Pair Of Complex Conjugate Eigenvalues Research Articles

Intermittency as a codimension-three phenomenon

We analyze the interaction of three Hopf modes and show that locally a bifurcation gives rise to intermittency between three periodic solutions. This phenomenon can occur naturally in three-parameter families. Consider a vector fieldf with an equilibrium and suppose that the linearization off about this equilibrium has three rationally independent complex conjugate pairs of eigenvalues on the imaginary axis. As the parameters are varied, generically three branches of periodic solutions bifurcate from the steady-state solution. Using Birkhoff normal form, we can approximatef close to the bifurcation point by a vector field commuting with the symmetry group of the three-torus. The resulting system decouples into phase amplitude equations. The main part of the analysis concentrates on the amplitude equations in R3 that commute with an action ofZ2+Z2+Z2. Under certain conditions, there exists an asymptotically stable heteroclinic cycle. A similar example of such a phenomenon can be found in recent work by Guckenheimer and Holmes. The heteroclinic cycle connects three fixed points in the amplitude equations that correspond to three periodic orbits of the vector field in Birkhoff normal form. We can considerf as being an arbitrarily small perturbation of such a vector field. For this perturbation, the heteroclinic cycle disappears, but an “invariant” region where it was is still stable. Thus, we show that nearby solutions will still cycle around among the three periodic orbits.

Complex and Backward-Wave Modes in Inhomogeneously and Anisotropically Filled Waveguides

A rigorous analysis of lossless inhomogeneously and anisotropically filled waveguides of arbitrarily shaped cross section is presented. The mode propagation constants squared appear as eigenvalues of a real infinite-dimensional characteristic matrix, which is, in the general case, nonsymmetric. Complex conjugate pairs of eigenvalues are then possible, which give rise to complex modes. Modes at cutoff are shown to be either TE or TM with real cutoff frequencies. An investigation of the power flow shows that backward-wave modes may exist as well. Different orthogonality relations are derived from which the power coupling between complex modes is investigated.

Spiralling invariant manifolds

A definition of the concept of a multidimensional spiralling manifold is studied. Manifolds of this type are shown to occur as invariant manifolds of flows containing hyperbolic restpoints with a pair of complex conjugate eigenvalues. It is shown that spiralling can be a mechanism for producing intersections of invariant manifolds.

Unfolding of degenerate Hopf bifurcation for supersonic flow past a pitching wedge

This paper investigates the stability and bifurcation behavior of a double-wedge aerofoil performing a pitching motion at high angles of attack. When a pair of complex conjugate eigenvalues crosses the imaginary axis of the eigenvalue plane, the trivial solution loses stability giving rise to a periodic solution, known as Hopf bifurcation, provided certain transversality conditions are not violated. The existence of degenerate Hopf bifurcation due to the violation of Hopf s transversality condition at certain critical values of the system parameters is shown. The behavior of the pitching motion near these critical values is examined by unfolding the degeneracies. For the supersonic double-wedge aerofoil, various parameters defining the bifurcation paths were numerically evaluated.

Steady-state and unstable behavior of a single-mode inhomogeneously broadened laser

This paper is devoted to the study of the steady-state solutions and their stability for inhomogeneously broadened, unidirectional single-mode ring lasers. For both Lorentzian and Gaussian inhomogeneous-broadening profiles we find that, for appropriate detuning of the laser cavity, as many as three nontrivial steady-state solutions may appear and provide a formal confirmation of a phenomenon termed mode splitting by Casperson and Yariv in 1970 [ Appl. Phys. Lett.17, 259 ( 1970)]. We show through stability arguments that bistability between trivial (zero-intensity) and nontrivial solutions is possible. This bistability appears experimentally accessible. We analyze the stability of the stationary solutions, especially in connection with its dependence on the detuning and pump parameters. The instability boundary in the plane of these two control parameters can present a fairly complicated structure with alternate ranges of stability and instability. In correspondence with certain points of the instability boundary, two complex-conjugate pairs of eigenvalues of the characteristic equation become simultaneously unstable. This situation is likely to produce spontaneous oscillations with two coexisting fundamental frequencies.

Symmetric and flat bifurcations: An oscillatory phenomenon

Two distinct bifurcation phenomena associated with non-linear autonomous lumpedparameter systems are analyzed fully in ann-dimensional state space. The analysis is asymptotic and is performed via an intrinsic harmonic balancing technique. In one case a pair of complex conjugate eigenvalues touches the imaginary axis tangentially without crossing it in violation of the main Hopf condition associated with the so-called Hopf bifurcation. It is shown that under the new condition a bifurcating family of limit cycles may or may not axist. If it exists, the family is represented by two paths in the parameteramplitude space which have finite slopes and are symmetrically located with respect to the amplitude axis. The second case is concerned with aflat Hopf bifurcation, and the limit cycles always exist. This phenomenon occurs when the Hopf condition, that a pair of complex conjugate eigenvalues crosses the imaginary axis with non-zero velocity, is satisfied but another significant coefficient vanishes.

Bifurcation to Quasi-Periodic Tori in the Interaction of Steady State and Hopf Bifurcations

Bifurcations to quasi-periodic tori in a two parameter family of vector fields are studied. At criticality, the vector field has an equilibrium point with a zero eigenvalue and a pair of complex conjugate eigenvalues. This situation has been studied by Langford, Iooss, Holmes and Guckenheimer. Here we provide explicitly computed conditions under which the stability of the secondary branch of tori, and whether the flow on them is quasiperiodic, can be determined. The results are applied to “Brusselator” system of reaction diffusion equations.

Breather-like solution of the Korteweg-de Vries equation

A new real singular solution of the Korteweg-de Vries equation is briefly described. It is shown that the spectrum of the associated linear problem consists of a pair of complex conjugate eigenvalues. The existence of constants of motion and eigenfunction normalization integral is also discussed.

Asymptotic properties of the Fokker-Planck equation near a critical point

A method of studying the asymptotic properties of the Fokker-Planck equation near a critical point in which two pairs of complex conjugate eigenvalues cross the imaginary axis is developed. The method consists in the construction of a non-linear coordinate transformation which transforms the drift term into a canonical form.

Flow Induced Bifurcations to Three-Dimensional Oscillatory Motions in Continuous Tubes

Three-dimensional motions of a cantilever tube carrying an incompressible fluid and having rotational symmetry about the vertical axis are examined for bifurcating oscillatory solutions. The system behavior depends on two parameters, $\rho $, the flow velocity and $\beta$, the mass ratio of the fluid and the tube. As the flow rate is increased past a critical value, the zero solution becomes unstable by a double pair of complex conjugate eigenvalues of the linearized operator crossing the imaginary axis. Using ideas from the center manifold theory and the method of integral averaging, the governing partial differential equations are reduced to a set of four nonlinear ordinary differential equations. It is then shown that the system bifurcates into two qualitatively distinct oscillatory motions. These oscillatory solutions themselves, for some values of the mass ratio $\beta$, bifurcate into more complex amplitude modulated oscillations.

On the Global Convergence of the Toda Lattice for Real Normal Matrices and Its Applications to the Eigenvalue Problem

The asymptotic behavior of the Toda lattice, when acting on real normal matrices, is studied. It is shown that the solution flow eventually converges to a diagonal block form where for a real eigenvalue the associated block is of size $1 \times 1$ with that eigenvalue as its element and for complex-conjugate pairs of eigenvalues the associated block is of size $2 \times 2$ with the real part as its diagonal elements and the (negative) imaginary part as its off-diagonal elements. This result generalizes the well-known asymptotic behavior of Jacobi matrices and is consistent with that from the $QR$-algorithm.

Periodic solutions of matrix Riccati equations in discrete time-invariant systems

The well known matrix Riccati equations arising in optimal filtering as well as control problems for discrete time-invariant systems are considered in this paper. Existence conditions are established both for general periodic solutions and for non-negative definite periodic solutions. It is shown that a pair of complex conjugate eigenvalues or a pair of real eigenvalues in a certain symplectic matrix produce periodic solutions. Furthermore, algorithms are obtained to derive such periodic solutions using the symplectic matrix, which require slight additional computations to deriving steady state solutions. Finally, numerical examples which possess periodic solutions are presented.

On the global convergence of the Eberlein method for real matrices

The global convergence proof of the column-and row-cyclic Eberlein diagonalization process for real matrices is given. The convergence to a fixed matrix in Murnaghan form is obtained with the well-known exception of complex-conjugate pairs of eigenvalues whose real parts are more than double.

Numerical Hopf bifurcation analysis in nonlinear ordinary and partial differential systems from chemical reactor theory

A code has been developed which will automatically locate and analyze points of Hopf bifurcation in autonomous ordinary differential systems. The code first locates critical value(s) v c of a user-specified parameter v (the bifurcation parameter) such that a stationary (equilibrium) solution x *( v) loses linear stability by virtue of a complex conjugate pair of eigenvalues. The code computes x *( v) during the location of v c . Then the code computes the various coefficients in a local approximation to the family of periodic solutions which arise, a process which involves computation of second and third partial derivatives by numerical differencing of the user-supplied Jacobian matrix. The current version of the code, called BIFOR2, is fully described in Hassard, Kazarinoff, and Wan, Theory and Applications of Hopf Bifurcation, Cambridge U.P., 1981. In this paper we demonstrate the code in applications to two systems drawn from chemical reactor theory. The first application is to a 4th order ordinary differential system modeling a coupled tank reactor. The second application is to a partial differential system modeling a catalyst particle system. These represent the first applications of the code to chemically reacting systems other than the Brusselator. The second application demonstrates how collocation methods may be used in conjunction with BIFOR2 to perform Hopf bifurcation analysis of partial differential systems.

Hopf Bifurcation Phenomena in Tubes Carrying a Fluid

Self-induced planar nonlinear oscillations of a cantilever tube carrying an incompressible fluid are investigated. The system behavior is governed by a nonself-adjoint integrodifferential equation and an ordinary differential equation which are coupled nonlinearly. The system behavior depends on three parameters,$\rho_0 $, the flow velocity,$\beta $, the mass ratio of the tube and the fluid and $\alpha $, a parameter related to the pressure loss in the tube. For small $\rho_0 $, the zero solution is asymptotically stable. As $\rho_0$ is increased, for each $\beta $, a critical value of $\rho 0,\rho 0 = \rho _{cr} $, is reached when the zero solution becomes unstable by a pair of complex conjugate eigenvalues of the linearized system crossing the imaginary axis. The system satisfies conditions for Hopf bifurcation and the zero solution bifurcates into periodic orbits. The bifurcated periodic solutions are determined using center-manifold and averaging ideas. In general, the system equations include changes in flow velocity introduced by the tube motions. However, based on the order of $\alpha $, two classes of problems are shown to exist. When $\alpha $ is very large compared to the size of the motion, it is shown that fluid velocity changes do not enter the first order theory and the results show that in this case only supercritical bifurcation occurs. In the second case, either sub or supercritical solutions are found to occur depending on the value of $\alpha $.

Single-station time-domain (SSTD) vibration testing technique: Theory and application

The single-station time-domain (SSTD) technique is proposed as a vibration testing method for the identification of the dynamic characteristics of complex structures. The theoretical part of the study shows that the technique works, in principle, for lumped mass systems and continuous systems when a finite number of modes are included in the response. The idea behind the method is to make use of free vibration time response data—such as acceleration, velocity, displacement, or strain at a station on the structure— to determine unknown natural frequencies and associated viscous damping ratios. The analysis procedure is based on the construction of a mathematical model for the vibrating system from the experimental response data. This model has a number of degrees of freedom equal to the number of modes excited. The solution of the resulting eigenvalue problem identifies the natural frequencies and their damping ratios. Each pair of complex conjugate eigenvalues and eigenvectors corresponds to an underdamped mode, whereas each pair of real eigenvalues and eigenvectors corresponds to a critically or overdamped mode. Also included in the analysis is a study of how to optimize the effects of different parameters introduced in the method, such as sampling rates, frequency bandwidth and number of modes, on the accuracy of the results. An experimental investigation is described, for the identification of the dynamic characteristics of real structures by the SSTD method, in which a single channel was used to measure, filter and record the free acceleration responses at an arbitrary point on the structure after an adequate period of excitation; the excitation need not be measured. With the SSTD method, one can minimize the amount of instrumentation required and more accurate results have been obtained than from other vibration testing techniques under similar circumstances. The results obtained from this laboratory study have indicated that the SSTD method is not sensitive to measurement noise or round-off errors and therefore it shows promise of becoming one of the best techniques for structural identification, especially when the system is complex, has closely spaced natural frequencies, and/or is a heavily damped structure.

On a mathematical problem of the theory of elastic stability

There is considered the problem of bifurcation of the self-oscillations from the equilibrium state for nonlinear differential equations of hyperbolic type. A general qualitative analysis is performed and formulas are presented to compute the parameters of periodic solutions. Loss of stability occurs in many nonconservative problems of elastic stability theory (see [1,3], for instance) because the pair of complex-conjugate eigenvalues belonging to the stability spectrum of the linearized problem intersects the imaginary axis under changes in some parameters and goes over into the right half-plane of the complex plane. In this case, the analysis of the appropriate mechanical problem in a nonlinear formulation will permit clarification of the soft or hard nature of loss of stability. One of the best known of such problems is nonlinear “panel flutter” at supersonic speeds, which is simulated by using the law of plane sections [4], The mathematical methods used in this latter problem reduce to the replacement of the nonlinear partial differential equations by their Galerkin approximations with subsequent application of the method of a small parameter of the harmonic balance method. As has been shown in [5], the results obtained by this means are often far from the correct one. A strict method of investigating appropriate nonlinear partial differential equations is proposed below. The mathematical apparatus to be used here is a combination of the method of integral manifolds and a method developed in [6]. The method of integral manifolds is not applicable here in pure form. This is related to the specific hyperbolic-type equations for which the properties of the solutions cannot assure the necessary smoothness of the invariant surface [7]. Let us note that the problem being solved below has actually been solved by V.V. Bolotin (see [1], p. 333).

A convergent Jacobi method for solving the eigenproblem of arbitrary real matrices

A global convergence proof of a Jacobi-type norm-reducing diagonalisation process for arbitrary real matrices is given. The convergence to a fixed matrix in Murnaghan form is obtained with the well-known exception of complex conjugate pairs of eigenvalues whose real parts are more than double.