Dynamics of a general jerky equation

Some classic nonlinear dynamical systems, such as Rössler's toroidal model, the Genesio model, and 19 Sprott's models, can be classified into seven distinct basic classes of jerky dynamics, labeled by [Formula: see text]. This paper is devoted to the dynamics of a general jerky equation which contains [Formula: see text] as parameters vary. It is shown that the system undergoes fold, Hopf, zero-Hopf, and Bogdanov–Takens bifurcations based on the center manifold theorem and normal form theory. Numerical simulations are also given to make the theoretical results visible and to detect more complicated dynamical behaviors, including degenerate Hopf bifurcation, fold bifurcation of cycle, and limit cycles. Especially, an apple-like attractive portrait is discovered near the zero-Hopf bifurcation point for the first time. Finally, according to the conclusions of the general jerky equation, exact conditions are summarized by two tables on how bifurcations will occur for [Formula: see text], respectively.

Dynamics of Attractively and Repulsively Coupled Elementary Chaotic Systems

We investigate an elementary model for doubly coupled dynamical systems that consists of two identical, mutually interacting minimal chaotic flows in the form of jerky dynamics. The coupling mechanisms allow for the simultaneous presence of attractive and repulsive interactions between the systems. Despite its functional simplicity, the model is capable of exhibiting diverse types of dynamical phenomena induced by the presence of the couplings. We provide an in-depth numerical investigation of the dynamics depending on the coupling strengths and the autonomous dynamical behavior of the subsystems. Partly, the dynamics of the system can be analytically understood using the Poincaré–Lindstedt method. An approximation of periodic orbits is carried out in the vicinity of a phase-flip transition that leads to deeper insights into the organization of the appearing dynamics in the parameter space. In addition, we propose a circuit that enables an electronic implementation of the model. A variation of the coupling mechanism to a coupling in conjugate variables leads to a regime of amplitude death.

On the Possibility of the Jerk Derivative in Electrical Circuits

A subclass of dynamical systems with a time rate of change of acceleration are called Newtonian jerky dynamics. Some mechanical and acoustic systems can be interpreted as jerky dynamics. In this paper we show that the jerk dynamics are naturally obtained for electrical circuits using the fractional calculus approach with order γ. We consider fractional LC and RL electrical circuits with 1⩽γ<2 for different source terms. The LC circuit has a frequency ω dependent on the order of the fractional differential equation γ, since it is defined as ω(γ)=ω0γγ1-γ, where ω0 is the fundamental frequency. For γ=3/2, the system is described by a third-order differential equation with frequency ω~ω03/2, and assuming γ=2 the dynamics are described by a fourth differential equation for jerk dynamics with frequency ω~ω02.

A Driver Modeling Based on the Preview-Follower Theory and the Jerky Dynamics

Based on the preview optimal simple artificial neural network driver model (POSANN), a new driver model, considering jerky dynamics and the tracing error between the real track and the planned path, is established. In this paper, the modeling for the driver-vehicle system is firstly described, and the relationship between weighting coefficients of driver model and system parameters is examined through test data. Secondly, the corresponding road test results are presented in order to verify the vehicle model and obtain the information on drive model and vehicle parameters. Finally, the simulations are carried out via CarSim. Simulation results indicate that the jerky dynamics need to be considered and the proposed new driver model can achieve a better path-following performance compared with the POSANN driver model.