We consider a multifunction $F:T\times X\to 2^E$, where T, X and E are separable metric spaces, with E complete. Assuming that F is jointly measurable in the product and a.e. lower semicontinuous in the second variable, we establish the existence of a selection for F which is measurable with respect to the first variable and a.e. continuous with respect to the second one. Our result is in the spirit of [11], where multifunctions of only one variable are considered.

In this paper we consider parametric nonlinear evolution inclusions driven by time-dependent subdifferentials. First we prove some continuous dependence results for the solution set (of both the convex and nonconvex problems) and for the set of solution-selector pairs (of the convex problem). Then we derive a continuous version of the “Filippov-Gronwall” inequality and using it, we prove the parametric relaxation theorem. An example of a parabolic distributed parameter system is also worked out...

This paper presents two theorems for designing controllers to achieve directional partial generalized synchronization (PGS) of two independent (chaotic) differential equation systems or two independent (chaotic) discrete systems. Two numerical simulation examples are given to illustrate the effectiveness of the proposed theorems. It can be expected that these theorems provide new tools for understanding and studying PGS phenomena and information encryption.

Further extension of the Levinson transformation theory is performed for partially dissipative periodic processes via the fixed point index. Thus, for example, the periodic problem for differential inclusions can be treated by means of the multivalued Poincaré translation operator. In a certain case, the well-known Ważewski principle can also be generalized in this way, because no transversality is required on the boundary.

Let $f(t,x)$ be a vector valued function almost periodic in $t$ uniformly for $x$, and let $\mathrm{m}od\left(f\right)=L_1\oplus L_2$ be its frequency module. We say that an almost periodic solution $x\left(t\right)$ of the system $$\dot{x}=f(t,x),t\in ℝ,x\in D\subset {ℝ}^n$$ is irregular with respect to $L_2$ (or partially irregular) if $(\mathrm{m}od\left(x\right)+L_1)\cap L_2=\left\{0\right\}$. Suppose that $f(t,x)=A\left(t\right)x+X(t,x),$ where $A\left(t\right)$ is an almost periodic $(n\times n)$-matrix and $\mathrm{m}od\left(A\right)\cap \mathrm{m}od\left(X\right)=\left\{0\right\}.$ We consider the existence problem for almost periodic irregular with respect to $\mathrm{m}od\left(A\right)$ solutions of such system. This problem is reduced to a similar problem for a system of smaller dimension, and sufficient conditions...

In this paper, the problem of passivity analysis for a class of uncertain stochastic neural networks with mixed delays and impulsive control is investigated. The mixed delays include constant delay in the leakage term, discrete and distributed delays. The discrete delays are assumed to be time-varying and belong to a given interval, which means that the lower and upper bounds of interval time-varying delays are available. By using Lyapunov stability theory, stochastic analysis, linear matrix inequality...

This paper is concerned with the structure of asymptotically stabilizing feedbacks for a nonlinear control system on ${ℝ}^n$. We first introduce a family of discontinuous, piecewise smooth vector fields and derive a number of properties enjoyed by solutions of the corresponding O.D.E's. We then define a class of “patchy feedbacks” which are obtained by patching together a locally finite family of smooth controls. Our main result shows that, if a system is asymptotically controllable at the origin,...

We implement a singularity theory approach, the path formulation, to classify ${𝔻}_3$-equivariant bifurcation problems of corank 2, with one or two distinguished parameters, and their perturbations. The bifurcation diagrams are identified with sections over paths in the parameter space of a ${𝔻}_3$-miniversal unfolding $F_0$ of their cores. Equivalence between paths is given by diffeomorphisms liftable over the projection from the zero-set of $F_0$ onto its unfolding parameter space. We apply our results to degenerate...

We present a three species model describing the degradation of substrate by two competing populations of microorganisms in a marine sediment. Considering diffusion to be the main transport process, we obtain a reaction diffusion system (RDS) which we study in terms of spontaneous pattern formation. We find that the conditions for patterns to evolve are likely to be fulfilled in the sediment. Additionally, we present simulations that are consistent with experimental data from the literature. We...

We study the coexistence of multiple periodic solutions for an analogue of the integrate-and-fire neuron model of two-neuron recurrent inhibitory loops with delayed feedback, which incorporates the firing process and absolute refractory period. Upon receiving an excitatory signal from the excitatory neuron, the inhibitory neuron emits a spike with a pattern-related delay, in addition to the synaptic delay. We present a theoretical framework to view...