Conservative chaos and invariant tori in the modified Sprott A system

  • Shijian CangEmail author
  • Yue Li
  • Wei Xue
  • Zenghui Wang
  • Zengqiang Chen
Original paper


Generally, there are few volume-conservative but not energy-conservative chaotic systems in the literature. By recomposing the skew-symmetric state matrix of the Sprott A system, a new volume-conservative chaotic system is coined in this paper. We mainly focus on investigating its dynamical behaviors that relay on the external excitation k, which directly determines the number of equilibria of the system. Without k, there exist two lines of equilibria that can be degenerated into two or four equilibria for the given nonzero initial conditions, while with k, the system is a no-equilibrium system that can produce conservative chaos and invariant tori for different initial conditions. Moreover, these rich dynamical behaviors are illustrated by several numerical techniques including time series, phase portraits, Poincaré sections, bifurcation diagrams and Lyapunov exponents.


Sprott A system Conservative chaos Invariant tori Generalized Hamiltonian system 



This work is partly supported by the National Natural Science Foundation of China (Grant No. 61873186), South African National Research Foundation (Grant Nos. 112142 and 112108), South African National Research Foundation Incentive Grant (No. 114911), and South African Eskom Tertiary Education Support Programme.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Product DesignTianjin University of Science and TechnologyTianjinPeople’s Republic of China
  2. 2.College of Electronic Information and AutomationTianjin University of Science and TechnologyTianjinPeople’s Republic of China
  3. 3.Department of Electrical and Mining EngineeringUniversity of South AfricaFloridaRepublic of South Africa
  4. 4.College of Artificial IntelligenceNankai UniversityTianjinPeople’s Republic of China

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