Existence Of Invariant Manifolds Research Articles

Theory and applications of equivariant normal forms and Hopf bifurcation for semilinear FDEs in Banach spaces

This paper is concerned with equivariant normal forms of semilinear functional differential equations (FDEs) in general Banach spaces. The analysis is based on the theory previously developed for autonomous delay differential equations and on the existence of invariant manifolds. We show that in the neighborhood of trivial solutions, variables can be chosen so that the form of the reduced vector field relies not only on the information of the linearized system at the critical point but also on the inherent symmetry. We observe that the normal forms give critical information about dynamical properties, such as generic local branching spatiotemporal patterns of equilibria and periodic solutions. As an important application of equivariant normal forms, we not only establish equivariant Hopf bifurcation theorem for semilinear FDEs in general Banach spaces, but also in a natural way derive criteria for the existence, stability, and bifurcation direction of branches of bifurcating periodic solutions. We employ these general results to obtain the existence of infinite many small-amplitude wave solutions for a delayed Ginzburg-Landau equation on a two-dimensional disk with the homogeneous Dirichlet boundary condition.

Algorithmic Reduction of Biological Networks with Multiple Time Scales

We present a symbolic algorithmic approach that allows to compute invariant manifolds and corresponding reduced systems for differential equations modeling biological networks which comprise chemical reaction networks for cellular biochemistry, and compartmental models for pharmacology, epidemiology and ecology. Multiple time scales of a given network are obtained by scaling, based on tropical geometry. Our reduction is mathematically justified within a singular perturbation setting. The existence of invariant manifolds is subject to hyperbolicity conditions, for which we propose an algorithmic test based on Hurwitz criteria. We finally obtain a sequence of nested invariant manifolds and respective reduced systems on those manifolds. Our theoretical results are generally accompanied by rigorous algorithmic descriptions suitable for direct implementation based on existing off-the-shelf software systems, specifically symbolic computation libraries and Satisfiability Modulo Theories solvers. We present computational examples taken from the well-known BioModels database using our own prototypical implementations.

Normal forms of C ∞ vector fields based on the renormalization group

The normal form theory for polynomial vector fields is extended to those for C∞ vector fields vanishing at the origin. Explicit formulas for the C∞ normal form and the near identity transformation that brings a vector field into its normal form are obtained by means of the renormalization group method. The dynamics of a given vector field such as the existence of invariant manifolds is investigated via its normal form. The C∞ normal form theory is applied to prove the existence of infinitely many periodic orbits of two dimensional systems, which is not shown from polynomial normal forms.

Near-Integrability and Recurrence in FPU Chains with Alternating Masses

An important aspect of understanding FPU chains is the existence of invariant manifolds (called “bushes”) in FPU chains. We will focus on the classical periodic FPU chain and on the FPU chain with alternating masses where we show that also in the alternating case nested manifolds (related to bushes) exist. The use of symmetries leads to the emergence of systems of n particles as invariant manifolds of systems with a multiple of n particles. This analysis is followed by examples of existence and stability of special invariant manifolds and phase-space dynamics in the case of 4 and 8 particles. These examples are typical for periodic FPU chains with 4n or 8n particles. It turns out that in the alternating case the dynamics is strongly affected by the choice of the alternating mass m. Normal form calculations help to identify quasi-trapping regions leading to delay of recurrence. The results suggest that equipartition of energy near stable equilibrium is improbable.

On the Stability of Toroidal Manifold for One Class of Dynamical Systems

We study the problem of stability of an invariant toroidal manifold for one class of linear extensions of a dynamical system on a torus. The obtained result is used to investigate the problem of existence of invariant manifold for a nonlinear system of differential equations.

On some qualitative properties of monotone linear extensions of dynamical systems

We study monotone linear extensions of dynamical systems. The problem of the existence of invariant manifolds and exponential separation is investigated for linear extensions that preserve the order structure. We also study the relationship between the monotonicity of linear extensions and the existence (weak regularity, quasiregularity) of bounded solutions of inhomogeneous linear extensions.

INVARIANT MANIFOLD WITH COMPLETE FOLIATIONS AND CHAOTIFICATION ANALYSIS FOR A KIND OF PDES WITH BOUNDARY COUPLING

In this paper, we are concerned with the existence of invariant manifolds and complete foliations for a class of PDEs with boundary coupling. Some new form of gap relative coupling and inequality conditions are obtained. Further, we prove the topological equivalence of the flows in the respective attractors between the system and its spatial discretization system (an ODE system). Finally, the chaotification of the system is discussed through an example and simulation is generated to illustrate the theoretical results.

Invariant Manifolds for Nonautonomous Systems with Application to One-Step Methods

In this paper we study the existence of invariant manifolds for a special type of nonautonomous systems which arise in the study of discretization methods. According to [10], a one-step scheme of step-size ɛ for an autonomous system can be interpreted as the ɛ-flow of a perturbed nonautonomous system. The perturbation is “rapidly forced” in the sense that it is periodic with respect to time with period ɛ. Assuming a saddle node for the autonomous system, we prove that these rapidly forced perturbations have center manifolds which exist in a uniform neighborhood and which converge to a center manifold of the autonomous system as ɛ tends to zero. Our results are applied to obtain a smooth continuation as well as estimates of the well known center manifolds for one-step schemes. They also form the basis for studying saddle-node homoclinic orbits under discretization.

On regularity of linear systems with a degenerate matrix by the derivative

New sufficient conditions for the existence of invariant manifolds are proved for linear systems with degenerate matrix by the derivative.

Smooth invariant foliations in infinite dimensional spaces

One of the most useful properties of dynamical systems is the existence of invariant manifolds and their invariant foliations near an equilibrium or a periodic orbit. These manifolds and foliations serve as a convenient setting to describe the qualitative behavior of the local flows, and in many cases they are useful tools for technical estimates which facilitate the study of the local bifurcation diagram (see [6]). Many other important concepts in dynamical systems are closely related to the invariant manifolds and foliations. In finite dimensional space, the relations among invariant manifolds, invariant foliations, l-lemma, linearization, and homoclinic bifurcation have been studied in [ll]. It is well known that if each leaf is used as a coordinate, the original system is completely decoupled and the linearization follows easily (for example, see [27, 221). As a motivation, let us consider a linear system in Rmfn

Intermittent phenomena in adaptive systems: A case study

Temporary and permanent intermittent behavior in a simple adaptive control system are studied based on the existence of invariant manifold and periodic solutions, as well as on averaging theory. The role of self-oscillations, which are periodic solutions of the closed-loop system not directly determined by the external excitation, is enhanced. Self-oscillations may constitute the limit set of intermittent solutions. Intermittency, which may persist with P.E. reference signals, is intimately related to the relative frequency content in the reference and the perturbations. The concepts are illustrated with simulations.

Intermittency and Self-Oscillations in Discrete Time Adaptive Systems

The basic causes of intermittency in deterministic adaptive control systems are discussed. The analysis is based on the existence of invariant manifold and limit sets (as periodic solutions), as well as on averaging theory. Temporary and permanent intermittency can thus be foreseen. It is shown that permanent intermittency is associated with a very special kind of periodic solution of the closed-loop system called self-oscillations. Moreover, under certain conditions intermittent solutions may asymptotically converge toward these periodic solutions which are not directly determined by the external excitation. An important by-product of the analysis is that a Persistently Exciting reference signal does not suffice to impede intermittency.

The local solvability of functional equations and the existence of invariant manifolds

The local solvability of functional equations and the existence of invariant manifolds

Existence of invariant manifolds of systems with delay

Existence of invariant manifolds of systems with delay