Complex dynamics of a sub-quadratic Lorenz-like system

Abstract

Motivated by the generic dynamical property of most quadratic Lorenz-type systems that the unstable manifolds of the origin tending to the stable manifold of nontrivial symmetrical equilibria forms a pair of heteroclinic orbits, this technical note reports a new 3D sub-quadratic Lorenz-like system: x˙=a(y−x)\dot{x}=a(y-x), y˙=cx3+dy−x3z\dot{y}=c\sqrt[3]{x}+{\rm{d}}y-\sqrt[3]{x}z and z˙=−bz+x3y\dot{z}=-bz+\sqrt[3]{x}y. Instead, the unstable manifolds of nontrivial symmetrical equilibria tending to the stable manifold of the origin creates a pair of heteroclinic orbits. This drives one to further investigate it and reveal its other hidden dynamics: Hopf bifurcation, invariant algebraic surfaces, ultimate bound sets, globally exponentially attractive sets, existence of homoclinic and heteroclinic orbits, singularly degenerate heteroclinic cycles, and so on. The main contributions of this work are summarized as follows: First, the ultimate boundedness of that system yields the globally exponentially attractive sets of it. Second, the existence of another heteroclinic orbits is also proved by utilizing two different Lyapunov functions. Finally, on the invariant algebraic surface z=34ax43z=\frac{3}{4a}\sqrt[3]{{x}^{4}}, the existence of a pair of homoclinic orbits to the origin, and two pairs of heteroclinic orbits to two pairs of nontrivial symmetrical equilibria is also proved by utilizing a Hamiltonian function. In addition, the correctness of the theoretical results is illustrated via numerical examples.

Keywords

• sub-quadratic lorenz-like system
• globally exponentially attractive set
• homoclinic orbit
• heteroclinic orbit
• lyapunov function