Revealing More Hidden Attractors from a New Sub-Quadratic Lorenz-Like System of Degree 6 5

Abstract

In the sense that the descending powers of some certain variables may widen the range of parameters of self-excited and hidden attractors, this technical note proposes a new three-dimensional Lorenz-like system of degree [Formula: see text]. In contrast to the previously studied one of degree [Formula: see text], the newly reported one creates more hidden Lorenz-like attractors coexisting with the unstable origin and a pair of stable node-foci in a broader range of parameters, which confirms the generalization of the second part of the celebrated Hilbert’s 16th problem once more. In addition, some other dynamics, i.e. Hopf bifurcation, the generic and degenerate pitchfork bifurcation, invariant algebraic surface, first integral, singularly degenerate heteroclinic cycle with nearby chaotic attractor, ultimate bounded set and existence of a pair of heteroclinic orbits, are discussed.

Globally Exponentially Attracting Sets and Heteroclinic Orbits Revealed

“…Notably, although the method involving the Lyapunov functions and the definitions of both the α-limit set and ω-limit set has been applied to sub-quadratic, cubic, and other Lorenz-like systems [23], the globally exponentially attracting sets of these are still unknown. In addition to this, one needs to consider other important problems, e.g., selfexcited or hidden conservative Lorenz-like chaotic flows, homoclinic orbits, entropy [29], real-world applications, etc.…”

“…In 2006, Li et al combined the Lyapunov function and the definitions of both the α-limit set and ω-limit set to prove the existence of heteroclinic orbits in the Chen system [6], which did not need to consider the mutual disposition of the stable and unstable manifolds of a saddle equilibrium. In light of this, the method has been applied in many other Lorenz-like systems [2][3][4][5][7][8][9]23,24]. Formally, a chaotic system is bounded, meaning that its dynamics remain inside an orbit, rather than escaping to infinity.…”