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Revisiting a newly reported modified Chen system by both the definitions of $ \alpha $-limit and $ \omega $-limit set, Lyapunov function and Hamiltonian function, this paper seized a multitude of pairs of potential heteroclinic orbits to (1) $ E_{0} $ and $ E_{\pm} $, or (2) $ E_{+} $ or (3) $ E_{-} $, and homoclinic and heteroclinic orbits on its invariant algebraic surface $ Q = z - \frac{x^{2}}{2a} = 0 $ with cofactor $ -2a $, which is not available in the existing literature to the best of our knowledge. Particularly, the theoretical conclusions were verified via numerical examples.

Aiming to explore the subtle connection between the number of nonlinear terms in Lorenz-like systems and hidden attractors, this paper introduces a new simple sub-quadratic four-thirds-degree Lorenz-like system, where x˙=a(y−x), y˙=cx−x3z, z˙=−bz+x3y, and uncovers the following property of these systems: decreasing the powers of the nonlinear terms in a quadratic Lorenz-like system where x˙=a(y−x), y˙=cx−xz, z˙=−bz+xy, may narrow, or even eliminate the range of the parameter c for hidden attractors, but enlarge it for self-excited attractors. By combining numerical simulation, stability and bifurcation theory, most of the important dynamics of the Lorenz system family are revealed, including self-excited Lorenz-like attractors, Hopf bifurcation and generic pitchfork bifurcation at the origin, singularly degenerate heteroclinic cycles, degenerate pitchfork bifurcation at non-isolated equilibria, invariant algebraic surface, heteroclinic orbits and so on. The obtained results may verify the generalization of the second part of the celebrated Hilbert’s sixteenth problem to some degree, showing that the number and mutual disposition of attractors and repellers may depend on the degree of chaotic multidimensional dynamical systems.

Motivated by the open problems on the global dynamics of the generalized four-dimensional Lorenz-like system, this paper deals with the existence of globally exponentially attracting sets and heteroclinic orbits by constructing a series of Lyapunov functions. Specifically, not only is a family of mathematical expressions of globally exponentially attracting sets derived, but the existence of a pair of orbits heteroclinic to S0 and S± is also proven with the aid of a Lyapunov function and the definitions of both the α-limit set and ω-limit set. Moreover, numerical examples are used to justify the theoretical analysis. Since the obtained results improve and complement the existing ones, they may provide support in chaos control, chaos synchronization, the Hausdorff and Lyapunov dimensions of strange attractors, etc.

In contrast to the coexistence of infinitely many pseudo and true singularly degenerate heteroclinic cycles with nearby two-scroll hyperchaotic Lorenz-like attractors coined in four-dimensional systems, this note reports the ones with nearby bifurcated chaotic attractors in a new 3D cubic Lorenz-like system. Except for the rabbit head/boat shape two-scroll attractors, there exist globally exponentially asymptotically stable parabolic type equilibria and a pair of heteroclinic orbits. In addition, numerical simulations also validate the correctness of the theoretical analysis.

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.