Stability Analysis Of Periodic Solutions Research Articles

Koopman–Hill stability computation of periodic orbits in polynomial dynamical systems using a real-valued quadratic harmonic balance formulation

In this paper, we generalize the Koopman–Hill projection method, which was recently introduced for the numerical stability analysis of periodic solutions, to be included immediately in classical real-valued harmonic balance (HBM) formulations. We incorporate it into the Asymptotic Numerical Method (ANM) continuation framework, providing a numerically efficient stability analysis tool for frequency response curves obtained through HBM. The Hill matrix, which carries stability information and follows as a by-product of the HBM solution procedure, is often computationally challenging to analyze with traditional methods. To address this issue, we generalize the Koopman–Hill projection stability method, which extracts the monodromy matrix from the Hill matrix using a matrix exponential, from complex-valued to real-valued formulations. In addition, we propose a differential recast procedure, which makes this real-valued Hill matrix immediately available within the ANM continuation framework. Using as an example a nonlinear von Kármán beam, we demonstrate that these modifications improve computational efficiency in the stability analysis of frequency response curves.

Stability Analysis of Periodic Solutions of Bonhoeffer–Van Der Pol System with Applied Impulse

Stability Analysis of Periodic Solutions of Bonhoeffer–Van Der Pol System with Applied Impulse

Sorting-free Hill-based stability analysis of periodic solutions through Koopman analysis

In this paper, we aim to study nonlinear time-periodic systems using the Koopman operator, which provides a way to approximate the dynamics of a nonlinear system by a linear time-invariant system of higher order. We propose for the considered system class a specific choice of Koopman basis functions combining the Taylor and Fourier bases. This basis allows to recover all equations necessary to perform the harmonic balance method as well as the Hill analysis directly from the linear lifted dynamics. The key idea of this paper is using this lifted dynamics to formulate a new method to obtain stability information from the Hill matrix. The error-prone and computationally intense task known by sorting, which means identifying the best subset of approximate Floquet exponents from all available candidates, is circumvented in the proposed method. The Mathieu equation and an n-DOF generalization are used to exemplify these findings.

Stability analysis of periodic solutions of the neutral-type neural networks with impulses and time-varying delays

This paper is concerned with a class of neutral-type neural networks with impulses and delays. By using continuation theorem due to Mawhin and constructing the appropriate Lyapunov-Krasovskii functional, several new sufficient conditions ensuring the existence and global exponential stability of the periodic solution are obtained. Moreover, a numerical example is provided to illustrate the main results. Our results can extend and improve some earlier publications.

Stability Analysis of an Industrial Blade Accounting for a Blade-Tip/Casing Nonlinear Interface

Abstract This paper investigates the local stability analysis of periodic solutions corresponding to the nonlinear vibration response of an industrial compressor blade, NASA rotor 37, on which are applied different types of nonlinearities. These solutions are obtained using a harmonic balance method-based approach presented in a previous paper. It accounts for unilateral contact and dry friction of the rotating blade against a rigid casing through a regularized penalty law. A Lanczos filtering technique is also employed to mitigate spurious oscillations related to the Gibbs phenomenon thus enhancing the robustness of the solver. In addition, a component mode synthesis technique is used to reduce the dimension of the numerical model. Stability assessment of the computed solutions relies on Floquet theory. It is performed through the computation of the monodromy matrix as well as Hill's method. Both methodologies are applied and thoroughly compared as the severity of the nonlinearity is gradually increased from a cubic spring to three-dimensional contact conditions on a deformed casing. While the presented results underline the applicability of both stability assessment methodologies for all types of nonlinearities, they also put forward the much higher computational effort required when computing the monodromy matrix. Indeed, it is shown that Hill's method yields converged results for significantly lower values of both the number of retained harmonics and the considered number of time steps thus making it a far more efficient method when dealing with industrial models. It is also underlined that the presented results are in excellent agreement with reference solution points obtained with time domain solution methods. Specific implementation tweaks that were found to be of critical importance in order to efficiently assess the stability of computed solutions are also detailed in order to provide a comprehensive view of the challenges inherent to such numerical developments.

Stability analysis of periodic solutions computed for blade-tip/casing contact problems

This article focuses on the stability assessment of mechanical systems featuring a nonlinear interface on which are applied unilateral contact constraints. The regularized-Lanczos harmonic balance method is used to compute periodic solutions for each system. For each considered mechanical system, the implementation of distinct stability assessment numerical algorithms (including the Newmark 2 n -pass method and Hill’s method using eigenvalue sorting) relying on Floquet theory allows for an in-depth investigation of their convergence and accuracy. From a single-degree of freedom impactor impacting a fixed obstacle to the industrial finite element model of a transonic compressor blade impacting a deformed casing, the gradually increasing complexity of the considered mechanical systems provides new insight on which stability assessment numerical algorithm may be the best suited when dealing with contact nonlinearities. From a numerical standpoint, attention is paid to specific developments that were identified as key for enhancing the numerical efficiency of the stability assessment process including the iterative solver, the computation of analytical derivatives, the scaling of the problem and the use of fast Fourier transforms. Particular attention is also paid to the influence of the Lanczos filtering procedure and its implications in terms of stability assessment. While it is found that both the Newmark 2 n -pass method and Hill’s method with eigenvalue sorting can be successfully applied on large industrial systems featuring a nonlinear interface, the faster convergence of Hill’s method is underlined and suggests it may be a better option for an accurate and efficient stability assessment on which Lanczos filtering has a negligible influence. • Local stability analysis of periodic solutions computed using a Harmonic Balance Method. (HBM) based numerical strategy for contact problems of increasing severity. • Analytical expression of blade-tip/casing contact forces related Jacobian matrices. • Numerical considerations for efficient HBM computations and in-depth comparison of time and frequency domain-based stability assessment algorithms. • Detailed application on both academical impactors and an industrial blade: NASA rotor 37. • Influence of Lanczos filtering over stability assessment.

A robust and efficient stability analysis of periodic solutions based on harmonic balance method and Floquet-Hill formulation

Harmonic balance method (HBM) incorporated with arc-length continuation and Floquet-Hill formulation now becomes an essential and strong tool for computing periodic solution branches and performing stability analysis, which can provide great insight into the responses of nonlinear dynamic systems. However, due to the redundancy of eigenvalues of Hill’s matrix over the actual Floquet exponents of periodic solutions and the distortion on the redundancy relation of the eigenvalues induced by order truncation of harmonics, it is not routine work to correctly select the actual Floquet exponents from the eigenvalues of Hill’s matrix in order to achieve correct stability and bifurcation analysis, especially the order of harmonics in HBM is low. In this work, a new robust strategy for Floquet exponents filtering is introduced, which is chosen based on the character of the real part (FEF-RP) of the Hill’s eigenvalues. Furthermore, an adaptive arc-length control for the arc-length continuation (ALC) is proposed, which is continuously governed by the minimum parameterized eigenvalue of the Jacobian matrix (ALC-MPE) and can provide smooth and efficient continuation of the turning points of periodic solutions. The study on a 4DOFs rotor–stator rubbing system and a 3DOFs damped Duffing oscillator system shows the feasibility of the proposed methods for arc-length control (ALC-MPE) and Floquet exponents filtering (FEF-RP).

Stability Analysis of Periodic Solutions in Alternately Advanced and Retarded Neural Network Models with Impulses

In this paper, the global exponential stability and periodicity are investigated for impulsive neural network models with Lipschitz continuous activation functions and piecewise alternately advanced and retarded argument of generalized argument (in short IDEPCAG). The sufficient conditions for the existence and uniqueness of periodic solutions of the model are established by applying fixed point theorem and the successive approximations method. By constructing suitable differential inequalities with piecewise alternately advanced and retarded argument, some sufficient conditions for the global exponential stability of the model are obtained. Typical numerical examples with simulations are utilized to illustrate the validity and improvement in less conservatism of the theoretical results.

Stability Analysis of Periodic Solutions of Van Der Pol-Duffing Forced Oscillator with Asymmetric Nonlinearities

Stability Analysis of Periodic Solutions of Van Der Pol-Duffing Forced Oscillator with Asymmetric Nonlinearities

Vibration Stability of the Impulse System of the Electric Drive Control

Purpose of reseach is Study of vibration stability of the impulse system of direct current electric drive in order to ensure operating modes with specified dynamic characteristics. Methods. The stability analysis of periodic solutions of differential equations with discontinuous right-hand side is reduced to the problem of studying local stability of fixed map points. Results. The analysis of stability is carried out depending on the supply voltage of the electric drive and the gain of the correcting link in the feedback circuit. It is revealed that the boundary of the stability region on the plane of the variable parameters has a pronounced extremum in the form of a maximum at the bifurcation point of codimension two, also called the resonance point 1: 2. On one side of this point, the stability region is bounded by the NeimarkSacker bifurcation line, and on the other, by the period-doubling bifurcation line. This means that with a change in the parameters, the radius of the stability region first increases, reaching a maximum at the resonance point 1: 2, and then decreases. This important conclusion can be used in optimization calculations. Conclusion. The analysis of the vibration stability of the impulse system of direct current electric drive, the behavior of which is described by differential equations of the discontinuous right-hand side, is carried out. The problem of finding periodic solutions to differential equations is reduced to the problem of finding fixed points of the map. The fixed points of the map satisfy a system of nonlinear equations, which was solved numerically by the NewtonRaphson method. The stability of periodic solutions of differential equations corresponds to the stability of fixed points of the corresponding map. The studies were carried out with variation of the gain in the feedback circuit and the supply voltage. It is revealed that the loss of a fixed point occurs through the supercritical Neimark-Sacker bifurcation, when the complex-conjugate pair of multipliers leaves the unit circle when the parameters change. However, with an increase in the supply voltage, the Neimark-Saker bifurcation boundary passes into the perioddoubling bifurcation boundary at the 1: 2 resonance point.

The Existence and Stability Analysis of Periodic Solution of Izhikevich Model

In this paper, based on the Izhikevich model, a more realistic hybrid impulsive neuron model combining model with state-dependent impulsive effects is proposed. By means of the theory of impulsive semidynamic system, the Poincare section and the ordinary differential equation geometry theory, the properties of the equilibrium points and the sufficient conditions for the existence and stability of different order-1 or order-2 periodic solutions of the system are derived near the equilibrium point or limit cycle. In addition, subthreshold bifurcation behavior which are saddle-node bifurcation, Andronov-Hopf bifurcation, Bogdano-Takens bifurcation and bifurcation behavior of the model with state-dependent impulsive effects are also studied. Final, the main results are illustrated by numerical simulations, and the bifurcation diagram shows that the system becomes more complicated due to the existence of impulses, and there exist bifurcations and chaos phenomena, indicating that the system has rich dynamic behavior.

Orbital Stability Analysis for Perturbed Nonlinear Systems and Natural Entrainment via Adaptive Andronov–Hopf Oscillator

Periodic orbits often describe desired state trajectories of dynamical systems in various engineering applications. Stability analysis of periodic solutions lays a foundation for control design to achieve convergence to a prescribed orbit. Here we consider a class of perturbed nonlinear systems with fast and slow dynamics and develop a novel averaging method for analyzing the local exponential orbital stability of a periodic solution. A framework is then proposed for feedback control design to stabilize a natural oscillation of an uncertain nonlinear system using a synchronous adaptive oscillator. The idea is applied to linear mechanical systems and a design theory is established. In particular, we propose a controller based on the Andronov–Hopf oscillator with additional adaptation mechanisms for estimating the unknown natural frequency and damping parameters. We prove that, with sufficiently slow adaptation, the estimated parameters locally converge to their true values and entrainment to the natural oscillation is achieved as part of an orbitally stable limit cycle. Numerical examples demonstrate that adaptation and convergence can in fact be fast.

Stability analysis for periodic solutions of fuzzy shunting inhibitory CNNs with delays

We consider fuzzy shunting inhibitory cellular neural networks (FSICNNs) with time-varying coefficients and constant delays. By virtue of continuation theorem of coincidence degree theory and Cauchy–Schwartz inequality, we prove the existence of periodic solutions for FSICNNs. Furthermore, by employing a suitable Lyapunov functional we establish sufficient criteria which ensure global exponential stability of the periodic solutions. Numerical simulations that support the theoretical discussions are depicted.

Stability Analysis of Periodic Solutions of Some Duffing’s Equations

In this paper, some stability results were reviewed. A suitable and complete Lyapunov function for the hard spring model was constructed using the Cartwright method. This approach was compared with the existing results which confirmed a superior global stability result. Our contribution relies on its application to high damping door constructions. (2010 Mathematics Subject Classification: 34B15, 34C15, 34C25, 34K13.)

Stability Analysis of Periodic Solution for a Complex‐valued Neural Networks With Bounded and Unbounded Delays

AbstractThis paper is devoted to investigating a class of complex‐valued neural networks with bounded and unbounded delays. By means of Mawhin's continuation theorem, some criteria on existence and uniqueness of periodic solution are established for the complex‐valued neural networks. By constructing an appropriate Lyapunov‐Krasovskii functional and M−matrix theory, some sufficient conditions are derived for the global exponential stability of periodic solutions to the complex‐valued neural networks. Finally, two numerical examples are given to show the effectiveness and merits of the present results.

Stability analysis for periodic solutions of the Van der Pol–Duffing forced oscillator

Based on the homotopy analysis method (HAM), the high accuracy frequency response curve and the stable/unstable periodic solutions of the Van der Pol-Duffing forced oscillator with the variation of the forced frequency are obtained and studied. The stability of the periodic solutions obtained is analyzed by use of Floquet theory. Furthermore, the results are validated in the light of spectral analysis and bifurcation theory.

A stability analysis of periodic solutions to the steady forced Korteweg–de Vries–Burgers equation

The stability of periodic solutions to the steady forced Korteweg–de Vries–Burgers (fKdVB) equation is investigated here. This family of periodic solutions was identified by Hattam and Clarke (2015) using a multi-scale perturbation technique. Here, Floquet theory is applied to the governing equation. Consequently, two criteria are found that determine when the periodic solutions are stable. This analysis is then confirmed by a numerical study of the steady fKdVB equation.

Instability and noise-induced thermalization of Fermi–Pasta–Ulam recurrence in the nonlinear Schrödinger equation

We investigate the spontaneous growth of noise that accompanies the nonlinear evolution of seeded modulation instability into Fermi–Pasta–Ulam recurrence. Results from the Floquet linear stability analysis of periodic solutions of the three-wave truncation are compared with full numerical solutions of the nonlinear Schrödinger equation. The predicted initial stage of noise growth is in a good agreement with simulations, and is expected to provide further insight into the subsequent dynamics of the field evolution after recurrence breakup.

Static properties of multiple-sine-Gordon systems

In this paper, we examine some basic properties of the multiple-sine-Gordon (MSG) systems, which constitute a generalization of the celebrated sine-Gordon (SG) system. We start by showing how MSG systems can be viewed as a general class of periodic functions. Next, periodic and step-like solutions of these systems are discussed in some details. In particular, we study the static properties of such systems by considering slope and phase diagrams. We also use concepts like energy density and pressure to characterize and distinguish such solutions. We interpret these solutions as an interacting many body system, in which kinks and antikinks behave as extended particles. Finally, we provide a linear stability analysis of periodic solutions which indicates short wavelength solutions to be stable.

Boundary Conditions for Numerical Stability Analysis of Periodic Solutions of Ordinary Differential Equations

This paper considers numerical methods for stability analyses of periodic solutions of ordinary differential equations. Stability of a periodic solution can be determined by the corresponding monodromy matrix and its eigenvalues. Some commonly used numerical methods can produce inaccurate results of them in some cases, for example, near bifurcation points or when one of the eigenvalues is very large or very small. This work proposes a numerical method using a periodic boundary condition for vector fields, which preserves a critical property of the monodromy matrix. Numerical examples demonstrate effectiveness and a drawback of this method.