Dynamic aspects of Sprott BC chaotic system

Marcos C. Mota; Regilene D. S. Oliveira

doi:10.3934/dcdsb.2020177

Article Contents

2021, Volume 26, Issue 3: 1653-1673. Doi: 10.3934/dcdsb.2020177

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Dynamic aspects of Sprott BC chaotic system

Marcos C. Mota and
Regilene D. S. Oliveira^,

Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Avenida Trabalhador São–carlense, 400, Centro, 13.566-590, São Carlos, SP, Brazil

^* Corresponding author: regilene@icmc.usp.br
^* Corresponding author: regilene@icmc.usp.br

Communicated by Dongmei Xiao

Received: March 31, 2019

Early access: June 2020

Published: March 2021

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Níıvel Superior - Brasil (CAPES) - Finance Code 001 and FAPESP grant number 2017/20854-5

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Abstract

In this paper we study global dynamic aspects of the quadratic system

$ \dot x = yz,\quad \dot y = x-y,\quad \dot z = 1-x(\alpha y+\beta x), $

where $ (x,y,z) \in \mathbb R^3 $ and $ \alpha, \beta \in[0,1] $ are two parameters. It contains the Sprott B and the Sprott C systems at the two extremes of its parameter spectrum and we call it Sprott BC system. Here we present the complete description of its singularities and we show that this system passes through a Hopf bifurcation at $ \alpha = 0 $. Using the Poincaré compactification of a polynomial vector field in $ \mathbb R^3 $ we give a complete description of its dynamic on the Poincaré sphere at infinity. We also show that such a system does not admit a polynomial first integral, nor algebraic invariant surfaces, neither Darboux first integral.

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Mathematics Subject Classification: Primary: 34C05, 34C45; Secondary: 37D10, 37D45.

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Figure 1. Local behavior of orbits around the finite singularities of Sprott B (in 1(A)) and Sprott C (in 1(B)) systems

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Figure 2. Phase portrait of system (3) on the Poincaré sphere. In Figure 2(A) there exist two closed curves filled up with singularities and one pair of distinguished singularities. These distinguished singularities possess two parabolic attractor sectors and two parabolic repelling sectors. In Figure 2(B) there exist one closed curve filled up with singularities and one pair of center type singularities

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Figure 3. Phase portrait of system (3) on the Poincaré sphere. In Figure 3(A) there exist a pair of cusp type singularities and a pair of node type singularities (being one attractor and other repelling). In Figure 3(B) there exist a pair of saddles, a pair of centers and a pair of nodes (being one attractor and other repelling)

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