Oscillations In Dynamical Systems Research Articles

An Innovative Approach in Inspecting a Damped Mathieu Cubic–Quintic Duffing Oscillator

Abstract Purpose The objective of the present study is to analyze a damped Mathieu–cubic quintic Duffing oscillator as a parametric nonlinear oscillatory dynamical system. This equation has multiple applications in diverse fields, including optics, quantum physics, and general relativity. There are multiple concerns related to periodic motion and the analysis of boundary-value problems with elliptic symmetries. The current effort aims to determine the frequency amplitude of parametric nonlinear issues. Method The non-perturbative approach (NPA) is employed to transform the nonlinear ordinary differential equation (ODE) into a linear equation. The derivation of the approximate solutions is achieved without relying on typical perturbation approaches, separate from the series expansion. Hence, the objective of this study is to depart from traditional perturbation methods and acquire approximated solutions for minor amplitude parametric components without imposing any limitations. Furthermore, the technique is extended to ascertain optimal solutions for the nonlinear large amplitude of fluctuation. Results The current approach allows for rapid estimation of the frequency-amplitude relationship in order to attain successive approximations of the solutions for parametric nonlinear fluctuations. A validation is obtained for the derived parametric equation, demonstrating a high level of agreement with the original equation. An analysis of stability behavior is conducted in multiple scenarios. In addition, the Floquet theory is used to examine the transition curves. Conclusion The current technique is characterized by its clear principles, making it practical, user-friendly, and capable of achieving exceptionally high numerical precision. The current approach is highly beneficial for addressing nonlinear parametric problems due to its ability to minimize algebraic complexity during implementation.

A direct method approach for data-driven inference of high accuracy adaptive phase-isostable reduced order models

Phase–amplitude reduction techniques have shown great promise for identifying analytically tractable reduced order models in applications involving strongly perturbed and strongly coupled oscillatory dynamical systems. However, efficient methods for inference of these reduced order models from data are still needed. In this work, a data-driven strategy for inference of the necessary terms comprising an adaptive phase-isostable reduced order model is proposed. This strategy requires no more data than the well-established direct method used for obtaining standard phase reduced models, i.e., the application of a series of pulse inputs and the subsequent examination of the relaxation to the underlying limit cycle. Illustrative examples are provided for a collection of numerical models where the resulting adaptive phase-isostable reduced order equations are substantially more accurate than models obtained using standard phase reduction.

Oscillations in Dynamic Systems with an Entropy Operator

This paper considers dynamic systems with an entropy operator described by a perturbed constrained optimization problem. Oscillatory processes are studied for periodic systems with the following property: the entire system has the same period as the process generated by its linear part. Existence and uniqueness conditions are established for such oscillatory processes, and a method is developed to determine their form and parameters. Also, the general case of noncoincident periods is analyzed, and a method is proposed to determine the form, parameters, and the period of such oscillations. Almost periodic processes are investigated, and existence and uniqueness conditions are proved for them as well.

Extracting Phase Coupling Functions between Collectively Oscillating Networks from Time-Series Data

Many real-world systems are often regarded as weakly coupled limit-cycle oscillators, in which each oscillator corresponds to a dynamical system with many degrees of freedom that have collective oscillations. One of the most practical methods for investigating the synchronization properties of such a rhythmic system is to statistically extract phase coupling functions between limit-cycle oscillators directly from observed time-series data. In Particular, using a method that combines phase reduction theory and Bayesian inference, the phase coupling functions can be extracted from the time-series data of even just one variable in each oscillatory dynamical system with many degrees of freedom. However, it remains unclear how the choice of the observed variables affects the statistical inference for the phase coupling functions. In this study, we examine the influence of observed variable types on the extraction of phase coupling functions using some typical dynamical elements under various conditions. We demonstrate that our method can consistently extract the macroscopic phase coupling functions between two phases representing collective oscillations in a fully locked state, regardless of the observed variable types; for example, even using one variable of any element in one system and the mean-field value over all the elements in another system. We also study the case of globally coupled phase oscillators in a partially locked state. Our results reveal directional asymmetry in the robustness of extracting the macroscopic phase coupling function between two networks. For instance, when an asynchronous oscillator in network $A$ and the macroscopic collective oscillation of network $B$ is observed, the macroscopic phase coupling function from network $A$ to network $B$ can be extracted more robustly than in the opposite direction.

New bifurcation diagrams based on hypothesis testing: Pseudo-periodic surrogates with correlation dimension as discriminating statistic

This paper discusses a possibility of using a new type of bifurcation diagrams that can be computed for time series being either solutions of nonlinear oscillatory dynamical systems or measured and recorded in a laboratory when no mathematical model is known. The obtained or measured noisy time-series in mechanical (and in other areas of engineering) dynamical systems are analyzed in a certain range of a slowly varying parameter and the added noise is uncorrelated with the dynamics of the system. In such a scenario with noise, typical well-known bifurcation diagrams cannot be used, so a new method of computing bifurcation diagrams via hypothesis testing from statistics is proposed - a method based on pseudo-periodic surrogates and correlation dimension as the discriminating statistics. An illustrative example based on the chaotic Rössler system is provided to show promising results of this new approach.

Spiral patterns and numerical bifurcation analysis in a three-component Brusselator model for chemical reactions

Pattern formation is a natural phenomenon that can be modeled by reaction–diffusion systems. Regular rotating spiral pattern formation and the instability of the spiral dynamics are very closely related to morphogenesis processes, cardiac arrhythmias, and chemical reactions. Since regular rotating spirals can emit periodic traveling waves from their core, hence the analysis of periodic traveling wave solutions of an oscillatory dynamical system is crucial to understand the instability of spiral patterns. To investigate the spiral patterns for chemical reactions, we consider a three-component Brusselator model. We first examine the local behavior of solutions of the model. Then we analyze the occurrence of periodic traveling wave solutions in a two-dimensional parameter plane. Next, we explore the stability outcome of periodic traveling wave solutions. It is observed that periodic traveling wave solutions change their stability, which is the Eckhaus type. The results are justified by a direct numerical simulation in one and two space dimensions. The regular and oscillatory periodic traveling wave solutions are revealed in the one-dimensional space dimension of the model by considering an Eckhaus bifurcation point. Moreover, we illustrate regular rotating spiral waves and the phenomenon for far-field breakup from the spiral waves in the two-dimensional space.

Generalization of the Concept of Bandwidth

In the sciences and engineering, there is no generally accepted definition of bandwidth beyond those used to characterize low-loss linear time-invariant systems of low order operating at steady-state. In fact, the concept of bandwidth is often subject to interpretation depending upon context and the requirements of a specific community. The focus of this work is to formulate this concept for a general class of passive oscillatory dynamical systems, including but not limited to mechanical, structural, acoustic, electrical, and optical. Typically, the (linearized) bandwidth of these systems is determined by the half-power (-3 dB) method, and the result is often referred to as “half-power bandwidth.” The fundamental assumption underlying this definition is that the system performance degrades once its power decreases by 50%; moreover, there are restrictive conditions, rarely met, that render a system amenable to the use of this approach, such as linearity, low-dimensionality, low-loss, and stationary output. Here the concept of root mean square (RMS) bandwidth is considered, justified by the Fourier uncertainty principle, to generalize the definition of bandwidth to encompass linear/nonlinear, single/multi-mode, low/high loss and time-varying/invariant oscillating systems. By tying the bandwidth of an oscillatory dynamical system directly to its dissipative capacity, one can formulate a definition based solely on its transient energy evolution, effectively circumventing the previous restrictions. Further, applications are given that highlight the limitations of the traditional half-power bandwidth; these include a Duffing oscillator with hardening nonlinearity, and a bi-stable, geometrically nonlinear oscillator with tunable hardening or softening nonlinearity. The resulting energy-dependent bandwidth computations are compatible with the nonlinear dynamics of these systems, since at low energies they recover the (linearized) half-power bandwidth, whereas at high energies they accurately capture the nonlinear physics. Moreover, the bandwidth computation is directly tied to nonlinear harmonic generation in the transient dynamics, so that the contributions to the bandwidth of the individual harmonics and of inter-harmonic targeted energy transfers can be directly quantified and studied. The new bandwidth definition proposed in this work has broad applicability and can be regarded as a generalization of the traditional linear half-power bandwidth which is used widely in the sciences and engineering.

Delay-induced uncertainty for a paradigmatic glucose-insulin model.

Medical practice in the intensive care unit is based on the assumption that physiological systems such as the human glucose-insulin system are predictable. We demonstrate that delay within the glucose-insulin system can induce sustained temporal chaos, rendering the system unpredictable. Specifically, we exhibit such chaos for the ultradian glucose-insulin model. This well-validated, finite-dimensional model represents feedback delay as a three-stage filter. Using the theory of rank one maps from smooth dynamical systems, we precisely explain the nature of the resulting delay-induced uncertainty (DIU). We develop a framework one may use to diagnose DIU in a general oscillatory dynamical system. For infinite-dimensional delay systems, no analog of the theory of rank one maps exists. Nevertheless, we show that the geometric principles encoded in our DIU framework apply to such systems by exhibiting sustained temporal chaos for a linear shear flow. Our results are potentially broadly applicable because delay is ubiquitous throughout mathematical physiology.

Analysis of input-induced oscillations using the isostable coordinate framework.

Many reduced order modeling techniques for oscillatory dynamical systems are only applicable when the underlying system admits a stable periodic orbit in the absence of input. By contrast, very few reduction frameworks can be applied when the oscillations themselves are induced by coupling or other exogenous inputs. In this work, the behavior of such input-induced oscillations is considered. By leveraging the isostable coordinate framework, a high-accuracy reduced set of equations can be identified and used to predict coupling-induced bifurcations that precipitate stable oscillations. Subsequent analysis is performed to predict the steady state phase-locking relationships. Input-induced oscillations are considered for two classes of coupled dynamical systems. For the first, stable fixed points of systems with parameters near Hopf bifurcations are considered so that the salient dynamical features can be captured using an asymptotic expansion of the isostable coordinate dynamics. For the second, an adaptive phase-amplitude reduction framework is used to analyze input-induced oscillations that emerge in excitable systems. Examples with relevance to circadian and neural physiology are provided that highlight the utility of the proposed techniques.

Evolution of populations with strategy-dependent time delays.

We study the effects of strategy-dependent time delays on the equilibria of evolving populations. It is well known that time delays may cause oscillations in dynamical systems. Here we report a novel behavior. We show that microscopic models of evolutionary games with strategy-dependent time delays lead to a new type of replicator dynamics. It describes the time evolution of fractions of the population playing given strategies and the size of the population. Unlike in all previous models, the stationary states of such dynamics depend continuously on time delays. We show that in games with an interior stationary state (a globally asymptotically stable equilibrium in the standard replicator dynamics), at certain time delays it may disappear or there may appear another interior stationary state. In the Prisoner's Dilemma game, for time delays of cooperation smaller than time delays of defection, there appears an unstable interior equilibrium, and therefore for some initial conditions the population converges to the homogeneous state with just cooperators.

Optimal Control of Oscillation Timing and Entrainment Using Large Magnitude Inputs: An Adaptive Phase-Amplitude-Coordinate-Based Approach

Given the high dimensionality and underlying complexity of many oscillatory dynamical systems, phase reduction is often an imperative first step in control applications where oscillation timing and...

Optogenetic Tuning Reveals Rho Amplification-Dependent Dynamics of a Cell Contraction Signal Network

SummaryLocal cell contraction pulses play important roles in tissue and cell morphogenesis. Here, we improve a chemo-optogenetic approach and apply it to investigate the signal network that generates these pulses. We use these measurements to derive and parameterize a system of ordinary differential equations describing temporal signal network dynamics. Bifurcation analysis and numerical simulations predict a strong dependence of oscillatory system dynamics on the concentration of GEF-H1, an Lbc-type RhoGEF, which mediates the positive feedback amplification of Rho activity. This prediction is confirmed experimentally via optogenetic tuning of the effective GEF-H1 concentration in individual living cells. Numerical simulations show that pulse amplitude is most sensitive to external inputs into the myosin component at low GEF-H1 concentrations and that the spatial pulse width is dependent on GEF-H1 diffusion. Our study offers a theoretical framework to explain the emergence of local cell contraction pulses and their modulation by biochemical and mechanical signals.

Exponentially Fitted Cubature Kalman Filter With Application to Oscillatory Dynamical Systems

This paper proposes a new nonlinear filtering technique under the Gaussian filtering approach, which is based on the numerical approximation of intractable integrals. The proposition is in the category of spherical-radial rule based Gaussian filtering (dominated by cubature Kalman filter), which is most commonly used in practical applications. The existing Gaussian filters with spherical-radial rule are accurate only for the systems modelled with polynomials of a certain order and suffer from poor estimation accuracy for the oscillatory systems. The proposed method, however, provides an efficient approach for estimation and filtering in oscillatory environment. It introduces an exponentially-fitted spherical-radial rule of numerical approximation, which is accurate for oscillatory functions. The exponentially-fitted spherical-radial rule is composed of a third-degree spherical-cubature rule and an exponentially-fitted Gauss-Laguerre quadrature rule. The proposed filter is named as exponentially-fitted cubature Kalman filter (ECKF). The estimation accuracy of the ECKF is analyzed for nonlinear filtering problems related to the Duffing and Coulomb oscillators in terms of root mean square error (RMSE). The RMSE analysis concludes an improved estimation accuracy for the proposed ECKF compared to the existing Gaussian filters.

Phase-amplitude reduction far beyond the weakly perturbed paradigm.

While phase reduction is a well-established technique for the analysis of perturbed limit cycle oscillators, practical application requires perturbations to be sufficiently weak thereby limiting its utility in many situations. Here, a general strategy is developed for constructing a set of phase-amplitude reduced equations that is valid to arbitrary orders of accuracy in the amplitude coordinates. This reduction framework can be used to investigate the behavior of oscillatory dynamical systems far beyond the weakly perturbed paradigm. Additionally, a patchwork phase-amplitude reduction method is suggested that is useful when exceedingly large magnitude perturbations are considered. This patchwork method incorporates the high-accuracy phase-amplitude reductions of multiple nearby periodic orbits that result from modifications to nominal parameters. The proposed method of high-accuracy phase-amplitude reduction can be readily implemented numerically and examples are provided where reductions are computed up to fourteenth order accuracy.

A data-driven phase and isostable reduced modeling framework for oscillatory dynamical systems.

Phase-amplitude reduction is of growing interest as a strategy for the reduction and analysis of oscillatory dynamical systems. Augmentation of the widely studied phase reduction with amplitude coordinates can be used to characterize transient behavior in directions transverse to a limit cycle to give a richer description of the dynamical behavior. Various definitions for amplitude coordinates have been suggested, but none are particularly well suited for implementation in experimental systems where output recordings are readily available but the underlying equations are typically unknown. In this work, a reduction framework is developed for inferring a phase-amplitude reduced model using only the observed model output from an arbitrarily high-dimensional system. This framework employs a proper orthogonal reduction strategy to identify important features of the transient decay of solutions to the limit cycle. These features are explicitly related to previously developed phase and isostable coordinates and used to define so-called data-driven phase and isostable coordinates that are valid in the entire basin of attraction of a limit cycle. The utility of this reduction strategy is illustrated in examples related to neural physiology and is used to implement an optimal control strategy that would otherwise be computationally intractable. The proposed data-driven phase and isostable coordinate system and associated reduced modeling framework represent a useful tool for the study of nonlinear dynamical systems in situations where the underlying dynamical equations are unknown and in particularly high-dimensional or complicated numerical systems for which standard phase-amplitude reduction techniques are not computationally feasible.

Szegő Quadrature Kalman Filter for Oscillatory Systems

Filtering problems with oscillatory system dynamics commonly appear in real-life. However, the existing Gaussian filters, like the unscented Kalman filter (UKF), cubature Kalman filter CKF, Gauss-Hermite filter (GHF) and cubature quadrature Kalman filter (CQKF), are accurate for the nonlinear systems with a particular order of polynomials only. This manuscript introduces a new Gaussian filter, which is accurate for oscillatory systems with 2π-period of oscillation. The proposed method is named as Szegö Quadrature Kalman Filter (SQKF). The SQKF transforms the intractable integrals that appear during the filtering over a unit circle. The transformed integral is approximated using the univariate Szegö quadrature rule. The univariate quadrature rule is extended in a multivariate domain using the product rule. Simulation results reveal an improved estimation accuracy for the SQKF in an oscillatory environment. The computational burden of the SQKF is similar to the GHF and higher than the UKF, CKF and CQKF.

Homotopy perturbation aided optimization procedure with applications to oscillatory fractional order nonlinear dynamical systems

This paper presents an approximate solution of nonlinear fractional differential equations (FDEs) that exhibit an oscillatory behavior by using a metaheuristic technique. The solutions of the governing equations are approximated by using homotopy perturbation method (HPM) along with the fractional derivative in the Caputo sense. The designed methodology is based on a weighted series of HPM in conjunction with a nature-inspired algorithm. The idea is instantly fascinated by the researchers on the consequent implementation of nature-inspired learning algorithms such as a Cuckoo search algorithm (CSA). The usage of CSA has accelerated the minimized search path of error to the convergent values of the solution. The validity and accuracy of the proposed technique are ascertained by calculating the approximate solution and the error norms which ensure the convergence of the approximation that can be further increased. The critical analysis is also provided by the numerical simulation of two different test models. Discussion of key points has been determined by the tabulation of numerical values and graphs. Comparative study of the results with known numerical technique is also performed.

Erratum: "Inferring the dynamics of oscillatory systems using recurrent neural networks" [Chaos 29, 063128 (2019)

First Page

Inferring the dynamics of oscillatory systems using recurrent neural networks.

We investigate the predictive power of recurrent neural networks for oscillatory systems not only on the attractor but in its vicinity as well. For this, we consider systems perturbed by an external force. This allows us to not merely predict the time evolution of the system but also study its dynamical properties, such as bifurcations, dynamical response curves, characteristic exponents, etc. It is shown that they can be effectively estimated even in some regions of the state space where no input data were given. We consider several different oscillatory examples, including self-sustained, excitatory, time-delay, and chaotic systems. Furthermore, with a statistical analysis, we assess the amount of training data required for effective inference for two common recurrent neural network cells, the long short-term memory and the gated recurrent unit.

Computing With Networks of Oscillatory Dynamical Systems

As we approach the end of the silicon road map, alternative computing models that can solve at-scale problems in the data-centric world are becoming important. This is accompanied by the realization that binary abstraction and Boolean logic, which have been the foundations of modern computing revolution, fall short of the desired performance and power efficiency. In particular, hard computing problems relevant to pattern matching, image and signal processing, optimizations, and neuromorphic applications require alternative approaches. In this paper, we review recent advances in oscillatory dynamical system-based models of computing and their implementations. We show that simple configurations of oscillators connected using simple electrical circuits can result in interesting phase and frequency dynamics of such coupled oscillatory systems. Such networks can be controlled, programmed, and observed to solve computationally hard problems. Although our discussion in this paper is limited to insulator-to-metal transition devices and spin-torque oscillators, the general philosophy of such a computing paradigm of “let physics do the computing” can be translated to other mediums as well, including micromechanical and optical systems. We present an overview of the mathematical treatments necessary to understand the time evolution of these systems and highlight the recent experimental results in this area that suggest the potential of such computational models.