No abstract

Compared with most known singularly degenerate heteroclinic cycles consisting of two different equilibria of a line or a curve, or two parallel lines of semi-hyperbolic equilibria, little seems to be noticed about the one that connects two perpendicular lines of semi-hyperbolic equilibria, i.e. [Formula: see text] and [Formula: see text], [Formula: see text], which is found in the mathematical chaos model: [Formula: see text], [Formula: see text], [Formula: see text] when [Formula: see text] and [Formula: see text]. Surprisingly, apple-shape attractors are also created nearby that kind of singularly degenerate heteroclinic cycles in the case of small [Formula: see text]. Further, some other rich dynamics are uncovered, i.e. global boundedness, Hopf bifurcation, limit cycles coexisting with one chaotic attractor, etc. We not only prove that the ultimate bound sets and globally exponentially attracting sets perfectly coincide under the same parameters, but also illustrate four limit cycles coexisting with one chaotic attractor, the saddle in the origin, and other two stable nontrivial node-foci, which are also trapped in the obtained globally exponentially attracting set, extending the recently reported results of the Lü-type subsystem. In addition, combining theoretical analysis and numerical simulation, the bidirectional forming mechanism of that kind of singularly degenerate heteroclinic cycles is illustrated, and their collapses may create three-scroll/apple-shape attractors, or limit cycles, etc. Finally, conservative chaotic flows are numerically found in the new system. We expect that the outcome of the study may provide a reference for subsequent research.

No abstract

Little seems to be considered about the globally exponentially asymptotical stability of parabolic type equilibria and the existence of heteroclinic orbits in the Lorenz-like system with high-order nonlinear terms. To achieve this target, by adding the nonlinear terms yz and $$x^{2}y$$ x 2 y to the second equation of the system, this paper introduces the new 3D cubic Lorenz-like system: $$\dot{x}=a(y - x)$$ x ˙ = a ( y - x ) , $$\dot{y}=b_{1}y+b_{2}yz+b_{3}xz+b_{4}x^{2}y$$ y ˙ = b 1 y + b 2 y z + b 3 x z + b 4 x 2 y , $$\dot{z}= -cz + y^{2}$$ z ˙ = - c z + y 2 , which does not belong to the generalized Lorenz systems family. In addition to giving rise to generic and degenerate pitchfork bifurcation, Hopf bifurcation, hidden Lorenz-like attractors, singularly degenerate heteroclinic cycles with nearby chaotic attractors, etc., one still rigorously proves that not only the parabolic type equilibria $$S_{x} = \{(x, x, \frac{x^{2}}{c})|x\in \mathbb {R}, c\ne 0\}$$ S x = { ( x , x , x 2 c ) | x ∈ R , c ≠ 0 } are globally exponentially asymptotically stable, but also there exists a pair of symmetrical heteroclinic orbits with respect to the z-axis, as most other Lorenz-like systems. This study may offer new insights into revealing some other novel dynamic characteristics of the Lorenz-like system family.