Dynamic Equations, Control Problems on Time Scales, and Chaotic Systems

Abstract

The unification of integral and differential calculus with the calculus of finite differences has been rendered possible by providing a formal structure to study hybrid discrete-continuous dynamical systems besides offering applications in diverse fields that require simultaneous modeling of discrete and continuous data concerning dynamic equations on time scales. Therefore, the theory of time scales provides a unification between the calculus of the theory of difference equations with the theory of differential equations. In addition, it has become possible to examine diverse application problems more precisely by the use of dynamical systems on time scales whose calculus is made up of unification and extension as the two main features. In the meantime, chaos theory comes to the foreground as a concept that a small change can result in a significant change subsequently, and thus, it is suggested that nonlinear dynamical systems which are apparently random are actually deterministic from simpler equations. Consequently, diverse techniques have been devised for chaos control in physical systems that change across time-dependent spatial domains. Accordingly, this Editorial provides an overview of dynamic equations, time-variations of the system, difference and control problems which are bound by chaos theory that is capable of providing a new way of thinking based on measurements and time scales. Furthermore, providing models that can be employed for chaotic behaviors in chaotic systems is also attainable by considering the arising developments and advances in measurement techniques, which show that chaos can offer a renewed perspective to proceed with observational data by acting as a bridge between different domains.

Keywords

• Dynamic equations on time scales
• Chaotic systems
• Nonlinearity and chaos
• Unification and extension
• Control problems
• Time variations of systems

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