Topological horseshoe in a single-scroll Chen system with time delay

Hai-Peng Ren*, Kun Tian, Celso Grebogi

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)
3 Downloads (Pure)


The proof that an attractor is chaotic is not trivial. A single-scroll attractor in the Chen system with time delay is investigated through both theory and simulation. The Chen system with time delay is infinite dimensional parameterized by the time delay τ. The detailed procedure operations for finding topological horseshoe in the delay differential equation are different from that one in ordinary differential equation. We show the existence of chaos by using both the topological horseshoe theory and its corollary, and the Smale horseshoe construction if the geometry of the attractor on the 2D plane section satisfies certain conditions. This paper presents both methods for establishing the presence of the horseshoe in a Chen system with time delay. In the first method, we select two quadrilaterals in the 2D transversal section, and calculate the relationship of the quadrilaterals under the map. In the second method, we select quadrilaterals in the neighborhood of a short unstable periodic orbit in the section and obtain the approximate location of the quadrilaterals under the map, yielding the Smale horseshoe. By using the above two methods, the geometrical expansion of the quadrilaterals under the map satisfies the Topological Horseshoe Corollary and also the Smale horseshoe construction, thus showing that the time-delayed single-scroll attractor is indeed chaotic.
Original languageEnglish
Article number109593
Number of pages7
JournalChaos, Solitons & Fractals
Early online date6 Jan 2020
Publication statusPublished - 1 Mar 2020

Bibliographical note

This work was supported in part by the NSFC (60804040), the Shaanxi Provincial Special Support Program for Science and Technology Innovation Leader.


  • Infinite dimensional system
  • Single-scroll hyper-chaotic attractor
  • Topological horseshoe
  • Smale horseshoe


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