Abstract
A conjugate Lorenz-like system which includes only two quadratic nonlinearities is proposed in this paper. Some basic properties of this system, such as the distribution of its equilibria and their stabilities, the Lyapunov exponents, the bifurcations are investigated by some numerical and theoretical analysis. The forming mechanisms of compound structures of its new chaotic attractors obtained by merging together two simple attractors after performing one mirror operation are also presented. Furthermore, some of its other complex dynamical behaviours, which include the existence of singularly degenerate heteroclinic cycles, the existence of homoclinic and heteroclinic orbits and the dynamics at infinity, etc, are formulated in detail. In the meantime, some problems deserving further investigations are presented.
Highlights
There has been an increasing interest in exploiting chaotic systemssince E.N
We study other complicate dynamical behaviours of system (1.3), mainly for the existence of singular orbits, such as singularly degenerate heteroclinic cycle, homoclinic and heteroclinic orbit, etc, which play some key roles in revealing the fascinating nature of system (1.3)
Recall that a singularly degenerate heteroclinic cycle consists of an invariant set formed by a line of equilibria together with a heteroclinic orbit connecting two of the equilibria
Summary
There has been an increasing interest in exploiting chaotic systems (such as the known Chen system [1], Rossler system [16], and Lusystems [13], etc.). Based on exploiting chaotic systems and the above system (1.2), we introduce in this paper the following 3D Lorenz-type system with six terms x = a(x − y),. By virtue of some numerical simulations, we find some complicate and interesting singular orbits in system (1.3), such as singularly degenerate heteroclinic cycles, homoclinic and heteroclinic orbits etc. These properties with such singular orbits of (1.3), to the best of our knowledge, have not been found in any known literature.
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