K. Huseyin

A perturbation analysis of interactive static and dynamic bifurcations

The instability behavior of a nonlinear autonomous system in the vicinity of a coincident critical point, which leads to interactions between static and dynamic bifurcations, is studied. The critical point considered is characterized by a simple zero and a pair of pure imaginary eigenvalues of the Jacobian, and the system contains two independent parameters. The static and dynamic bifurcations and quasiperiodic motions resulting from the interaction of the bifurcation modes and the associated invariant tori are analyzed by a novel unification technique that is based on an intrinsic perturbation procedure. Divergence boundary, dynamic bifurcation boundary, secondary bifurcations, and invariant tori are determined explicitly. Two illustrative examples concerning control systems are presented. >

An intrinsic method of harmonic analysis for non-linear oscillations (a perturbation technique)

Abstract An intrinsic method of harmonic analysis for application to problems concerning non-linear periodic oscillations is presented. The method is designed to overcome the observed shortcomings of the conventional method of harmonic balance, and it is described with the aid of an example which was used earlier to demonstrate the inconsistency of the results obtained through the method of harmonic balance. The new technique is a perturbation method and it eliminates the danger of omitting any possible contributions of various harmonics to a particular approximation by engaging all the necessary harmonics and excluding the others automatically. Thefore the new approach yields ordered forms of consistent approximations for the non-linear problem without requiring knowledge about the number of harmonics a priori, in contrast to the conventional technique.

On the analysis of hopf bifurcations

Abstract The oscillatory instability and the family of limit cycles associated with a general autonomous dynamical system described by n nonlinear first order differential equations and an independently assignable scalar parameter are examined via an intrinsic method of harmonic analysis. The method is essentially a variation of the classical method of “harmonic balancing”, and is designed to eliminate the drawbacks and shortcomings associated with the latter. Indeed, the new approach yields consistent approximations for the nonlinear dynamical bifurcation problem under consideration through a systematic perturbation procedure. It has thus been possible to derive explicit, formula-type expressions for the post-critical family of the periodic solutions, frequency of oscillations and the path which represents the family bifurcating from a flutter-critical point on an initially stable equilibrium path. The results are available to be used directly in the analysis of specific problems which fall within the scope of the formulation, without actually performing much analysis. Two illustrative examples are provided.

Static and dynamic bifurcations associated with a double-zero eigenvalue

This paper is concerned with the static and dynamic bifurcation phenomena exhibited by a nonlinear autonomous system in the vicinity of a double-zero eigenvalue. It is demonstrated analytically that such a coincident critical point is often located on the intersection of the divergence and flutter boundaries of the system. The analysis is performed via a new ‘unification technique’ which is based on an intrinsic perturbation procedure. The new approach is capable of yielding information about stability of solutions as well as incipient and secondary bifurcations. Indeed, the stability properties of the system and secondary Hopf bifurcations are discussed conveniently, and explicit results concerning post-critical solutions are presented. An illustrative example is analysed.

Bifurcations associated with a double zero and a pair of pure imaginary eigenvalues

Interactive static and dynamic bifurcations associated with a nonlinear autonomous system and the stability properties of various solutions are explored. Attention is focused on the vicinity of a compound critical point where the Jacobian of the system exhibits a double zero eigenvalue of index one and a pair of pure imaginary eigenvalues. The system under consideration has three independent parameters, and depending on the route followed in the parameter space, the system may exhibit static bifurcations, Hopf bifurcations, secondary Hopf bifurcations, and bifurcations into two or three dimensional tori. The general analysis is based on a perturbation method which combines an intrinsic harmonic balancing technique with a certain unification procedure. This approach leads to bifurcation equations and simplified differential equations governing the local dynamics in the vicinity of the compound critical point. A control system is analyzed for illustration.

Standard forms of the eigenvalue problems associated with gyroscopic systems

Abstract The eigenvalue problem arising in the free vibration and stability analysis of gyroscopic systems is associated with a λ-matrix in which λ as well as its square appears. The characteristic polynomial of a non-dissipative gyroscopic system however, is a function of λ 2 , and an equivalent standard eigenvalue formulation involving only λ 2 as an eigenvalue, therefore, seems to be a more appropriate representation of the physical system. Such forms and the related transformations are discussed herein. Similarly, when the loading parameter also appears explicitly in the λ-matrix, an equivalent double-eigenvalue problem involving λ 2 and the loading parameter as eigenvalues is generated. The extremum properties of the Rayleigh Quotient leads to a convenient proof of an upper bound theorem on λ 2 . The use of left and right eigenvectors and the new double-eigenvalue formulation allows for the establishment of a flutter condition similar to one obtained for circulatory systems earlier. An example illustrating some of the concepts is presented.

An intrinsic multiple-scale harmonic balance method for non-linear vibration and bifurcation problems

Abstract An intrinsic multiple-time-scale harmonic balance method for the analyses of non-linear periodic oscillations and bifurcation problems is developed. The new approach combines selected features of the intrinsic harmonic balancing (introduced earlier) and multiple-time-scaling in an effort to produce a more efficient technique for non-linear analysis. This is achieved by retaining advantageous aspects of both methods and eliminating some of the observed disadvantages. Thus, the new approach has the following features: (1) it produces ordered forms of consistent approximations for the solutions of a non-linear problem such that the number of perturbations required for a particular approximation is less than that required by the intrinsic harmonic balancing; (2) it leads to simplified differential equations governing the local dynamics which readily yield the stability properties of the solutions; (3) it does not require the solution of differential equations at each perturbation step, as in the conventional multiple-time-scaling method, and the need to eliminate secular terms does not arise. The application of the new procedure is illustrated on several examples.

On bifurcations into nonresonant quasi-periodic motions

Abstract The stability, instability, and bifurcaton behaviour of a nonlear autonomous system in the vicinity of a compound critical point is studied in detail. The critical point is characterized by two distinct pairs of pure imaginary eigenvalues of the Jacobian, and the system is described by two independent parameters. Tne analysis is based on a generalized perturbation procedure which employs multiple-time-scale Fourier series and embraces the intrinsic harmonic balancing and unification technique introduced earlier. This more comprehensive perturbation approach leads to explicit asymptotic results concerning periodic and nonresonant quasiperiodic motions which take place on an invariant torus. An electrical network is analyzed to illustrate the direct applicability of the analytical results.

The multiple-parameter perturbation technique for the analysis of non-linear systems

Abstract A special perturbation technique, employing several independent perturbation parameters, is outlined in general terms. The procedure is then illustrated with reference to multiple-parameter non-linear stability problems of a conservative structural field. Successive approximations to the associated stability boundary are obtained in parametric form. This is equivalent to the general mathematical problem of locating the points on a non-linear surface at which points the Hessian of a potential function vanishes. Such a problem can arise in relation to other branches of mechanics and applied mathematics and the technique described in this paper is readily applicable. A shallow circular arch subjected to combined uniform pressure and concentrated load at the apex is analysed for illustration.

Non-linear bifurcation analysis of non-gradient systems☆

Abstract The post-divergence behaviour of non-gradient systems is studied through a perturbation approach, attention being restricted to equilibrium paths in the neighbourhood of a critical point. Simple and coincident critical points are treated separately. Various characteristic phenomena are explored in general terms by identifying certain distinct properties of the Jacobian matrix in a state-space formulation. It is demonstrated that the well-known asymmetric and symmetric points of bifurcation can also arise in non-gradient systems and the conditions giving rise to each of these phenomena are discussed. An illustrative example is presented.