Dynamical properties of a novel one dimensional chaotic map

<abstract><p>In this paper, a novel one dimensional chaotic map $ K(x) = \frac{\mu x(1\, -x)}{1+ x} $, $ x\in [0, 1], \mu &gt; 0 $ is proposed. Some dynamical properties including fixed points, attracting points, repelling points, stability and chaotic behavior of this map are analyzed. To prove the main result, various dynamical techniques like cobweb representation, bifurcation diagrams, maximal Lyapunov exponent, and time series analysis are adopted. Further, the entropy and probability distribution of this newly introduced map are computed which are compared with traditional one-dimensional chaotic logistic map. Moreover, with the help of bifurcation diagrams, we prove that the range of stability and chaos of this map is larger than that of existing one dimensional logistic map. Therefore, this map might be used to achieve better results in all the fields where logistic map has been used so far.</p></abstract>

Symmetric Coexisting Attractors in a Novel Memristors-Based Chuas Chaotic System

This paper introduces a charge-controlled memristor based on the classical Chuas circuit. It also designs a novel four-dimensional chaotic system and investigates its complex dynamics, including phase portrait, Lyapunov exponent spectrum, bifurcation diagram, equilibrium point, dissipation and stability. The system appears as single-wing, double-wings chaotic attractors and the Lyapunov exponent spectrum of the system is symmetric with respect to the initial value. In addition, symmetric and asymmetric coexisting attractors are generated by changing the initial value and parameters. The findings indicate that the circuit system is equipped with excellent multi-stability. Finally, the circuit is implemented in Field Programmable Gate Array (FPGA) and analog circuits.

Dynamic Response and Chaotic Behavior of a Controllable Flexible Robot

Flexible robots with controllable mechanisms have advantages over common tandem robots in vibration magnitude, residual vibration time, working speed, and efficiency. However, abnormal vibration can sometimes occur during their use, affecting their normal operation. In order to better understand the causes of this abnormal vibration, our work takes a controllable flexible robot as a research object, and uses a combination of Lagrangian and finite element methods to establish its nonlinear elastic dynamics. The effectiveness of the model is verified by comparing the frequency of the numerical calculation and the test. The time-domain diagram, phase diagram, Poincaré map, and maximum Lyapunov exponent of the elastic motion of the robot wrist are studied, and the chaotic phenomena in the system are identified through the phase diagram, Poincaré map, and the maximum Lyapunov exponent. The relationship between the parameters of the robot motion and the maximum Lyapunov exponent is discussed, including trajectory angular speed and radius. The results show that chaotic behavior exists in the controllable flexible robot, and that trajectory angular speed and radius all have an influence on the chaotic motion, which provides a theoretical basis for further research on the control and optimal design of the mechanism.

Chaotic dynamics in Parkfield region, California and characteristics earthquakes

Based on about 75000 earthquakes in the California region detected through Parkfield network during the years 1969-1987, the occurrence of chaos was examined by two different approaches, namely, strange at tractor dimension and the Lyapunov exponent. The strange at tractor dimension was found as 6.3 in this region suggesting atleast 7 parameters for earthquake predictability. Small positive Lyapunov exponent of 0.045 provided further evidence for deterministic chaos in the region which showed strong dependence on the initial conditions. Implications of chaotic dynamics on characteristic Parkfield earthquakes has been discussed. The strange at tractor dimension in the region could be representative for the Transform type of plate boundary which is lower than that reported for continent collision type of plate boundary which is lower than that reported for continent collision type of plate boundary near Hindukush northwest Himalayan region.

Hamiltonian energy computation and complex behavior of a small heterogeneous network of three neurons: circuit implementation

Brain functions are sometimes emulated using some analog integrated circuits based on the organizational principle of natural neural networks. Neuromorphic engineering is the research branch devoted to the study and realization of such circuits with striking features. In this contribution, a novel small network of three neurons is introduced and investigated. The model is built from the coupling between two 2D Hindmarsh–Rose neurons through a 2D FitzHugh–Nagumo neuron. Thus, a heterogeneous coupled network is obtained. The biophysical energy released by the network during each electrical activity is evaluated. In addition, nonlinear analysis tools such as two-parameter Lyapunov exponent, bifurcation diagrams, the graph of the largest Lyapunov exponent, phase portraits, time series, as well as the basin of attractions are used to numerically investigate the network. It is found that the model can experience hysteresis justified by the simultaneous existence of three distinct electrical activities using the same set of parameters. Finally, the circuit implementation of the network is addressed in PSPICE to further support the obtained results.

Hyperchaos in a Bose-Hubbard Chain with Rydberg-Dressed Interactions

We study the chaos and hyperchaos of Rydberg-dressed Bose–Einstein condensates (BECs) in a one-dimensional optical lattice. Due to the long-range, soft-core interaction between the dressed atoms, the dynamics of the BECs are described by the extended Bose-Hubbard model. In the mean-field regime, we analyze the dynamical stability of the BEC by focusing on the ground state and localized state configurations. Lyapunov exponents of the two configurations are calculated by varying the soft-core interaction strength, potential bias, and length of the lattice. Both configurations can have multiple positive Lyapunov exponents, exhibiting hyperchaotic dynamics. We show the dependence of the number of the positive Lyapunov exponents and the largest Lyapunov exponent on the length of the optical lattice. The largest Lyapunov exponent is directly proportional to areas of phase space encompassed by the associated Poincaré sections. We demonstrate that linear and hysteresis quenches of the lattice potential and the dressed interaction lead to distinct dynamics due to the chaos and hyperchaos. Our work is relevant to current research on chaos as well as collective and emergent nonlinear dynamics of BECs with long-range interactions.

Classification of Chaotic Squeak and Rattle Vibrations by CNN Using Recurrence Pattern

The chaotic squeak and rattle (S&R) vibrations in mechanical systems were classified by deep learning. The rattle, single-mode, and multi-mode squeak models were constructed to generate chaotic S&R signals. The repetition of nonlinear signals generated by them was visualized using an unthresholded recurrence plot and learned using a convolutional neural network (CNN). The results showed that even if the signal of the S&R model is chaos, it could be classified. The accuracy of the classification was verified by calculating the Lyapunov exponent of the vibration signal. The numerical experiment confirmed that the CNN classification using nonlinear vibration images as the proposed procedure has more than 90% accuracy. The chaotic status and each model can be classified into six classes.

Dynamics analysis of a gyrostat system with intermittent forcing

The dynamical characteristics of a gyrostat system with intermittent forcing are investigated, the main work and contributions are given as follows: (1) The gyrostat system with an intermittent forcing is studied, and its dynamical characteristics are investigated by the corresponding Lyapunov exponent spectrums and bifurcation diagrams with respect to the amplitude of intermittent forcing. The modified gyrostat system exists chaotic motion when the amplitude of intermittent forcing belongs to a certain interval, and it can be at a state of stable point or periodic motion by the design of amplitude. (2) The gyrostat system with multiple intermittent forcings is also investigated through the combination of Lyapunov exponent spectrums and bifurcation diagrams, and it behaves periodic motion or chaotic motion when the amplitude or forcing width is different. (3) By the selection of parameters in intermittent forcings, the modified gyrostat system is at a state of stable point, periodic motion or chaotic motion. Numerical simulations verify the feasibility and effectiveness of the modified gyrostat system.