2014/03/17 by Hassan Sedaghat, Sedaghat, H.
Computer Science · Mathematics · Physics and Astronomy · #34A30 #34A34 #37B55 #39A10 #Advanced Differential Equations and Dynamical Systems #Chaos control and synchronization #Chaotic Dynamics (nlin.CD) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Dynamics and Pattern Formation
paper · pdf · doi:10.48550/arxiv.1403.3995
openalex publication_date 2014/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A typical system of k difference (or differential) equations can be compressed, or folded into a difference (or ordinary differential) equation of order k. Such foldings appear in control theory as the canonical forms of the controllability matrices. They are also used in the classification of systems of three nonlinear differential equations with chaotic flows by examining the resulting jerk functions. The solutions of the higher order equation yield one of the components of the system's k-dimensional orbits and the remaining components are determined from a set of associated passive equations. The folding algorithm uses a sequence of substitutions and inversions along with index shifts (for difference equations) or higher derivatives (for differential equations). For systems of two difference or differential equations this compression process is short and in some cases yields second-order equations that are simpler than the original system. For all systems, the folding algorithm yields detailed amount of information about the structure of the system and the interdependence of its variables. As with two equations, some special cases where the derived higher order equation is simpler to analyze than the original system are considered.