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.

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 [

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 [

Equilibrium points are connected with criteria and theorems that determine the existence of chaotic behavior of a system (Melnikov function, Shilnikov chaos, etc.) [

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 [

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 [

The system has rich dynamics. In general it has a chaotic behavior but for certain initial values and different values for the parameter

In Section

We study a nonlinear system with cubic nonlinearity:

We used many tools to analyze numerically the above system: Bifurcation diagrams, Poincaré maps (for

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

Bifurcation diagrams for different initial conditions taken for

Lyapunov Characteristic Exponents for different initial conditions.

Furthermore, from the study of the LCEs we detected that for many initial conditions and values of

Lyapunov Characteristic Exponents for different initial conditions where transient hyperchaotic behavior is detected.

In what follows we will present three examples of the dynamics of the system, for different values of the parameter

Poincaré section for

For example, for the initial conditions (

Lyapunov Characteristic Exponents for

Trajectories for

Trajectories on the 3D space

For the initial conditions (

Lyapunov Characteristic Exponents for

Trajectories for

Trajectories on the 3D space

Poincaré section for

For the initial conditions (

Lyapunov Characteristic Exponents for

Trajectories for

Trajectories on the 3D space

Also for the initial conditions (

Lyapunov Characteristic Exponents for

Trajectories for

Trajectories on the 3D space

Poincaré section for

For the initial conditions (

Lyapunov Characteristic Exponents for

Trajectories for

Trajectories on the 3D space

Circuital design of chaotic systems plays a crucial role in the field of nonlinear science not only for providing a simple experimental confirmation of phenomena related to nonlinear dynamics but also due to its applications in many engineering approaches, such as secure communication, signal processing, random bit generator, or path planning for autonomous mobile robot [

In this work, an electronic circuit (Figure

The schematic of the circuit that emulates the proposed dynamical system (

The designed circuit is implemented in the electronic simulation package Cadence OrCAD and the obtained results are displayed in Figures

Simulation phase portraits produced by the OrCAD and the respective ones produced by the system’s arithmetic integration, for

Simulation phase portraits produced by the OrCAD and the respective ones produced by the system’s arithmetic integration, for

We study a nonlinear system with cubic nonlinearity and no equilibrium point through numerical simulations and confirm that the system has rich dynamics.

Specifically, the system has, in general, chaotic behavior. A transient hyperchaotic (two positive LCEs) behavior is also detected.

Also, for different initial conditions and different values of the parameter

Finally, the designed nonlinear electronic circuit emulates very well the proposed system’s dynamic behavior.

The authors declare that there is no conflict of interests regarding the publication of this paper.