Parallel Dynamical Systems: Characterization, Chaos Preservation, and Synchronization Applications

Document Type : Original Article

Authors
Jafar hasanmojaver 1 ΒΆ
Ali Ebadian 1
Bahman Rezaei 1
sohrab karimi 2
1 Department of Mathematics, Urmia University
2 Department of Mathematics, Ferdowsi University of Mashhad
10.22034/maco.5.2.12
Abstract
This paper introduces and rigorously analyzes the concept of parallel dynamical systems, a novel framework for understanding and achieving synchronization
in chaotic systems. For any given dynamical system defined by ordinary differential
equations, we construct a corresponding parallel system through a scaling transformation of phase space variables. We prove rigorously that the parallel system
preserves all fundamental chaotic properties of the original system, including sensitive dependence on initial conditions, topological transitivity, and density of periodic
orbits. This theoretical foundation enables a controller-free synchronization method
where systems naturally synchronize through appropriate initial condition scaling.
Numerical simulations of the Lorenz system validate our theoretical predictions,
demonstrating perfect synchronization and practical applicability

Keywords

Subjects

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Mathematical Analysis and Convex Optimization
Volume 5, Issue 2
December 2025