Hopf bifurcations in time-delay systems with band-limited feedback

We investigate the steady-state solution and its bifurcations in time-delay systems with band-limited feedback. This is a first step in a rigorous study concerning the effects of AC-coupled components in nonlinear devices with time-delayed feedback. We show that the steady state is globally stable for small feedback gain and that local stability is lost, generically, through a Hopf bifurcation for larger feedback gain. We provide simple criteria that determine whether the Hopf bifurcation is supercritical or subcritical based on the knowledge of the first three terms in the Taylor-expansion of the nonlinearity. Furthermore, the presence of double-Hopf bifurcations of the steady state is shown, which indicates possible quasiperiodic and chaotic dynamics in these systems. As a result of this investigation, we find that AC-coupling introduces fundamental differences to systems of Ikeda-type [Ikeda et al., Physica D 29 (1987) 223-235] already at the level of steady-state bifurcations, e.g. bifurcations exist in which limit cycles are created with periods other than the fundamental ``period-2'' mode found in Ikeda-type systems.Comment: 32 pages, 5 figures, accepted for publication in Physica D: Nonlinear Phenomen

Stability and drift of underwater vehicle dynamics: Mechanical systems with rigid motion symmetry

This paper develops the stability theory of relative equilibria for mechanical systems with symmetry. It is especially concerned with systems that have a noncompact symmetry group, such as the group of Euclidean motions, and with relative equilibria for such symmetry groups. For these systems with rigid motion symmetry, one gets stability but possibly with drift in certain rotational as well as translational directions. Motivated by questions on stability of underwater vehicle dynamics, it is of particular interest that, in some cases, we can allow the relative equilibria to have nongeneric values of their momentum. The results are proved by combining theorems of Patrick with the technique of reduction by stages. This theory is then applied to underwater vehicle dynamics. The stability of specific relative equilibria for the underwater vehicle is studied. For example, we find conditions for Liapunov stability of the steadily rising and possibly spinning, bottom-heavy vehicle, which corresponds to a relative equilibrium with nongeneric momentum. The results of this paper should prove useful for the control of underwater vehicles

Stability of Nonlinear Control Systems by the Second Method of Liapunov

This report investigates the stability of autonomous closed-loop control systems containing nonlinear elements. An n-th order nonlinear autonomous system is described by a set of n first order differential equations of the type dxi/dt=xi(x1, x2, …xn) i=1,2,…n. Liapunov\u27s second (direct) method is used in the stability analysis of such systems. This method enables one to prove that a system is stable (or unstable) if a function V=V (x1, x2, … xn) can be found which, together with its time derivative, satisfies the requirements of Liapunov\u27s stability (or instability) theorems. At the present time, there are no generally applicable straight forward procedures available for constructing these Liapunov\u27s functions. Several Liapunov\u27s functions, applicable to systems described in the canonic form of differential equations, have been reported in the literature. In this report, it is shown that any autonomous closed-loop system containing a single nonlinear element can be described by canonic differential equations. The stability criteria derived from the Liapunov\u27s functions for canonic systems give sufficient and not necessary conditions for stability. It is known that these criteria reject many systems which are actually stable. The reasons why stable systems are sometimes rejected by these simplified stability criteria are investigated in the report. It is found that a closed-loop system will always be rejected by these simplified stabi1ity criteria if the root locus of the transfer function G(s), representing the linear portion of the system, is not confined to the left-half of the s-plane for all positive values of the loop gain. A pole-shifting technique and a zero-shifting technique, extending -the applicability of the simplified stability criteria to systems that are stable for sufficiently high and/or sufficiently low values of the loop gain, are proposed in this report. New simplified stability criteria have been developed which incorporate the changes in the canonic form of differential equations caused by the application of the zero-shifting technique. Other methods of constructing Liapunov\u27s functions for nonlinear control systems are presented in Chapter III, These include the work of Pliss, Aizerman and Krasovski. Numerous other procedures, which have been reported in literature, apply to only very special cases of automatic control systems. No attempt has been made to account for all of these special cases and the presentation of methods of constructing Liapunov’s functions is limited to only those which are more generally applicable. A pseudo-canonic transformation has been developed which enables one to find stability criteria of canonic systems without the use of complex variables. The results of this research indicate that the second method of Liapunov is a very powerfuI tool of exact stability analysis of nonlinear systems. Additional research, especially in the direction of the methods of construction of Liapunov’s functions, will not only yield new analysis and synthesis procedures, but also will aid in arriving at a set of meaningful performance specifications for nonlinear control systems

Aeroelastic stability of coupled flap-lag motion of hingeless helicopter blades at arbitrary advance ratios

Equations for large amplitude coupled flap-lag motion of a hingeless elastic helicopter blade in forward flight are derived. Only a torsionally rigid blade excited by quasi-steady aerodynamic loads is considered. The effects of reversed flow together with some new terms due to radial flow are included. Using Galerkin's method the spatial dependence is eliminated and the equations are linearized about a suitable equilibrium position. The resulting system of homogeneous periodic equations is solved using multivariable Floquet-Liapunov theory, and the transition matrix at the end of the period is evaluated by two separate methods. Computational efficiency of the two numerical methods is compared. Results illustrating the effects of forward flight and various important blade parameters on the stability boundaries are presented

A guide of the application of the liapunov direct method to flight control systems

Application of Liapunov direct method to flight control system

The Gierer-Meinhardt system on a compact two-dimensional Riemannian Manifold: Interaction of Gaussian curvature and Green's function

In this paper, we rigorously prove the existence and stability of single-peaked patterns for the singularly perturbed Gierer-Meinhardt system on a compact two-dimensional Riemannian manifold without boundary which are far from spatial homogeneity. Throughout the paper we assume that the activator diffusivity is small enough. We show that for a threshold ratio of the activator diffusivity and the inhibitor diffusivity, the Gaussian curvature and the Green's function interact. A convex combination of the Gaussian curvature and the Green's function together with their derivatives are linked to the peak locations and the o(1) eigenvalues. A nonlocal eigenvalue problem (NLEP) determines the O(1) eigenvalues which all have negative part in this case.RGC of Hong Kon

Stability in Hamiltonian systems : applications to the restricted three-body problem

The N-body problem is a classical famous problem which has attracted a lot of attention. It consists of describing the complete behavior of all solutions of the equations of motions for a given initial condition. Still related to this kind of problem Euler in 1772 describe the three-body problem in his eort to study the motion of the moon. Later on Jacobi in 1836 brought forward the main mathematical interest in an even more specic part of the three body problem, namely the one which is reduced to a conservative two degrees of freedom problem. This has somehow brought up an extensive study on mechanics. Despite of all this eort of great mathematicians, in general the N-body problem for N> 2 is still unsolved