Periodic orbits of diffusionless Lorenz system

Abstract

The strange attractor of a chaotic system is composed of numerous periodic orbits densely covered. The periodic orbit is the simplest invariant set except for the fixed point in the nonlinear dynamic system, it not only reflects all the characteristics of the chaotic motion, but also is closely related to the amplitude generation and change of chaotic system. Therefore, it is of great significance to obtain the periodic orbits in order to analyze the dynamical behaviors of the complex system. In this paper, we study the periodic orbits of the diffusionless Lorenz equations which are derived in the limit of high Rayleigh and Prandtl numbers. A new approach to establishing one-dimensional symbolic dynamics is proposed, and the periodic orbits based on a topological structure are systematically calculated. We use the variational method to locate the cycles, which is proposed to explore the periodic orbits in high-dimensional chaotic systems. The method not only preserves the robustness characteristics of most of other methods, such as the Newton descent method and multipoint shooting method, but it also has the characteristics of fast convergence when the search process is close to the real cycle in practice. In order to apply the method, a rough loop guess must be made first based on the entire topology for the cycle to be searched, and then the variational algorithm will bring the initial loop guess to evolving toward the real periodic orbit in the system. In the calculations, the Newton descent method is used to achieve stability. Two cycles can be used as basic building blocks for initialization, searching for more complex cycles with multiple circuits around the two fixed points requires more delicate initial conditions; otherwise, it will probably lead to nonconvergence. We can initialize the loop guess for longer cycles constructed by cutting and gluing the short, known cycles. For this system, such a method yields quite a good systematic initial guess for longer cycles. Even if we deform the orbit manually into a closed loop, the variational method still shows its powerfulness for good convergence. The topological classification based on the entire orbital structure is shown to be effective. Furthermore, the deformation of periodic orbits with the change of parameters is discussed, which provides a route to the periods of cycles. The present research may provide a method of performing systematic calculation and classification of periodic orbits in other similar chaotic systems.

Unstable periodic orbits analysis in the generalized Lorenz-type system

“…This method is not only suitable for the determination of periodic orbits but also for the homoclinic and heteroclinic orbits [32]. In the previous work, the periodic orbits in various chaotic systems were calculated efficiently using the variational method [33][34][35][36], which illustrates the practicability of this method in the GLTS.…”

Dynamics, periodic orbits of a novel four-dimensional hyperchaotic system with hidden attractors

“…Continuous changes to a single parameter can cause sudden and rapid reactions, ultimately altering the motion properties of the system. This phenomenon is commonly referred to as bifurcation in dynamics [24]. In other words, quantitative change leads to qualitative change, so the qualitative or topological changes in the behaviour of the dynamic system will occur, which means the occurrence of bifurcation.…”

Dynamic Analysis of a Novel 3D Chaotic System with Hidden and Coexisting Attractors: Offset Boosting, Synchronization, and Circuit Realization

“…When the parameters of the system change, the first return map of the system will also be altered accordingly, which may no longer be a 1D unimodal map, but have multiple branches, thus requiring more symbols to encode periodic orbits. In this case, it is more convenient and effective to establish symbolic dynamics based on the topological structure of orbits [49][50][51], such as the number of rotations between periodic orbits and equilibrium points. Furthermore, continuous deformation of the cycles with the change of parameters can also be explored by the variational method, which can help us judge the parameter values when the number of cycles or stability changes, and thus confirm the corresponding bifurcation phenomenon [52][53][54].…”