Periodic solutions for second-order difference equations with quadratic–supquadratic condition

In this paper, we consider the existence of multiple periodic solutions for a class of second-order difference equations with quadratic–supquadratic growth condition at infinity. Moreover, we give three examples to illustrate our main result.

Bistable and oscillatory dynamics of Nicholson's blowflies equation with Allee effect

<p style='text-indent:20px;'>The bistable dynamics of a modified Nicholson's blowflies delay differential equation with Allee effect is analyzed. The stability and basins of attraction of multiple equilibria are studied by using Lyapunov-LaSalle invariance principle. The existence of multiple periodic solutions are shown using local and global Hopf bifurcations near positive equilibria, and these solutions generate long transient oscillatory patterns and asymptotic stable oscillatory patterns.</p>

Multiple Periodic Solutions of a Class of Fractional Laplacian Equations

In this paper, we study the existence of multiple periodic solutions for the following fractional equation:(-\Delta)^{s}u+F^{\prime}(u)=0,\qquad u(x)=u(x+T)\quad x\in\mathbb{R}.For an even double-well potential, we establish more and more periodic solutions for a large period T. Without the evenness of F we give the existence of two periodic solutions of the problem. We make use of variational arguments, in particular Clark’s theorem and Morse theory.

Multiple Periodic Solutions and Fractal Attractors of Differential Equations with n-Valued Impulses

Ordinary differential equations with n-valued impulses are examined via the associated Poincaré translation operators from three perspectives: (i) the lower estimate of the number of periodic solutions on the compact subsets of Euclidean spaces and, in particular, on tori; (ii) weakly locally stable (i.e., non-ejective in the sense of Browder) invariant sets; (iii) fractal attractors determined implicitly by the generating vector fields, jointly with Devaney’s chaos on these attractors of the related shift dynamical systems. For (i), the multiplicity criteria can be effectively expressed in terms of the Nielsen numbers of the impulsive maps. For (ii) and (iii), the invariant sets and attractors can be obtained as the fixed points of topologically conjugated operators to induced impulsive maps in the hyperspaces of the compact subsets of the original basic spaces, endowed with the Hausdorff metric. Five illustrative examples of the main theorems are supplied about multiple periodic solutions (Examples 1–3) and fractal attractors (Examples 4 and 5).

Existence of Multiple Periodic Solutions for Cubic Nonautonomous Differential Equation

In this article, approaches to estimate the number of periodic solutions of ordinary differential equation are considered. Conditions that allow determination of periodic solutions are discussed. We investigated focal values for first-order differential nonautonomous equation by using the method of bifurcation analysis of periodic solutions from a fine focus Z=0. Keeping in focus the second part of Hilbert’s sixteenth problem particularly, we are interested in detecting the maximum number of periodic solution into which a given solution can bifurcate under perturbation of the coefficients. For some classes like C7,7,C8,5,C8,6,C8,7, eight periodic multiplicities have been observed. The new formulas ξ10 and ϰ10 are constructed. We used our new formulas to find the maximum multiplicity for class C9,2. We have succeeded to determine the maximum multiplicity ten for class C9,2 which is the highest known multiplicity among the available literature to date. Another challenge is to check the applicability of the methods discussed which is achieved by presenting some examples. Overall, the results discussed are new, authentic, and novel in its domain of research.

Dynamic Complexity in a Prey-Predator Model with State-Dependent Impulsive Control Strategy

In this paper, an ecological model described by a couple of state-dependent impulsive equations is studied analytically and numerically. The theoretical analysis suggests that there exists a semitrivial periodic solution under some conditions and it is globally orbitally asymptotically stable. Furthermore, using the successor function, we study the existence, uniqueness, and stability of order-1 periodic solution, and the boundedness of solution is also presented. The relationship between order-k successor function and order-k periodic solution is discussed as well, thereby giving the existence condition of an order-3 periodic solution. In addition, a series of numerical simulations are carried out, which not only support the theoretical results but also show the complex dynamics in the model further, for example, the coexistence of multiple periodic solutions, chaos, and period-doubling bifurcation.