Center manifolds without a phase space

Abstract

We establish center manifold theorems that allow one to study the bifurcation of small solutions from a trivial state in systems of functional equations posed on the real line. The class of equations includes most importantly nonlinear equations with nonlocal coupling through convolution operators as they arise in the description of spatially extended dynamics in neuroscience. These systems possess a natural spatial translation symmetry but local existence or uniqueness theorems for a spatial evolution associated with this spatial shift or even a well motivated choice of phase space for the induced dynamics do not seem to be available, due to the infinite range forward-and backward-coupling through nonlocal convolution operators. We perform a reduction relying entirely on functional analytic methods. Despite the nonlocal nature of the problem, we do recover a local differential equation describing the dynamics on the set of small bounded solutions, exploiting that the translation invariance of the original problem induces a flow action on the center manifold. We apply our reduction procedure to problems in mathematical neuroscience, illustrating in particular the new type of algebra necessary for the computation of Taylor jets of reduced vector fields.Center-manifold reductions have become a central tool to the analysis of dynamical systems. The very first results on center manifolds go back to the pioneering works of Pliss [21] and Kelley [16] in the finitedimensional setting. In the simplest context, one studies differential equations in the vicinity of a nonhyperbolic equilibrium,The basic reduction establishes that the set of small bounded solutions u(t), t ∈ R, sup |u(t)| < δ ≪ 1, is pointwise contained in a manifold, that is, u(t) ∈ W c for all t. This manifold is a subset of phase space, W c ⊂ R n , contains the origin, 0 ∈ W c , and is tangent to E c , the generalized eigenspace associated with purely imaginary eigenvalues of f ′ (0). As a consequence, the flow on W c can be projected onto E c , to yield a reduced vector field. The reduction to this lower-dimensional ODE then allows one to describe solutions qualitatively, even explicitly in some cases. Of course, the method applies to higher-order differential equation, which one simply writes as first-order equation in a canonical fashion. Extensions to infinite-dimensional dynamical systems were pursued soon after; see for instance [12].Starting with the work of Kirchgässner [17], such reductions have been extended to systems with u ∈ X , a Banach space, where the initial value problem is not well-posed: For most initial conditions u 0 , there does not exist a local solution u(t), 0 ≤ t < δ, say. Local solutions do exist however for all initial conditions on a finite-dimensional center-manifold, and much of the theory is quite analogous to the finite-dimensional case; see [25]. In these theories, one can typically split the phase space in infinite-dimensional linear spaces where solutions to the linearized equation either decay or grow, and a finite-d...

Spatial Hamiltonian Identities for Nonlocally Coupled Systems

“…and v has support supp v Ă R r´1, 1s, then ωpv, wq " 0, for any w. We are however able to show that ω is locally non-degenerate on a center manifold, giving a (non-canonical) symplectic structure and reduced Hamiltonian vector fields and thereby extending the results from [20]. Section 4 considers the effect of (spatially) dissipative terms, such as equations of the forḿ…”

“…This would put the nonlocal equations in the context of Hamiltonian PDEs, as considered for example in [29], possibly allowing for global Lyapunov center theorems and KAM theory. A first step in this direction could be an extension of center manifold theory [20] to neighborhoods of periodic solutions, comparable to [35], where new questions arise, both in terms of regularity constraints necessary for the reduction procedure and in terms of the reduced dynamics.…”

“…A first approach casts systems with finite-range interactions as ill-posed dynamical systems on an infinite-dimensional phase space, much like elliptic equations posed on a cylinder, and uses dynamical systems techniques, in particular invariant manifolds, to study the dynamics; see for instance [24,25,26,32]. A different avenue evokes dynamical systems techniques without ever setting up a phase space for a pointwise description, but rather adapting techniques from dynamical systems by reducing to basic functional analytic aspects; see [19] for geometric singular perturbation techniques, [20] for center manifold reductions, and [40] for bifurcation methods. The present work can be viewed as continuing this latter approach, providing a framework for exploiting Hamiltonian and symplectic concepts.…”

Bifurcation to Coherent Structures in Nonlocally Coupled Systems

“…The restriction to kernels with rational Fourier transform is clearly restrictive, excluding for instance Gaussians, and one naturally wonders if similar results hold outside of this class. The more recent results in [13] answer this question in the affirmative, for x ∈ R and K, K exponentially localized, such thatK is analytic in a strip of the complex plane | Re ξ| < η.…”

Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions

“…In [22], Faye and Scheel studied equations such as (1.9) for discretisations with infinite-range interactions featuring exponential decay in the coupling strength. They circumvented the need to use a state space as in [35], which enabled them to construct pulses to (1.9) for arbitrary discretisation distance h. Very recently [23], they developed a center manifold approach that allows bifurcation results to be obtained for neural field equations. In this paper, we also construct pulse solutions to equations such as (1.9), but under weaker assumptions on the decay rate of the couplings.…”