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

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.