Symmetry-breaking, amplitude control and constant Lyapunov exponent based on single parameter snap flows

In this paper, we introduce a novel snap system with a unique parameterized piecewise quadratic nonlinearity in the form $$\psi _{n} \left( x \right) =x-n\left| x \right| -\mathrm{d}x\left| x \right|$$where n controls the symmetry of the system and serves as total amplitude control of the signals. The model is described by a continuous time 4D autonomous system (ODE) with smooth conditional nonlinearity. We study the chaos mechanism with respect to system parameters both in the symmetric and asymmetric modes of oscillations by exploiting bifurcation diagrams, basin of attractions and phase portraits as main tools. In particular, for$$n=0$$, the system displays a perfect symmetry and develops rich dynamics including period doubling sequences, merging crisis, hysteresis, and coexisting multiple symmetric attractors. For$$n\ne 0$$, the system is non-symmetric and the space magnetization induced more complex and striking effects including asymmetric double scroll strange attractors, parallel branches, and asymmetric basin boundary leads to many coexisting asymmetric stable states and so on. Apart from all these complex and rich phenomena, many others including offset-boosting with total amplitude control, and antimonotonicity are also presented. Finally, Pspice based simulations of the proposed system are included

Analysis of a No Equilibrium Linear Resistive-Capacitive-Inductance Shunted Junction Model, Dynamics, Synchronization, and Application to Digital Cryptography in Its Fractional-Order Form

A linear resistive-capacitive-inductance shunted junction (LRCLSJ) model obtained by replacing the nonlinear piecewise resistance of a nonlinear resistive-capacitive-inductance shunted junction (NRCLSJ) model by a linear resistance is analyzed in this paper. The LRCLSJ model has two or no equilibrium points depending on the dc bias current. For a suitable choice of the parameters, the LRCLSJ model without equilibrium point can exhibit regular and fast spiking, intrinsic and periodic bursting, and periodic and chaotic behaviors. We show that the LRCLSJ model displays similar dynamical behaviors as the NRCLSJ model. Moreover the coexistence between periodic and chaotic attractors is found in the LRCLSJ model for specific parameters. The lowest order of the commensurate form of the no equilibrium LRCLSJ model to exhibit chaotic behavior is found to be 2.934. Moreover, adaptive finite-time synchronization with parameter estimation is applied to achieve synchronization of unidirectional coupled identical fractional-order form of chaotic no equilibrium LRCLSJ models. Finally, a cryptographic encryption scheme with the help of the finite-time synchronization of fractional-order chaotic no equilibrium LRCLSJ models is illustrated through a numerical example, showing that a high level security device can be produced using this system

Effects of symmetry-breaking on the dynamics of the Shinriki’s oscillator

We investigate the dynamics of the well-known Shinriki’s oscillator both in its symmetric and asymmetric modes of operation. Instead of using the classical approximate cubic model of the Shinriki’s oscillator, we propose a close form model by exploiting the Shockley exponential diode equation. The proposed model takes into account the intrinsic characteristics of semiconductor diodes forming the nonlinear part (i.e., the positive conductance). We address the realistic issue of symmetry-breaking by considering different numbers of diodes within the two branches of the positive conductance. The dynamics of the system is investigated by exploiting conventional nonlinear analysis tools such as bifurcation diagrams, phase-space trajectories plots, basins of attractions, and graphs of Lyapunov exponent as well. In the symmetric mode of operation, the system experiences coexisting symmetric attractors, period-doubling route to chaos, and merging crisis. The symmetry breaking analysis yields two asymmetric coexisting bifurcation branches (i.e., asymmetric bi-stability) each of which exhibits its own sequence of bifurcations to chaos when monitoring the main control parameter. In this special mode, the merging process never occurs; instead, one of the bifurcation branches vanishes when decreasing the control parameter beyond a critical value following a crisis event. The theoretical results are validated by carrying out laboratory experimental studies of the physical circuit

Fractional-order two-component oscillator: stability and network synchronization using a reduced number of control signals

In this paper, a fractional-order version of a chaotic circuit made simply of two non-idealized components operating at high frequency is presented. The fractional-order version of the Hopf bifurcation is found when the bias voltage source and the fractional-order of the system increase. Using Adams–Bashforth–Moulton predictor–corrector scheme, dynamic behaviors are displayed in two complementary types of stability diagrams, namely the two-parameter Lyapunov exponents and the isospike diagrams. The latest being a more fruitful type of stability diagrams based on counting the number of spikes contained in one period of the periodic oscillations. These two complementary types of stability diagrams are reported for the first time in the fractional-order dynamical systems. Furthermore, a new fractional-order adaptive sliding mode controller using a reduced number of control signals was built for the stabilization of a fractional-order complex dynamical network. Two examples are shown on a fractional-order complex dynamical network where the nodes are made of fractional-order two-component circuits. Firstly, we consider an ideal channel, and secondly, a non ideal one. In each case, increasing of the coupling strength leads to the phase transition in the fractional-order complex network

Coexisting attractors and bursting oscillations in IFOC of 3-phase induction motor

The dynamics of indirect field oriented control (IFOC) of 3-phase induction motor is studied in this paper. The dynamical behaviors of the studied system are performed using bifurcation diagrams, maximum Lyapunov exponent plots, phase portraits, and isospike diagram. The numerical simulation results reveal that the IFOC of 3-phase induction motor displays coexistence of attractors for the same set of IFOC of 3-phase induction motor parameters, periodic and chaotic bursting oscillations. Basins of attraction of different competing attractors are plotted showing complex basin boundaries. The numerical simulation finding are validated by the OrCAD-Spice results

Multistability Analysis and Function Projective Synchronization in Relay Coupled Oscillators

Regions of stability phases discovered in a general class of Genesio−Tesi chaotic oscillators are proposed. In a relatively large region of two-parameter space, the system has coexisting point attractors and limit cycles. The variation of two parameters is used to characterize the multistability by plotting the isospike diagrams for two nonsymmetric initial conditions. The parameters window in which the jerk system exhibits the unusual and striking feature of multiple attractors (e.g., coexistence of six disconnected periodic chaotic attractors and three-point attraction) is investigated. The second aspect of this study presents the synchronization of systems that act as mediators between two dynamical units that, in turn, show function projective synchronization (FPS) with each other. These are the so-called relay systems. In a wide range of operating parameters; this setup leads to synchronization between the outer circuits, while the relaying element remains unsynchronized. The results show that the coupled systems can achieve function projective synchronization in a determined time despite the unpredictability of the scaling function. In the coupling path, the outer dynamical systems show finite-time synchronization of their outputs, that is, displaying the same dynamics at exactly the same moment. Further, this effect is rather general and it has a wide range of applications where sustained oscillations should be retained for proper functioning of the systems