On Local Bifurcations in Neural Field Models with Transmission Delays

Neural field models with transmission delay may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results

Delays in Open String Field Theory

We study the dynamics of light-like tachyon condensation in a linear dilaton background using level-truncated open string field theory. The equations of motion are found to be delay differential equations. This observation allows us to employ well-established mathematical methods that we briefly review. At level zero, the equation of motion is of the so-called retarded type and a solution can be found very efficiently, even in the far light-cone future. At levels higher than zero however, the equations are not of the retarded type. We show that this implies the existence of exponentially growing modes in the non-perturbative vacuum, possibly rendering light-like rolling unstable. However, a brute force calculation using exponential series suggests that for the particular initial condition of the tachyon sitting in the false vacuum in the infinite light-cone past, the rolling is unaffected by the unstable modes and still converges to the non-perturbative vacuum, in agreement with the solution of Hellerman and Schnabl. Finally, we show that the growing modes introduce non-locality mixing present with future, and we are led to conjecture that in the infinite level limit, the non-locality in a light-like linear dilaton background is a discrete version of the smearing non-locality found in covariant open string field theory in flat space.Comment: 48 pages, 14 figures. v2: References added; Section 4 augmented by a discussion of the diffusion equation; discussion of growing modes in Section 4 slightly expande

Galerkin approximations for the optimal control of nonlinear delay differential equations

Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE's solution. Optimal controls computed from the Pontryagin's maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation.Comment: 29 pages. This is a sequel of the arXiv preprint arXiv:1704.0042

Flat systems, equivalence and trajectory generation

Flat systems, an important subclass of nonlinear control systems introduced via differential-algebraic methods, are defined in a differential geometric framework. We utilize the infinite dimensional geometry developed by Vinogradov and coworkers: a control system is a diffiety, or more precisely, an ordinary diffiety, i.e. a smooth infinite-dimensional manifold equipped with a privileged vector field. After recalling the definition of a Lie-Backlund mapping, we say that two systems are equivalent if they are related by a Lie-Backlund isomorphism. Flat systems are those systems which are equivalent to a controllable linear one. The interest of such an abstract setting relies mainly on the fact that the above system equivalence is interpreted in terms of endogenous dynamic feedback. The presentation is as elementary as possible and illustrated by the VTOL aircraft

Delay Equations and Radiation Damping

Starting from delay equations that model field retardation effects, we study the origin of runaway modes that appear in the solutions of the classical equations of motion involving the radiation reaction force. When retardation effects are small, we argue that the physically significant solutions belong to the so-called slow manifold of the system and we identify this invariant manifold with the attractor in the state space of the delay equation. We demonstrate via an example that when retardation effects are no longer small, the motion could exhibit bifurcation phenomena that are not contained in the local equations of motion.Comment: 15 pages, 1 figure, a paragraph added on page 5; 3 references adde

Controlling Mackey--Glass chaos

The Mackey--Glass equation, which was proposed to illustrate nonlinear phenomena in physiological control systems, is a classical example of a simple looking time delay system with very complicated behavior. Here we use a novel approach for chaos control: we prove that with well chosen control parameters, all solutions of the system can be forced into a domain where the feedback is monotone, and by the powerful theory of delay differential equations with monotone feedback we can guarantee that the system is not chaotic any more. We show that this domain decomposition method is applicable with the most common control terms. Furthermore, we propose an other chaos control scheme based on state dependent delays.Comment: accepted in Chaos: An Interdisciplinary Journal of Nonlinear Scienc