Implementation of multi-step differential transformation method for hyperchaotic Rossler system

Abstract

The behavior of dynamical systems such as circuits, mechanical devices, population growth, fluid dynamics, and weather are studied in a field of mathematics known as chaos theory. A chaotic system shows sensitivity to any tiny change in the initial conditions, so it has unpredictable behavior. In other words, it is impossible to predict the future behavior of a chaos system unless the initial conditions are entirely known and accurate. The first chaotic system, known as the Lorenz system, was discovered in 1963 by Lorenz for solving equations describing atmospheric flows [1]. Inspiring from Lorenz, Otto Rossler proposed a chaos system, known as Rossler system or Rossler attractor, resulted from studying a chemical reaction system[2]. This system contains three prototype first-order diferential equations with three dynamical variables in defining the phase space and three parameters. This system has been thoroughly studied by many researchers; (see [3, 4, 5] for more details). Continuing his work, In 1979 Rossler proposed another dynamical system which was made of four first order differential equations as following (see [6]).

Hyperchaotic Rossler system is the first four dimensional hyperchaotic system. This system contains four prototype ordinary differential equations as the following