Unique Periodic Orbits for Delayed Positive Feedback and the Global Attractor

The delay differential equation $$\dot x(t) = - \mu x(t) + f(x(t - r)), r = r(x(t))$$ with μ>0 and smooth real functions f, r satisfying f(0)=0, f′<0, and r(0)=1 models a system governed by state-dependent delayed negative feedback and instantaneous damping. For a suitable R≥1 the solutions generate a semiflow F on a compact subset LK of C([−R, 0], ℝ). F leaves invariant the subset S of φ∈LK with at most one sign change on all subintervals of [−R, 0] of length one. The induced semiflow on S has a global attractor \(A.A\)\{0} coincides with the set of segments of bounded globally defined slowly oscillating solutions. If \(A\)≠{0}, then \(A\) is homeomorphic to the closed unit disk, and the unit circle corresponds to a periodic orbit.

We consider a system of delay differential equations with a delayed excitatory feedback loop and instantaneous damping. We develop the Cao–Krisztin–Walther technique and establish the existence and uniqueness of periodic solutions with prescribed oscillation frequencies (characterized by the values of a discrete Lyapunov functional). We then use the Poincare-Bendixson theorem due to Mallet-Paret and Sell to show that the global attractor of such a system is the union of the unstable sets of stationary points and periodic orbits.

In this paper, some properties of a strictly A A -coupled-expanding map in compact subsets of a metric space are studied, where A A is a transition matrix. It is shown that this map has a compact invariant set on which it is topologically semi-conjugate to the subshift for A A . If the subshift for A A has positive topological entropy, then the map is chaotic in the sense of Li-Yorke. Moreover, in the one-dimensional case, the map is at most two-to-one conjugate to the subshift for A A and chaotic in the sense of Devaney.

In this article we study the operator version of a first order in time partial differential inclusion as well as its time discretization obtained by an implicit Euler scheme. This technique, known as the Rothe method, yields the semidiscrete trajectories that are proved to converge to the solution of the original problem. While both the time continuous problem and its semidiscretization can have nonunique solutions we prove that, as times goes to infinity, all trajectories are attracted towards certain compact and invariant sets, so-called global attractors. We prove that the semidiscrete attractors converge upper-semicontinuously to the global attractor of time continuous problem.

Real functions for representation of rigid solids

In this paper, we study some features of global behavior of the four‐dimensional superficial bladder cancer model with Bacillus Calmette‐Guérin (BCG) immunotherapy described by Bunimovich‐Mendrazitsky et al. in 2007 with the help of localization analysis of its compact invariant sets. Its dynamics is defined by the BCG treatment and by densities of three cells populations: effector cells, tumor infected cells by BCG, and tumor uninfected cells. We find upper bounds for ultimate dynamics of the whole state vector in the positive orthant and also under condition that there are no uninfected tumor cells. Further, we prove the existence of the bounded positively invariant domain in both of these two situations. Finally, by using these assertions, we derive our main result: sufficient conditions of global asymptotic stability of the tumor‐free equilibrium point in the positive orthant. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Bounds for a domain containing all compact invariant sets of the system describing the laser–plasma interaction

In this paper, using the Gromov–Hausdorff distances between two global attractors (which may be in disjoint phase spaces) and two semi‐dynamical systems introduced by Lee et al. (2020), we consider the continuous dependence of the global attractors and the stability of the semi‐dynamical systems on global attractors induced by the Brinkman–Forchheimer equation under variation of the domain. The results of this paper improve on previous results, which can compare any two systems in different phase spaces without the process of “pull‐backing” the perturbed systems to the original domain.

For an autonomous model of the motion of a nonlinear viscous fluid, we study the limiting behavior of its weak solutions as time tends to infinity. Namely, the existence of weak solutions on the positive half-axis is established, the trajectory space corresponding to the solutions of this model is determined, and the existence of a minimum trajectory attractor and, then, a global attractor in the phase space is proved using the theory of trajectory spaces. Thus, it turns out that whatever the initial state of the system describing the model is, it is “forgotten” over time and the global attractor is infinitely approached.

Global attractors for singular perturbations of the Cahn–Hilliard equations

It is well known that simple and complex dynamics of a nonlinear system are separated by a localizing set that contains all compact invariant sets and corresponds to a function in the phase space of the system. This separation means that, in the complement of the localizing set, the trajectory behavior of the system admits a standard description in the form of several variants, while, in the localizing set, the trajectory behavior of the system can be very complex, for example, chaotic. Domains in the localizing set with a similar standard behavior are indicated, and an external estimate of the domain with complex dynamics is sequentially refined. Two examples are considered.

We study the delay dieren tial equation _ x(t) = x (t) +f(x(t 1)) with 0 and C 1 -smooth real functions f satisfying f(0) = 0 and f 0 < 0. For a set of and f, we determine the number of periodic orbits, and describe the structure of the global attractor as the union of the strong unstable sets of the periodic orbits and of the stationary point 0.

2D piecewise affine models approximate real continuous dynamics up to invariant sets

Strange attractors and wandering domains near a homoclinic cycle to a bifocus

Global attractivity and oscillation in a nonlinear periodic delay difference equation