Input-Output Mechanism of the Discrete Chaos Extension

In this chapter the extension of chaos in difference equations is discussed. The theoretical results are based on chaos in the sense of Devaney and period-doubling cascades. The existence of homoclinic and heteroclinic orbits is rigorously proved, and a theoretical control technique for the extended chaos is proposed. The results are supported with the aid of simulations. Arbitrarily high-dimensional chaotic discrete-time dynamical systems can be designed by means of the presented technique. A discrete gonorrhea model is utilized to generate chaotic behavior in population dynamics.


Input-Output Mechanism of the Discrete Chaos Extension
Akhmet, Marat (Springer International Publishing, 2016-01-01)
In this paper, the authors analyze the extension of chaotic dynamics to a particular transformation of a discrete dynamical system. The main result consists in showing that starting from a chaotic map (input), the state variable obtained by adding a linear map and a continuous function of the chaotic state (output) is chaotic as well. These results are based on Devaney's definition of chaos and, for this purpose, this definition is extended to collections of sequences. Several examples are presented to show...
Persistence of Li-Yorke chaos in systems with relay
Akhmet, Marat; Kashkynbayev, Ardak (University of Szeged, 2017-01-01)
It is rigorously proved that the chaotic dynamics of the non-smooth system with relay function is persistent even if a chaotic perturbation is applied. We consider chaos in a modified Li-Yorke sense such that there are infinitely many almost periodic motions embedded in the chaotic attractor. It is demonstrated that the system under investigation possesses countable infinity of chaotic sets of solutions. An example that supports the theoretical results is represented. Moreover, a chaos control procedure bas...
Almost periodicity in chaos
Akhmet, Marat (2018-01-01)
Periodicity plays a significant role in the chaos theory from the beginning since the skeleton of chaos can consist of infinitely many unstable periodic motions. This is true for chaos in the sense of Devaney [1], Li-Yorke [2] and the one obtained through period-doubling cascade [3]. Countable number of periodic orbits exist in any neighborhood of a structurally stable Poincaré homoclinic orbit, which can be considered as a criterion for the presence of complex dynamics [4]-[6]. It was certified by Shilniko...
The Feynman path integral quantization of constrained systems
Muslih, S; Guler, Y (1997-01-01)
The Feynman path integral for constrained systems is constructed using the canonical formalism introduced by Guler. This approach is applied to a free relativistic particle and Christ-Lee model.
Multisymplectic Schemes for the Complex Modified Korteweg-de Vries Equation
AYDIN, AYHAN; Karasözen, Bülent (2008-09-20)
In this paper, the multisymplectic formulation of the CMKdV(complex modified Korteweg-de Vries equation) is derived. Based on the multisymplectic formulation, the eight-point multisymplectic Preissman scheme and a linear-nonlinear multisymplectic splitting scheme are developed. Both methods are compared numerically with respect to the conservation of local and global quantities of the CMKdV equation.
Citation Formats
M. Akhmet, “Input-Output Mechanism of the Discrete Chaos Extension,” 2014, vol. 15, Accessed: 00, 2020. [Online]. Available: