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Although the axis-symmetric singularly degenerate heteroclinic cycles with nearby bifurcated axis-symmetric Lorenz-like attractors in axis-symmetric Lorenz-like system family were intensively studied in past decades, the scenario with asymmetric cycles has not been investigated. This paper revisits a simple Lorenz-like system, and illustrates asymmetric singularly degenerate heteroclinic cycles with nearby bifurcated asymmetric Lorenz-like attractors. In addition, it is proved that there exists a pair of asymmetric heteroclinic orbits to stable origin, and a pair of unstable and stable nontrivial asymmetric equilibria, in contrast to the axis/centro-symmetric ones to the unstable origin and a pair of stable nontrivial axis/centro-symmetric equilibria in most reported Lorenz-like systems. Moreover, this system casts a mirror image of the aforementioned dynamics from the parameter $-d$ to $d$.

In this paper, we design and explore a novel 3D autonomous nonlinear system with logarithmic nonlinearity. It is framed from the Lorenz model by replacing the variable $y$ in the second equation by the nonlinear logarithmic term $ln|y|$. The bi-parametric dynamics of the modeled system is presented. Studies reveal the emergence of shrimp-shaped periodic structures in chaotic regime with period-adding phenomenon, and coexisting periodic-chaotic attractors, which do not occur in the Lorenz model. The stability, bifurcation patterns, and chaotic behaviors of the system are explored with the aid of bifurcation diagrams, phase portraits and Lyapunov exponents. The complexity of the basin sets for the coexisting attractors is also studied.

This paper analyzes the synchronization control of new chaotic systems called Lorenz-like systems. Based on the Lyapunov stability theory, an adaptive controller and a parameter update rule are designed. It is proved that the controller and update rule not only achieve self-synchronization of Lorenz-like systems but can also make the Lorenz-like system asymptotically synchronized with the Rössler system, and further identify the uncertain system parameters. Numerical simulations have shown the effectiveness of the adaptive controller.

This article introduces a new chaotic system of three-dimensional quadratic autonomous ordinary differential equations, which can display (i) two 1-scroll chaotic attractors simultaneously, with only three equilibria, and (ii) two 2-scroll chaotic attractors simultaneously, with five equilibria. Several issues such as some basic dynamical behaviors, routes to chaos, bifurcations, periodic windows, and the compound structure of the new chaotic system are then investigated, either analytically or numerically. Of particular interest is the fact that this chaotic system can generate a complex 4-scroll chaotic attractor or confine two attractors to a 2-scroll chaotic attractor under the control of a simple constant input. Furthermore, the concept of generalized Lorenz system is extended to a new class of generalized Lorenz-like systems in a canonical form. Finally, the important problems of classification and normal form of three-dimensional quadratic autonomous chaotic systems are formulated and discussed.

For Lorenz-like system, conditions of Shilnikov chaos are obtained. Algorithms for computation of bifurcational values of parameters, which correspond to homoclinic trajectories are formulated. Modified Lorenz, Chen and Lu systems are considered.

In this paper, two methods are proposed to construct multi-wing chaotic attractors based on a Lorenz-like autonomous chaotic system. The first is switching method which can directly multiply the wings of the system without segment linearization of the system. The second is coordinate transition method which is related to the initial value of the system. Moreover, theoretical analysis and simulation results show the effectiveness of these two methods.

For Tigan system existence conditions of homoclinic trajectories are obtained. Algorithms for computing bifurcation values of parameters corresponding to homoclinic trajectories are formulated.

In this report, by the numerical continuation method, we visualize and connect hidden chaotic sets in the Glukhovsky–Dolzhansky, Lorenz and Rabinovich systems using a certain path in the parameter space of a Lorenz-like system.

As reported in the existing literature, wing attractors are confined to 1D $n$-wing attractors, 2D $n\times m$-grid wing attractors. In this paper, we break this limitation and generate 3D $n\times m\times l$-grid multi-wing chaotic attractors (GMWCAs). The 3D GMWCAs are produced via the following three steps: (1) applying rotation transformation to a double-wing Lorenz-like system to ensure that its saddle-focus equilibria with index 2 are located on the plane $y=0$; (2) extending the wing attractors of the transformed Lorenz-like system along the $x$-axis to have mirror symmetry; (3) introducing stair switching functions to increase the number of saddle-focus equilibria with index 2 along the $y$-axis and $z$-axis. Furthermore, some basic dynamical properties of the 3D chaotic system, including equilibria, symmetry, dissipativity, Lyapunov exponents and bifurcation diagram, are investigated and a module-based unified circuit diagram is designed. The effectiveness of this approach is confirmed by both numerical simulations and electrical circuit experiment.

In this article using an analytical method called Fishing principle we obtain the region of parameters, where the existence of a homoclinic orbit to a zero saddle equilibrium in the Lorenz-like system is proved. For a qualitative description of the different types of homoclinic bifurcations, a numerical analysis of the obtained region of parameters is organized, which leads to the discovery of new bifurcation scenarios.

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 $\frac{6}{5}$. In contrast to the previously studied one of degree $\frac{4}{3}$, 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.