A Chaotification model based on sine and cosecant functions for enhancing chaos

Chaotic maps with higher chaotic complexity are urgently needed in many application scenarios. This paper proposes a chaotification model based on sine and cosecant functions (CMSC) to improve the dynamic properties of existing chaotic maps. CMSC can generate a new map with higher chaotic complexity by using the existing one-dimensional (1D) chaotic map as a seed map. To discuss the performance of CMSC, the chaos properties of CMSC are analyzed based on the mathematical definition of the Lyapunov exponent (LE). Then, three new maps are generated by applying three classical 1D chaotic maps to CMSC respectively, and the dynamic behaviors of the new maps are analyzed in terms of fixed point, bifurcation diagram, sample entropy (SE), etc. The results of the analysis demonstrate that the new maps have a larger chaotic region and excellent chaotic characteristics.

Dynamic Behaviors Analysis of a Novel Fractional-Order Chua’s Memristive Circuit

This paper proposed a novel fractional-order Chua’s memristive circuit. Firstly, a fractional-order mathematical model of a diode bridge generalized memristor with RLC filter cascade is established, and simulations verify that the fractional-order generalized memristor satisfies the basic characteristics of a memristor. Secondly, the capacitor and inductor in Chua’s chaotic circuit are extended to the fractional order, and the fractional-order generalized memristor is used instead of Chua’s diode to establish the fractional-order mathematical model of chaotic circuit based on RLC generalized memristor. By studying the stability analysis of the equilibrium point and the influence of the circuit parameters on the system dynamics, the dynamic characteristics of the proposed chaotic circuit are theoretically analyzed and numerically simulated. The results show that the proposed fractional-order memristive chaotic circuit has gone through three states: period, bifurcation, and chaos, and a narrow period window appears in the chaotic region. Finally, the equivalent circuit method is adopted in PSpice to realize the construction of the fractional-order capacitance and inductance, and the simulation of the fractional-order memristive chaotic circuit is completed. The results further verify the correctness of the theoretical analysis.

Analysis of Nonlinear Dynamic Characteristics of a Mechanical-Electromagnetic Vibration System with Rubbing

In this study, a rotor-bearing-runner system (RBRS) considering multiple nonlinear factors is established, and the complex nonlinear dynamic behavior of the coupling system is studied. The effects of excitation current, radial stiffness, and friction coefficient on dynamic characteristics are analyzed by numerical simulation. The research results show that the dynamic properties of the coupling system caused by different nonlinear factors are interactional. With the changes of different parameters, the RBRS presents multiple motion states, including periodic-n, quasi-periodic, and chaotic motion. The increase of the excitation current Ij has a certain inhibitory effect on the response amplitude of the system and makes the motion state of the system more complex, the chaotic motion wider, and the jump discontinuity enhanced. With the increase of radial stiffness kr, the motion complexity of the coupling system increases, the chaotic region increases, the response amplitude increases, and the vibration intensity increases. With the increase of the friction coefficient μ, the chaotic region increases first and decreases, the different motions alternate frequently, and the response amplitude gradually increases. This study can not only help to understand the dynamic characteristics of RBRS, but also help the stable operation of the generator set.

Modeling and Analysis of Chaos and Bifurcations for the Attitude System of a Quadrotor Unmanned Aerial Vehicle

A model of the attitude system for a quadrotor unmanned aerial vehicle (QUAV), assumed to be a rigid body, is developed. For specific parameter configurations, a chaotic region with a saddle and two stable node-focus equilibrium points is identified. The chaotic model provides an important reference for dynamic analysis and a challengeable task of controller design once the flight enters the chaotic region of parameters. The pitchfork bifurcation of the equilibrium points is provided. Rich dynamics of the system are revealed by two bifurcation regions, which demonstrates the diversity of the flight behaviors as the parameters vary. One bifurcation analysis is with respect to the speed of the front propeller and the speed difference of the front and left propellers, and another one is with respect to the speed of the front propeller and moment of inertia. The dynamic characteristics of the QUAV are further verified by the Casimir power bifurcations. The trajectories of three settings with different structural parameters are analyzed in detail. The stability of the QUAV is found to be enhanced for certain optimized values of the structural parameters. Finally, using the Casimir power and Lagrange multiplier method, a supremum bound of the chaotic attractor is presented.

A Route to Chaos in the Boros–Moll Map

The Boros–Moll map appears as a subsystem of a Landen transformation associated to certain rational integrals and its dynamics is related to their convergence. In the paper, we study the dynamics of a one-parameter family of maps which unfold the Boros–Moll one, showing that the existence of an unbounded invariant chaotic region in the Boros–Moll map is a peculiar feature within the family. We relate this singularity with a specific property of the critical lines that occurs only for this special case. In particular, we explain how the unbounded chaotic region in the Boros–Moll map appears. We especially explain the main contact/homoclinic bifurcations that occur in the family. We also report some other bifurcation phenomena that appear in the considered unfolding.

Secure Optical QAM Transmission Using Chaos Message Masking

This work presents the joint use of advance modulation scheme i. e. m-QAM and optical chaos to combine the best of the two advantages i. e. higher data rates and security. A semiconductor laser diode is driven into chaotic region using direct modulation scheme and 4-QAM signal is added by Chaos message masking (CMS). The chaotically masked data stream is transmitted over an optical communication link to investigate the propagation issues and synchronization of chaos at the receiver. The transmitted chaos is synchronized at the receiver to unmask the QAM stream and binary data by using subtraction rule and conventional QAM demodulator, respectively. The deterioration of constellation diagrams and bit error rate found dependent upon transmitter/receiver synchronization and link parameters. The use of 4-QAM chaotic scheme is extendable to m-QAM and is applicable to long haul, short haul point to point and for passive optical networks.

Stability of a Mathematical Model with Piecewise Constant Arguments for Tumor-Immune Interaction Under Drug Therapy

This paper studies a mathematical model for the interaction between tumor cells and Cytotoxic T lymphocytes (CTLs) under drug therapy. We obtain some sufficient conditions for the local and global asymptotical stabilities of the system by using Schur–Cohn criterion and the theory of Lyapunov function. In addition, it is known that the system without any treatment may undergo Neimark–Sacker bifurcation, and there may exist a chaotic region of values of tumor growth rate where the system exhibits chaotic behavior. So it is important to narrow the chaotic region. This may be done by increasing the intensity of the treatment to some extent. Moreover, for a fixed value of tumor growth rate in the chaotic region, a threshold value [Formula: see text] is predicted of the treatment parameter [Formula: see text]. We can see Neimark–Sacker bifurcation of the system when [Formula: see text], and the chaotic behavior for tumor cells ends and the system becomes locally asymptotically stable when [Formula: see text].