Pair Of Imaginary Eigenvalues Research Articles

Bifurcation Stabilization with Local Output Feedback

Local output feedback stabilization with smooth nonlinear controllers is studied for parameterized nonlinear systems for which the linearized system possesses either a simple zero eigenvalue or a pair of imaginary eigenvalues and the bifurcated solution is unstable at the critical value of the parameter. It is assumed that the unstable mode corresponding to the critical eigenvalue of the linearized system is not linearly controllable. Results are established for bifurcation stabilization using output feedback where the critical mode can be either linearly observable or linearly unobservable. The stabilizability conditions are characterized in explicit forms that can be used to synthesize stabilizing controllers. The results obtained in this paper are applied to rotating stall control for axial flow compressors as an application example.

On the construction of real canonical forms of Hamiltonian matrices whose spectrum is an imaginary pair

If A is a Hamiltonian matrix and P a symplectic matrix, the product P −1 AP is a Hamiltonian matrix. In this paper, we consider the case where the matrix A has a pair of imaginary eigenvalues and develop an algorithm which finds a matrix P such that the matrix P −1 AP has a particularly simple form, a canonical form.

Time-reversible and equivariant pitchfork bifurcation

In this note we describe a new phenomenon in steady-state bifurcations of time-reversible and equivariant vector fields. When the eigenspace associated with a pair of imaginary eigenvalues that passes through zero onto the real line is separable from the nullspace associated with the fixed zeros of the spectrum, we prove that a pitchfork bifurcation occurs.

Irregular dynamics and homoclinic orbits to Hamiltonian saddle centers

We consider 4-dimensional, real, analytic Hamiltonian systems with a saddle center equilibrium (related to a pair of real and a pair of imaginary eigenvalues) and a homoclinic orbit to it. We find conditions for the existence of transversal homoclinic orbits to periodic orbits of long period in every energy level sufficiently close to the energy level of the saddle center equilibrium. We also consider one-parameter families of reversible, 4-dimensional Hamiltonian systems. We prove that the set of parameter values where the system has homoclinic orbits to a saddle center equilibrium has no isolated points. We also present similar results for systems with heteroclinic orbits to saddle center equilibria. © 1997 John Wiley & Sons, Inc.

Irregular dynamics and homoclinic orbits to Hamiltonian saddle centers

We consider 4-dimensional, real, analytic Hamiltonian systems with a saddle center equilibrium (related to a pair of real and a pair of imaginary eigenvalues) and a homoclinic orbit to it. We find conditions for the existence of transversal homoclinic orbits to periodic orbits of long period in every energy level sufficiently close to the energy level of the saddle center equilibrium. We also consider one-parameter families of reversible, 4-dimensional Hamiltonian systems. We prove that the set of parameter values where the system has homoclinic orbits to a saddle center equilibrium has no isolated points. We also present similar results for systems with heteroclinic orbits to saddle center equilibria. © 1997 John Wiley & Sons, Inc.

Bifurcations of mode equations: spin waves

We study the bifurcations of the “spin-wave” equations that describe the parametric pumping of collective modes in magnetic media. We discuss the following dynamical phenomena: (i) sequential excitation of modes via zero-eigenvalue bifurcations; (ii) Hopf bifurcations followed (or not) by Feigenbaum cascades of period doublings; (iii) local and global homoclinic phenomena. Two new organizing centers for “routes to chaos” are identified; in the classification given by Guckenheimer and Holmes, one is the codimension-two local bifurcation, with one pair of imaginary eigenvalues and a zero eigenvalue, to which many dynamical consequences are known; the second, global homoclinic bifurcations associated to splitting of separatrices, in the limit where the system can be considered a Hamiltonian subjected to weak dissipation and forcing.

On the bifurcations of non-resonant solutions in autonomous systems

The behaviour of a non-linear system represented by a set of autonomous ordinary differential equations containing two real parameters is considered. At a certain critical value of these parameters it is assumed that the linearized system has two pairs of imaginary eigenvalues which are not even nearly commensurable. This non-resonant case is analyzed via a perturbation technique, and asymptotic formulate type results are obtained.

Adiabatic elimination for classical fermionic systems

Presents the reduction to a normal form of a Grassmannian differential equation describing a classical fermionic system in the neighbourhood of an instability. As an application of the method presented the author shows that the anticommuting version of the normal form associated with (a) the Hopf bifurcation (a simple pair of imaginary eigenvalues) leads to the overdamped Grassmann harmonic oscillator, and (b) the resonant Hopf bifurcation 1: 1 (a double pair of semisimple imaginary eigenvalues) corresponds exactly to the Thirring model for Grassmann solitons.

A Hopf bifurcation theorem for difference equations approximating a differential equation

If an ordinary differential equation is discretizised near an asymptotically stable stationary solution with a pair of imaginary eigenvalues by Euler's method with constant step lengthh, small invariant attracting cycles of radiusO(h1/2) will appear. This Hopf bifurcation theorem is applied to prove the existence of limit cycles in certain difference equations occurring in biomathematics (hypercycle, two loci-two alleles) and is also extended to general Runge—Kutta methods.