modified the phase space of the classical van der Pol oscillator by adding two fixed points: a saddle and a node. In addition, a damping term asymmetric with respect to the voltage was introduced. Introduction of these new features into the van der Pol oscillator allowed to change the firing frequency of the pacemaker node without changing the length of the refractory period – an important physiological detail. We also show different ways of changing the pacemaker rhythm. A comparison of the properties of the signal obtained from our model with the features of the action potentials measured by other groups is made.

Ventilatory support: a dynamical systems approach.

Misunderstanding of the dynamical behavior of the ventilatory system, especially under assisted ventilation, may explain the problems encountered in ventilatory support monitoring. Proportional assist ventilation (PAV) that theoretically gives a breath by breath assistance presents instability with high levels of assistance. We have constructed a mathematical model of interactions between three objects: the central respiratory pattern generator modelled by a modified Van der Pol oscillator, the mechanical respiratory system which is the passive part of the system and a controlled ventilator that follows its own law. The dynamical study of our model shows the existence of two crucial behaviors, i.e. oscillations and damping, depending on only two parameters, namely the time constant of the mechanical respiratory system and a cumulative interaction index. The same result is observed in simulations of spontaneous breathing as well as of PAV. In this last case, increasing assistance leads first to an increase of the tidal volume (VT), a further increase in assistance inducing a decrease in VT, ending in damping of the whole system to an attractive fixed point. We conclude that instabilities observed in PAV may be explained by the different possible dynamical behaviors of the system rather than changes in mechanical characteristics of the respiratory system.

On the complex dynamics of intracellular ganglion cell light responses in the cat retina.

We recorded intracellular responses from cat retinal ganglion cells to sinusoidal flickering lights, and compared the response dynamics with a theoretical model based on coupled nonlinear oscillators. Flicker responses for several different spot sizes were separated in a "smooth" generator (G) potential and corresponding spike trains. We have previously shown that the G-potential reveals complex, stimulus-dependent, oscillatory behavior in response to sinusoidally flickering lights. Such behavior could be simulated by a modified van der Pol oscillator. In this paper, we extend the model to account for spike generation as well, by including extended Hodgkin-Huxley equations describing local membrane properties. We quantified spike responses by several parameters describing the mean and standard deviation of spike burst duration, timing (phase shift) of bursts, and the number of spikes in a burst. The dependence of these response parameters on stimulus frequency and spot size could be reproduced in great detail by coupling the van der Pol oscillator and Hodgkin-Huxley equations. The model mimics many experimentally observed response patterns, including non-phase-locked irregular oscillations. Our findings suggest that the information in the ganglion cell spike train reflects both intraretinal processing, simulated by the van der Pol oscillator, and local membrane properties described by Hodgkin-Huxley equations. The interplay between these complex processes can be simulated by changing the coupling coefficients between the two oscillators. Our simulations therefore show that irregularities in spike trains, which normally are considered to be noise, may be interpreted as complex oscillations that might carry information.

Closed-loop suppression of chaos in nonlinear driven oscillators

This paper discusses the suppression of chaos in nonlinear driven oscillators via the addition of a periodic perturbation. Given a system originally undergoing chaotic motions, it is desired that such a system be driven to some periodic orbit. This can be achieved by the addition of a weak periodic signal to the oscillator input. This is usually accomplished in open loop, but this procedure presents some difficulties which are discussed in the paper. To ensure that this is attained despite uncertainties and possible disturbances on the system, a procedure is suggested to perform control in closed loop. In addition, it is illustrated how a model, estimated from input/output data, can be used in the design. Numerical examples which use the Duffing-Ueda and modified van der Pol oscillators are included to illustrate some of the properties of the new approach.

RETRIEVING DYNAMICAL INVARIANTS FROM CHAOTIC DATA USING NARMAX MODELS

This paper is concerned with the estimation of dynamical invariants from relatively short and possibly noisy sets of chaotic data. In order to overcome the difficulties associated with the size and quality of the data records, a two-step procedure is investigated. Firstly NARMAX models are fitted to the data. Secondly, such models are used to generate longer and cleaner time sequences from which dynamical invariants such as Lyapunov exponents, correlation dimension, the geometry of the attractors, Poincaré maps and bifurcation diagrams can be estimated with relative ease. An additional advantage of this procedure is that because the models are global and have a simple structure, such models are amenable for analysis. It is shown that the location and stability of the fixed points of the original systems can be analytically recovered from the identified models. A number of examples are included which use the logistic and Hénon maps, Duffing and modified van der Pol oscillators, the Mackey-Glass delay system, Chua’s circuit, the Lorenz and Rössler attractors. The identified models of these systems are provided including discrete multivariable models for Chua’s double scroll, Lorenz and Rössler attractors which are used to reconstruct the trajectories in a three-dimensional state space.

Reaction-diffusion processes in a one-dimensional array of active non-linear circuits

A linear array of coupled modified van der Pol oscillators has been studied both experimentally and numerically. A fixed width pulse was used to study the velocity of pulse-like travelling wave propagation in the chain and its dependence on the system parameters. We have shown that for a sufficiently large array the velocity tends to a limiting value that is independent of the array's length. The velocity monotonically decreases with the interval between pulses and it is inversely proportional to the square root of the coupling resistance. These results can be understood by using an analogy with the propagating waves in a high non-linear reaction-diffusion system, and specifically in an active excitable media.

The origins of chaos in a modified Van der Pol oscillator

An experimental and theoretical investigation of a modified Van der Pol oscillator has led to the discovery of degenerate bifurcations which control the global dynamics of the system. A two-dimensional parameter space for the oscillator is studied and the measured loci of critical points, where a qualitative change of behaviour is observed, are given in detail. It is shown explicitly, by analysis of the experimental data, that chaos arises through the Silnikov mechanism.

Experimental estimation of parameters used in a limit‐cycle‐oscillator model of spontaneous otoacoustic emissions: Effects of aspirin administration on time constants for suppression

Limit‐cycle‐oscillator models of spontaneous otoacoustic emissions have been used to provide qualitative descriptions of the interactions of spontaneous emissions with external tones and their response to aspirin administration. The parameters of several classes of such models are estimated from data on the relaxation dynamics of spontaneous emissions, associated with the onset of, and recovery from, suppression by external tones. Aspirin administration is used to control the emission levels, with the relaxation time being observed to increase with decreasing emission level. The models considered include a modified Van der Pol oscillator, with the possibility of linear passive damping in the limit of large displacements, and an oscillator with nonlinear time‐delayed elasticity. [Work supported by NIDCD.]