Vera Thümmler

Continuation of Low-Dimensional Invariant Subspaces in Dynamical Systems of Large Dimension

We present a continuation method for low-dimensional invariant subspaces of a parameterized family of large and sparse real matrices. Such matrices typically occur when linearizing about branches of steady states in dynamical systems that are obtained by spatial discretization of time-dependent PDE’s. The main interest is in subspaces that belong to spectral sets close the imaginary axis. Our continuation procedure provides bases of the invariant subspaces that depend smoothly on the parameter as long as the continued spectral subset does not collide with another eigenvalue. Generalizing results from [32] we show that this collision generically occurs when a real eigenvalue from the continued spectral set meets another eigenvalue from outside to form a complex conjugate pair. Such a situation relates to a turning point of the subspace problem and and we develop a method to inflate the subspace at such points.

Local and Global Stability Analysis of an Unsupervised Competitive Neural Network

Unsupervised competitive neural networks (UCNN) are an established technique in pattern recognition for feature extraction and cluster analysis. A novel model of an unsupervised competitive neural network implementing a multi—time scale dynamics is proposed in this paper. The global asymptotic stability of the equilibrium points of this continuous—time recurrent system whose weights are adapted based on a competitive learning law is mathematically analyzed. The proposed neural network and the derived results are compared with those obtained from other multi—time scale architectures.

Local uniform stability of competitive neural networks with different time-scales under vanishing perturbations

We prove local uniform stability for a class of parametric uncertain competitive multi-time scale neural networks and determine the conditions under which stability holds as simple relationships between the neural parameters. It is assumed that the resulting parametric perturbations are only limited by their bounds. The stability conditions are established based on Gershgorins Theorem and at the same time a more realistic upper bound is obtained than with the conservative method for the fast time scale associated with the neural activity state.

Grassmannian spectral shooting

We present a new numerical method for computing the pure-point spectrum associated with the linear stability of coherent structures. In the context of the Evans function shooting and matching approach, all the relevant information is carried by the flow projected onto the underlying Grassmann manifold. We show how to numerically construct this projected flow in a stable and robust manner. In particular, the method avoids representation singularities by, in practice, choosing the best coordinate patch representation for the flow as it evolves. The method is analytic in the spectral parameter and of complexity bounded by the order of the spectral problem cubed. For large systems it represents a competitive method to those recently developed that are based on continuous orthogonalization. We demonstrate this by comparing the two methods in three applications: Boussinesq solitary waves, autocatalytic travelling waves and the Ekman boundary layer.

Computing stability of multidimensional traveling waves

We present a numerical method for computing the pure-point spectrum associated with the linear stability of multidimensional traveling fronts to parabolic nonlinear systems. Our method is based on the Evans function shooting approach. Transverse to the direction of propagation we project the spectral equations onto a finite Fourier basis. This generates a large, linear, one-dimensional system of equations for the longitudinal Fourier coefficients. We construct the stable and unstable solution subspaces associated with the longitudinal far-field zero boundary conditions, retaining only the information required for matching, by integrating the Riccati equations associated with the underlying Grassmannian manifolds. The Evans function is then the matching condition measuring the linear dependence of the stable and unstable subspaces and thus determines eigenvalues. As a model application, we study the stability of two-dimensional wrinkled front solutions to a cubic autocatalysis model system. We compare our ...

Continuation of Invariant Subspaces for Parameterized Quadratic Eigenvalue Problems

We consider quadratic eigenvalue problems with large and sparse matrices depending on a parameter. Problems of this type occur, for example, in the stability analysis of spatially discretized and parameterized nonlinear wave equations. The aim of the paper is to present and analyze a continuation method for invariant subspaces that belong to a group of eigenvalues, the number of which is much smaller than the dimension of the system. The continuation method is of predictor-corrector type, similar to the approach for the linear eigenvalue problem in [Beyn, Kles, and Thummler, Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, Berlin, 2001], but we avoid linearizing the problem, which will double the dimension and change the sparsity pattern. The matrix equations that occur in the predictor and the corrector step are solved by a bordered version of the Bartels-Stewart algorithm. Furthermore, we set up an update procedure that handles the transition from real to complex conjugate eigenvalues, which occurs when eigenvalues from inside the continued cluster collide with eigenvalues from outside. The method is demonstrated on several numerical examples: a homotopy between random matrices, a fluid conveying pipe problem, and a traveling wave of a damped wave equation.

Phase Conditions, Symmetries and PDE Continuation

Integral phase conditions were first suggested by E.J. Doedel as an efficient tool for computing periodic orbits in dynamical systems. In general, phase conditions help in eliminating continuous symmetries as well as in reducing the effort for adaptive meshes during continuation. In this paper we discuss the usefulness of phase conditions for the numerical analysis of finiteand infinite-dimensional dynamical systems that have continuous symmetries. The general approach (called the freezing method) will be presented in an abstract framework for evolution equations that are equivariant with respect to the action of a (not necessarily compact) Lie group. We show particular applications of phase conditions to periodic, heteroclinic and homoclinic orbits in ODEs, to relative equilibria and relative periodic orbits in PDEs as well as to time integration of equivariant PDEs.

Freezing Multipulses and Multifronts

We consider nonlinear time dependent reaction diffusion systems in one space dimension that exhibit multiple pulses or multiple fronts. In an earlier paper two of the authors developed the freezing method that allows us to compute a moving coordinate frame in which, for example, a traveling wave becomes stationary. In this paper we extend the method to handle multifronts and multipulses traveling at different speeds. The solution of the Cauchy problem is decomposed into a finite number of single waves, each of which has its own moving coordinate system. The single solutions satisfy a system of partial differential algebraic equations coupled by nonlinear and nonlocal terms. Applications are provided to the Nagumo and the FitzHugh–Nagumo systems. We justify the method by showing that finitely many traveling waves, when patched together in an appropriate way, solve the coupled system in an asymptotic sense. The method is generalized to equivariant evolution equations and is illustrated by the complex Ginzbu...

Numerical Approximation of Relative Equilibria for Equivariant PDEs

We prove convergence results for the numerical approximation of relative equilibria of parabolic systems in one space dimension. These systems are special examples of equivariant evolution equations. We use finite differences on a large interval with appropriately chosen boundary conditions. Moreover, we consider the approximation of isolated eigenvalues of finite multiplicity of the linear operator which arises from linearization at the equilibrium as well as the approximation of the corresponding invariant subspace. The results in this paper which are a generalization of the results in [V. Thummler, Numerical Analysis of the Method of Freezing Traveling Waves, Ph.D. thesis, Bielefeld University, 2005] are illustrated by numerical computations for the cubic quintic Ginzburg-Landau equation.

A continuation framework for invariant subspaces and its application to traveling waves

We present a continuation method for low-dimensional invariant subspaces of a parameterized family of large and sparse matrices. Such matrices typically occur when linearizing about branches of steady states in reaction-diffusion equations. Our continuation method provides bases of the invariant subspaces depending smoothly on the parameter. Prom these we can compute the corresponding eigenvalues efficiently. The predictor and the corrector step are reduced to solving bordered matrix equations of Sylvester type. For these equations we develop a bordered version of the Bartels-Stewart algorithm. The numerical techniques are used to study the stability problem for traveling waves in two examples: the Ginzburg-Landau and the FitzHugh-Nagumo system. In these cases there always exists a simple or multiple eigenvalue zero while the remaining eigenvalues determine the stability. We demonstrate the difficulties of separating these critical eigenvalues from clusters of eigenvalues that are generated by the essential spectrum of the continuous problem.