Stability Analysis Of Periodic Solutions Research Articles

Robust LFR-based technique for stability analysis of periodic solutions

Linear matrix inequality based techniques, most often used for robust analysis of linear systems, are applied to the stability analysis of periodic equilibrium trajectories of nonlinear systems. Results are derived by linear-fractional representation of the nonlinearities and taking into account parametric uncertainties in the same time. Numerically exploitable formulas are obtained by discretization. An academic example illustrates the entire methodology.

PERIODIC SOLUTIONS OF DIFFERENTIAL ALGEBRAIC EQUATIONS WITH TIME DELAYS: COMPUTATION AND STABILITY ANALYSIS

This paper concerns the computation and local stability analysis of periodic solutions to semi-explicit differential algebraic equations with time delays (delay DAEs) of index 1 and index 2. By presenting different formulations of delay DAEs, we motivate our choice of a direct treatment of these equations. Periodic solutions are computed by solving a periodic two-point boundary value problem, which is an infinite-dimensional problem for delay DAEs. We investigate two collocation methods based on piecewise polynomials: collocation at Radau IIA and Gauss–Legendre nodes. Using the obtained collocation equations, we compute an approximation to the Floquet multipliers which determine the local asymptotic stability of a periodic solution. Based on numerical experiments, we present orders of convergence for the computed solutions and Floquet multipliers and compare our results with known theoretical convergence results for initial value problems for delay DAEs. We end with examples on bifurcation analysis of delay DAEs.

Non-linear oscillations of milling

The principal features of two mathematical models that can be used to study non-linear oscillations of a workpiece - tool system during a milling operation are presented and explained in this article. These models are non-linear, non-homogeneous, delay-differential systems with time-periodic coefficients. In the treatment presented here, the sources of non-linearities are the multiple regenerative effect and the loss-of-contact effect. The time-delay effect is taken into account, and the dependence of this delay effect on the feed rate is modelled. A variable time delay is introduced to capture the influence of the feed-rate in one of the models. Two formulations that can be used to carry out stability analysis of periodic solutions are presented. The models presented and the stability-analysis formulations are relevant for predicting and understanding chatter in milling.

INVESTIGATION OF TRIPLE PENDULUM WITH IMPACTS USING FUNDAMENTAL SOLUTION MATRICES

In this work a triple pendulum with damping, external forcing and impacts is investigated. We integrate the obtained system of governing equations between two successive impacts, and the discontinuity points are detected (by halving time step until the required precision is obtained). For each impact time, the state of the system is transformed using the extended restitution coefficient rule. The theory of Aizerman and Gantmakher is used to calculate the fundamental solution matrices in the analyzed system exhibiting discontinuities. Fundamental matrices are used during calculation of Lyapunov exponents, in stability analysis of periodic solution (Floquet multipliers) and in shooting method of finding periodic orbits. Then some examples for three coupled identical rods with horizontal barrier are shown.

Computation and Stability Analysis of Limit Cycles in Piecewise Linear Hybrid Systems

Hybrid systems often exhibit periodic behaviours. Therefore, the computation and the stability analysis of hybrid limit cycles is an important problem. Recently, several techniques have been proposed for the stability analysis of periodic solutions. However, all of these assume that the cycle has previously been computed, which is not the easiest part of the work. In this paper, we present a method for the numerical computation of limit cycles in piecewise linear hybrid systems. It uses the concept of hybrid Poincare map and is an extension of the algorithms existing for smooth dynamical systems. An example is treated in order to show the results that can be obtained in practice.

Dynamics Investigation of Three Coupled Rods with a Horizontal Barrier

This report is a part of the larger project of non-linear dynamics investigation of three coupled physical pendulums with damping and with arbitrary situated barriers, and externally driven. The set of differential equations and the set of algebraic inequalities (representing a barrier) governing the motion of three coupled rods are presented in the non-dimensional form. The system of governing equations is integrated between two successive impacts, and the discontinuity points are detected (by halving time step until a required precision is obtained). In each impact time, the state of the system is transformed using the extended restitution coefficient rule. The theory of Aizerman and Gantmakher is used to calculate the fundamental solution matrices in the analyzed system exhibiting discontinuities. The fundamental matrices are used during calculation of Lyapunov exponents, during stability analysis of periodic solutions (Floquet multipliers) and in shooting method applied to detect and trace periodic orbits. Some examples for three coupled identical rods with horizontal barrier are reported.

IMPROVED NUMERICAL FLOQUET MULTIPLIERS

This paper studies numerical methods for linear stability analysis of periodic solutions in codes for bifurcation analysis of small systems of ordinary differential equations (ODEs). Popular techniques in use today (including the AUTO97 method) produce very inaccurate Floquet multipliers if the system has very large or small multipliers. These codes compute the monodromy matrix explicitly or as a matrix pencil of two matrices. The monodromy matrix arises naturally as a product of many matrices in many numerical methods, but this is not exploited. In this case, all Floquet multipliers can be computed with very high precision by using the periodic Schur decomposition and corresponding algorithm [Bojanczyk et al., 1992]. The time discretisation of the periodic orbit becomes the limiting factor for the accuracy. We present just enough of the numerical methods to show how the Floquet multipliers are currently computed and how the periodic Schur decomposition can be fitted into existing codes but omit all details. However, we show extensive test results for a few artificial matrices and for two four-dimensional systems with some very large and very small Floquet multipliers to illustrate the problems experienced by current techniques and the better results obtained using the periodic Schur decomposition. We use a modified version of AUTO97 [Doedel et al., 1997] in our experiments.

Nonlinear computation in DIVA — methods and applications

Methods for one-parameter continuation and stability analysis of periodic solutions as well as two-parameter continuation of Hopf and limit bifurcation points have been added to the dynamic flowsheet simulation environment DIVA. They are specially tailored for large sparse systems of differential algebraic equations with arbitrary structural properties usually arising in dynamic flowsheet simulation of chemical processes and plants. The application of these methods combined with the other capabilities of such an integrated tool for nonlinear system analysis is demonstrated for two different types of processes. The first is the so-called circulation-loop reactor, which has been used for catalytic combustion of waste air. Through periodic operation of the reactor, abatement of small amounts of volatile organic compounds (VOCs) in waste air is possible autothermally or with minimum supply of energy. Control strategies are studied for extending the region of desired periodic operating points, when the amount of VOCs is too small for autonomous periodic operation. The second application is concerned with the nonlinear dynamic behaviour of reactive distillation processes for the production of fuel ethers MTBE and TAME. In particular, the existence of multiple steady states is analyzed for the practical important case, when pre-reaction is carried out prior to the reactive distillation column. Further, these results are compared with the case without pre-reaction and a physical interpretation is given.

Instability and bifurcation near the symplectic 1 :4 resonance

In coupled mechanical systems with an underlying Hamiltonian structure, the stability analysis-of periodic solutions or periodically forced equilibria-leads via a Poincare section to an interated symplectic map. In this paper, a two-parameter family of symplectic maps on is considered when the linearization has a loss of stability through a collision of Floquet multipliers at ±i. Two approaches to the problem are considered:first, a bifurcation analysis on configuration space which leads to a complete local theory for bifurcation and stability of period-4 points in the two-parameter family. The second approach is to define a model vector field whose. time-1 map approximates the dynamics near the instability. The model vector field derived here has independent interest as the truncated normal form for a -equivariant Hamiltonian vector field on with a double-zero eigenvalue. A spinning double-orthogonal pendulum with this instability is also presented.

Geometric Stability Analysis for Periodic Solutions of the Swift-Hohenberg Equation

In this paper we describe invariant geometrical ~structures in the phase space of the Swift-Hohenberg equation in a neighborhood of its periodic stationary states. We show that in spite of the fact that these states are only marginally stable (i.e., the linearized problem about these states has continuous spectrum extending all the way up to zero), there exist finite dimensional invariant manifolds in the phase space of this equation which determine the long-time behavior of solutions near these stationary solutions. In particular, using this point of view, we obtain a new demonstration of Schneider's recent proof that these states are nonlinearly stable.

Stability analysis of periodic solutions in non-linear autonomous circuits: a discrete time approach

International Journal of Circuit Theory and ApplicationsVolume 24, Issue 4 p. 511-517 Correspondence Stability analysis of periodic solutions in non-linear autonomous circuits: a discrete time approach Joan M. Miró-Sans, Joan M. Miró-Sans UPC-Department of Signal Theory and Communications, Sor Eulalia d'Anzizu s/n, E-08034 Barcelona, SpainSearch for more papers by this authorPere Palà-Schönwälder, Pere Palà-Schönwälder UPC-Department of Signal Theory and Communications, Sor Eulalia d'Anzizu s/n, E-08034 Barcelona, SpainSearch for more papers by this authorOrestes Mas-Casals, Orestes Mas-Casals UPC-Department of Signal Theory and Communications, Sor Eulalia d'Anzizu s/n, E-08034 Barcelona, SpainSearch for more papers by this author Joan M. Miró-Sans, Joan M. Miró-Sans UPC-Department of Signal Theory and Communications, Sor Eulalia d'Anzizu s/n, E-08034 Barcelona, SpainSearch for more papers by this authorPere Palà-Schönwälder, Pere Palà-Schönwälder UPC-Department of Signal Theory and Communications, Sor Eulalia d'Anzizu s/n, E-08034 Barcelona, SpainSearch for more papers by this authorOrestes Mas-Casals, Orestes Mas-Casals UPC-Department of Signal Theory and Communications, Sor Eulalia d'Anzizu s/n, E-08034 Barcelona, SpainSearch for more papers by this author First published: July 1996 https://doi.org/10.1002/(SICI)1097-007X(199607/08)24:4<511::AID-CTA894>3.0.CO;2-SCitations: 3AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume24, Issue4July 1996Pages 511-517 RelatedInformation

Stability analysis of periodic solutions in non‐linear autonomous circuits: a discrete time approach

Stability analysis of periodic solutions in non‐linear autonomous circuits: a discrete time approach

Stability analysis of periodic solutions in nonautonomous systems with hysteretic elements

AbstractThis paper considers the case where hysteresis is given by the Preisach model and discusses the stability analysis for the periodic solution in the nonautonomous system containing hysteretic elements. The behavior of the system containing hysteretic elements is affected by the history of the system until it arrives at the steady state.In this paper, it is shown first that even periodic solutions with the same hysteretic loop may have different variational systems for the steady‐state solution if the histories until arriving at the steady state are different. Next, two kinds of stability criteria are considered based on the difference of the history up to the steady state and the stability against the external disturbance is investigated. The dependence of the stability of the solution on the bifurcation of the periodic solution also is discussed. Finally, examples of numerical analysis are presented to verify the result of discussions.

Dynamic stability of periodic solutions of large scale nonlinear systems

A method suitable for the stability analysis of periodic solutions of large nonlinear systems is presented. The stability behaviour is inferred from a Galerkin approximation of the associated Poincaré map. The practical value of the method is shown by applying it to the nonlinear finite element model of a large scale wind energy convertor.