Chaotification of Impulsive Systems by Perturbations

In this paper, we present a new method for chaos generation in nonautonomous impulsive systems. We prove the presence of chaos in the sense of Li-Yorke by implementing chaotic perturbations. An impulsive Duffing oscillator is used to show the effectiveness of our technique, and simulations that support the theoretical results are depicted. Moreover, a procedure to stabilize the unstable periodic solutions is proposed

Replication of period-doubling route to chaos in impulsive systems

In this study, we investigate the dynamics of impulsive systems driven by a chaotic system. It is shown that the response impulsive system replicates the sensitivity and the period-doubling cascade of the drive. Illustrative examples that support the theoretical results are provided

Replication of period-doubling route to chaos in impulsive systems

In this study, we investigate the dynamics of impulsive systems driven by a chaotic system. It is shown that the response impulsive system replicates the sensitivity and the period-doubling cascade of the drive. Illustrative examples that support the theoretical results are provided

Topological entropy for impulsive differential equations

A positive topological entropy is examined for impulsive differential equations via the associated Poincaré translation operators on compact subsets of Euclidean spaces and, in particular, on tori. We will show the conditions under which the impulsive mapping has the forcing property in the sense that its positive topological entropy implies the same for its composition with the Poincaré translation operator along the trajectories of given systems. It allows us to speak about chaos for impulsive differential equations under consideration. In particular, on tori, there are practically no implicit restrictions for such a forcing property. Moreover, the asymptotic Nielsen number (which is in difference to topological entropy a homotopy invariant) can be used there effectively for the lower estimate of topological entropy. Several illustrative examples are supplied

Complex Dynamics of an Adnascent-Type Game Model

The paper presents a nonlinear discrete game model for two oligopolistic firms whose products are adnascent. (In biology, the term adnascent has only one sense, “growing to or on something else,” e.g., “moss is an adnascent plant.” See Webster's Revised Unabridged Dictionary published in 1913 by C. & G. Merriam Co., edited by Noah Porter.) The bifurcation of its Nash equilibrium is analyzed with Schwarzian derivative and normal form theory. Its complex dynamics is demonstrated by means of the largest Lyapunov exponents, fractal dimensions, bifurcation diagrams, and phase portraits. At last, bifurcation and chaos anticontrol of this system are studied

Perturbed Li-Yorke homoclinic chaos

It is rigorously proved that a Li–Yorke chaotic perturbation of a system with a homoclinic orbit creates chaos along each periodic trajectory. The structure of the chaos is investigated, and the existence of infinitely many almost periodic orbits out of the scrambled sets is revealed. Ott–Grebogi–Yorke and Pyragas control methods are utilized to stabilize almost periodic motions. A Duffing oscillator is considered to show the effectiveness of our technique, and simulations that support the theoretical results are depicted