Journal of Nonlinear Dynamics

Volume 2015 (2015), Article ID 257923, 13 pages

http://dx.doi.org/10.1155/2015/257923

## The Dynamics of a Cubic Nonlinear System with No Equilibrium Point

Physics Department, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece

Received 28 May 2015; Revised 2 July 2015; Accepted 9 July 2015

Academic Editor: Huai-Ning Wu

Copyright © 2015 J. O. Maaita et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study the dynamics of a three-dimensional nonlinear system with cubic nonlinearity and no equilibrium points with the use of Poincaré maps, Lyapunov Exponents, and bifurcations diagrams. The system has rich dynamics: chaotic behavior, regular orbits, and 3-tori periodicity. Finally, the proposed system is also reported to verify electronic circuit modeling feasibility.

#### 1. Introduction

A lot of work has been done in the field of dynamical system and many systems (Lorenz, Chua, Duffing, Van der pol, Sprott, and many others) have been exhaustively studied. The dynamics of such systems are well known and their properties are used in mechanical and electrical applications and experiments [1–4].

In the last two decades a new field of dynamical systems has been “discovered” and attracts the attention of scientists: dynamical systems with no equilibrium points or with conjugate equilibrium points.

Equilibrium points are important because their stability determines the dynamics of the system [5–7]. In particular, a stable equilibrium point is a point for which the trajectories around it remain close for small perturbations. On the other hand, an unstable equilibrium point is a point for which the trajectories around it escape even for small perturbations and remove the system from its initial state.

Equilibrium points are connected with criteria and theorems that determine the existence of chaotic behavior of a system (Melnikov function, Shilnikov chaos, etc.) [8]. The loss of equilibrium points means that the conventional Shilnikov criteria cannot be applied to prove the chaos in the flow.

A dynamical system with no equilibrium points is categorized as chaotic system with hidden attraction because the loss of equilibrium points means that its basin of attraction does not intersect with small neighborhoods of any equilibrium points.

Sprott (1994) was the first to introduce a simple flow with no equilibrium points [2]. Since then many researcher have introduced many systems with no equilibrium points or with conjugate equilibrium points [9–15].

In this work we study a modified version of the initial Sprott model with a cubic nonlinearity and a constant parameter .

We made a numerical study of the system and used tools such as Poincaré maps, Lyapunov Characteristic Exponents, bifurcations diagrams [16, 17].

The system has rich dynamics. In general it has a chaotic behavior but for certain initial values and different values for the parameter the system may have regular orbits (quasiperiodic or periodic). It is important to note here that for some values of initial conditions we detected transient hyperchaotic behavior of the system.

In Section 2 we analyze the system and present the behavior of the system for different values of the constant parameter and different values of the initial conditions. In Section 3 we present an electronic circuit that implements the above nonlinear system and finally conclude in Section 4 of the paper.

#### 2. Analysis

We study a nonlinear system with cubic nonlinearity: where is the parameter of the system. As it is obvious, since , the system has no equilibrium point.

We used many tools to analyze numerically the above system: Bifurcation diagrams, Poincaré maps (for , and Lyapunov Characteristic Exponents. We simulated the system for many thousands of different initial conditions and different values of the parameter .

The numerical work was done with the help of Mathematica and the programming Languages C and True Basic by using the classical fourth-order Runge-Kutta method.

As we see from the bifurcations diagrams (Figure 1) the system is in general chaos. This is confirmed by the calculations of the Lyapunov Characteristic Exponents (LCEs) (Figure 2) where we see that for many different initial conditions and different values of the parameter there are one positive LCE, one negative LCE, and one LCE that equals zero. This confirms the chaotic behavior of the system for these initial conditions.