This work presents a multiscroll generator system, which addresses the issue by the implementation of 9-level saturated nonlinear function, SNLF, being modified with a new control parameter that acts as a bifurcation parameter. By means of the modification of the newly introduced parameter, it is possible to control the number of scrolls to generate. The proposed system has richer dynamics than the original, not only presenting the generation of a global attractor; it is capable of generating monostable and bistable multiscrolls. The study of the basin of attraction for the natural attractor generation (9-scroll SNLF) shows the restrictions in the initial conditions space where the system is capable of presenting dynamical responses, limiting its possible electronic implementations.

Over the last few years, the development and implementation of chaotic oscillators have been extensively studied, taking a special interest in the generation of systems with multiscrolls in their phase space, such as the Lorenz [

An example of this kind of systems can be described within the theory of unstable dissipative systems, UDS [

In recent years, the design and control of systems with multiple scrolls have been a subject of interest for the scientific community, having a great impact in their application, such as secure communication systems, neuronal modeling, and generation of pseudorandom systems [

This work is structured as follows: the first section contains an introduction that describes previous works and theoretical principles of the system. The second section shows the UDS definition and the description of the multiscroll generator system. In the third section, the methodology and results of the studied system are shown. The analysis of the bifurcation diagrams exhibits the coexistence of two attractors for fixed set parameters, bistability, which is illustrated by the construction of the corresponding basins of attractions. The main conclusions are shown in the last section.

In the same spirit as [

A system can be considered as an UDS type I if their equilibrium points correspond to a hyperbolic-saddle-node, i.e., one eigenvalue is negative real (dissipative component) and the other two are complex conjugated with positive real part (unstable and oscillatory component), where the sum of the components must be less than zero. By other side, an UDS type II, eq. (

The multiscroll generator system studied is described by a set of three coupled differential equations that makes use of the definition of a saturated nonlineal function, SNLF, as a method for the scroll generation [

Considering the previous condition, it is possible to examine the behavior of the equilibrium points by sweeping the control parameter and finding the operation zone where their eigenvalues are consistent with an UDS I definition. The control parameter variation is developed by means of the characteristic polynomial of the system described in eq. (

Eigenvalues analysis as a function of the control parameter

It is well known [

Figure

(a) Saturated nonlineal function with

The SNLF contemplated for this study is constructed based on [

Considering the restrictions of the system, eq. (

The dynamical system, eq. (

Analyzing the dynamical system, Figure

Bifurcation diagram for the dynamical control parameter (a)

The 9-scroll natural attractor region

Bifurcation diagram for the dynamical control parameter

Space phase for each attractor identified into the bifurcation diagram, Figure

Table

Behavior analysis of Figures

Attractor type | Bistability | Fig. | |
---|---|---|---|

0.0310–0.0320 | Single-wing deformed | No | Figure |

0.0320–0.0335 | Single-wing | No | Figure |

0.0335–0.0350 | Single-wing deformed | No | Figure |

0.0350–0.0420 | Single-wing with 3 equilibrium points | No | Figure |

0.0420–0.0500 | 3-scroll | No | Figure |

0.0500–0.0510 | 5-scroll | No | Figure |

0.0510–0.0520 | 7-scroll | No | Figure |

0.0520–0.0530 | 8-scroll | No | Figure |

0.0530–0.0570 | 9-scroll (natural attractor) | No | Figure |

0.0570–0.0590 | Single-wing | Yes | Figure |

0.0590–0.0610 | Coherent single-wing | Yes | Figure |

0.0610–0.0670 | Single-wing | Yes | Figure |

0.0670–0.0690 | Single-wing | Yes | Figure |

0.0690–0.0740 | Coherent single-wing | Yes | Figure |

0.0740–0.0780 | Coherent single-wing | Yes | Figure |

0.0780–0.0820 | Double-scroll with 3 equilibrium points | No | Figure |

With the premise that every UDS I system has large basins of attraction [

An analysis of the dynamical response of the natural system attractor (nine multiscrolls) in a bidimensional space

Bidimensional space

As in [

(a) Ration curve of the 9-scroll operation zone and (b) normalized operation zone for every dynamical value.

As a result of the bifurcation diagrams shown in Figures

(a–c) Basins of attraction for the coexisting coherent single-wing attractor. (d–i) Coexisting attractors for each basin of attraction. (j–l) Natural system attractor. (m–o) Return time distribution calculated by the Poincare section for the natural system attractor. Columns from left to right correspond to

Figures

Analyzing the resulting basins of coexisting attractors, Figures

In this work, a multiscroll system of three-dimensional autonomous equations with a parameter

The described model, in addition to involving a relatively easy implementation [

The data used to support the findings of this study are available from the corresponding author upon request.

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

J. L. E. M. acknowledges CONACYT for the financial support (National Fellowship CVU-706850, No. 582124) and the University of Guadalajara, CULagos (Mexico). This work was supported by the University of Guadalajara under the project “research laboratory equipment for academic groups in optoelectronics from CULAGOS”, R-0138/2016, Agreement RG /019/2016-UdeG, Mexico. The authors acknowledge J.O. Esqueda-de la Torre for his help in improving the grammar.