Chaotic system with bondorbital attractors

Two kinds of chaotic attractors with nontrivial topologies are found in a 4D autonomous continuous dynamical system. Since the equilibria of the system are located on both sides of the basic structures of these attractors, and the basic structures of these attractors are bond orbits, we call them bondorbital attractors. They have fractional dimensions and their Kaplan–Yorke dimensions are greater than 3. The generation mechanisms of the two types of attractors are explored and analyzed based on the Shilnikov’s theorems. The type I attractors generated by system with parameter \(P_{1}\) possess coexistence features, and the type II attractors generated by system with parameter \(P_{2}\) have the ability to realize consecutive bond orbits. Furthermore, the type I attractors have continuous attracting basins with diagonal distribution and can be caught by means of a method of shorting capacitors in hardware experiments, whereas the type II attractors possess discrete basins of attraction and are difficult to be captured in hardware experiments. The difference between the two types of bondorbital attractors and traditional self-excited attractors in generation method is analyzed, and we also verify that they are not hidden attractors. Based on the step function sequence f(x, M, N), the type II attractors with at most (\(N+M+1\))-fold structures can be generated in the system with parameter \(P_{2}\). Two sets of symmetric specific initial conditions are used to verify that the system with parameter \(P_{2}\) can generate bondorbital attractors with fourfold and fivefold basic structures based on f(x, 2, 2). Some characteristics of the two classes of bondorbital attractors are listed in tabular form.