Exponent Diagrams Research Articles

Enhancing Ikeda Time Delay System by Breaking the Symmetry of Sine Nonlinearity

In the present contribution, an asymmetric central contraction mutation (ACCM) model is proposed to enhance the Ikeda time delay system. The modified Ikeda system model is designed by introducing a superimposed tanh function term into the sine nonlinearity term. Stability and Hopf bifurcation characteristics of the system are analyzed theoretically. Numerical simulations, carried out in terms of bifurcation diagrams, Lyapunov exponents spectrum, phase portraits, and two‐parameter (2D) largest Lyapunov exponent diagrams are employed to highlight the complex dynamical behaviors exhibited by the enhanced system. The results indicate that the modified system has rich dynamical behaviors including limit cycle, multiscroll hyperchaos, chaos, and hyperchaos. Moreover, as a major outcome of this paper, considering the fragile chaos phenomenon, the ACCM‐Ikeda time delay system has better dynamical complexity and larger connected chaotic parameter spaces (connectedness means that there is no stripe corresponding to nonchaotic dynamics embedded in the chaos regions).

Nonlinear vibrations of carbon fiber reinforced polymer laminated cylindrical shell under non-normal boundary conditions with 1:2 internal resonance

In this paper, the nonlinear vibrations of a carbon fiber reinforced polymer (CFRP) laminated cylindrical shell are investigated with 1:2 internal resonance, primary parametric resonance and 1/2 subharmonic resonance. The radial line load and axial excitation are acting on the two free ends of the CFRP laminated cylindrical shell. The partial differential governing equations of motion for the CFRP laminated cylindrical shell are derived by utilizing the von-Karman type nonlinear geometric relationship, the first-order shear deformation theory and Hamilton principle. Galerkin method is used to obtain two-degrees-of-freedom nonlinear ordinary differential governing equations of motion along the radial displacement of the CFRP laminated cylindrical shell under the non-normal boundary conditions. The ordinary differential governing equations are solved as a system of four-dimensional average equations by the second-order approximate multiple scale method. The first two order dimensionless natural frequencies versus the ratio of length to thickness L/h, the frequency-response curves, the force-response curves, the bifurcation diagrams, the three-dimensional bifurcation diagrams, the maximum Lyapunov exponent diagrams, the phase portraits, the time history diagrams and the Poincare maps are obtained by numerical calculations. The influences of the radial line load, the axial excitation as well as the detuning parameter of the CFRP laminated cylindrical shell on the 1:2 internal resonant behaviors are investigated.

Reversals of period doubling, coexisting multiple attractors, and offset boosting in a novel memristive diode bridge-based hyperjerk circuit

In this paper, a new memristive diode bridge-based RC hyperjerk circuit is proposed. This new memristive hyperjerk oscillator (MHO) is obtained from the autonomous 4-D hyperjerk circuit (Leutcho et al. in Chaos Solitons Fractals 107:67–87, 2018) by replacing the nonlinear component (formed by two antiparallel diodes) with a first order memristive diode bridge. The circuit is described by a fifth-order continuous time autonomous (‘elegant’) hyperjerk system with smooth nonlinearities. The dynamics of the system is investigated in terms of equilibrium points and stability, phase portraits, bifurcation diagrams and two-parameter Lyapunov exponents diagrams. The numerical analysis of the model reveals interesting behaviors such as period-doubling, chaos, offset boosting, symmetry recovering crisis, antimonotonicity (i.e. concurrent creation and destruction of periodic orbits) and several coexisting bifurcations as well. One of the most attractive features of the new MHO considered in this work is the presence of several coexisting attractors (e.g. coexistence of two, three, four, five, six, seven, or nine attractors) for some suitable sets of system parameters, depending on the choice of initial conditions. Accordingly, the distribution of initial conditions related to each coexisting attractor is computed to highlight different basins of attraction. Laboratory experimental measurements are carried out to verify the theoretical analysis.

Two bifurcation routes to multiple chaotic coexistence in an inertial two-neural system with time delay

In this article, we establish an inertial two-neural system with time delay and illustrate the stable coexistence of three chaotic attractors that arise via two different bifurcation routes, i.e., the period-doubling and quasi-periodic bifurcations. So, we firstly analyze the system equilibria by nullcline curves. By the pitchfork/saddle-node bifurcation of the trivial/nontrivial equilibria, the system parameter ( $$c_{1}$$ , $$c_{2})$$ -plane is divided into the different regions having the different number of equilibrium. Further, the trivial and nontrivial equilibria will lose their stability and bifurcate into periodic orbits as the effect of time delay. The system has the stable coexistence of two periodic orbits near the nontrivial equilibria. For some delayed regions, the system illustrates the stability switching, i.e., the dynamic behaviors lost, retrieved, and lastly lost their stability with increase in delay. Using the Hopf–Hopf bifurcation analysis, we find a quasi-periodic orbit surrounded by the trivial equilibrium. Lastly, based on numerical simulations, such as phase portrait, Poincare section, Lyapunov exponent, and one-dimensional bifurcation diagram, we further investigate the dynamical evolution of the periodic and quasi-periodic orbits. The results show that the neural system presents the multiple stable coexistence with three chaotic attractors by the different bifurcation routes, i.e., the period-doubling and quasi-periodic bifurcations.

A family of fully integrated CMOS chaos generators with strictly 1-D linear-piecewise chaos maps

A family of three complementary metal-oxide semiconductor (CMOS) chaos generators is presented. The proposed scheme starts by obtaining strictly 1-D piecewise-linear chaos maps using a stroboscopic sampling method. The chaotic behavior is then analyzed with the help of numerical simulations using Mathematica. As required for practical applications, the robustness of the derived chaos maps was also confirmed by analyzing bifurcation and Lyapunov exponent diagrams. For verification, a tent-shape chaos generator was implemented using the 0.18- $$\upmu $$ m CMOS process. The simulation as well as measurement results confirmed the proposed scheme. The key advantages of the proposed chaos generators with varied fabrication parameters, including compact size, low power consumption, simple achievement of chaos, and robust chaos generation, recommend them as a solution in design of fully integrated chaotic circuits for applications where noise-like signal sources are required.

A new 4D chaotic system with hidden attractor and its engineering applications: Analog circuit design and field programmable gate array implementation

Nowadays, designing chaotic systems with hidden attractor is one of the most interesting topics in nonlinear dynamics and chaos. In this paper, a new 4D chaotic system is proposed. This new chaotic system has no equilibria, and so it belongs to the category of systems with hidden attractors. Dynamical features of this system are investigated with the help of its state-space portraits, bifurcation diagram, Lyapunov exponents diagram, and basin of attraction. Also a hardware realisation of this system is proposed by using field programmable gate arrays (FPGA). In addition, an electronic circuit design for the chaotic system is introduced.

Dynamic analysis of a unique jerk system with a smoothly adjustable symmetry and nonlinearity: Reversals of period doubling, offset boosting and coexisting bifurcations

We investigated the dynamics of a novel jerk system generalized with a single parametric nonlinearity in the form φk(x)=0.5(exp(kx)-exp(-x)). The form of nonlinearity is interesting in the sense that the corresponding circuit realization involves only off-the shelf electronic components such as resistors, semiconductor diodes and operational amplifiers. Parameter k (i.e. a control resistor) serves to smoothly adjust the shape of the nonlinearity, and hence the symmetry of the system. In particular, for k=1, the nonlinearity is a hyperbolic sine, and thus the system is point symmetric about the origin. For k≠1, the system is non-symmetric. To the best of the authors’ knowledge, this interesting feature is unique and has not yet been discussed in chaotic systems. The dynamic analysis of the model involves a preliminary study of basic properties such as the nature of the fixed point, the transitions to chaos (bifurcation diagrams), the phase portraits as well as the Lyapunov exponent diagrams. When monitoring the system parameters, some striking phenomena are found including period doubling cascade bifurcation, reverse bifurcations, merging crises, offset boosting, coexisting bifurcations and hysteresis. Several windows in the parameters’ space are depicted in which the novel jerk system displays a plethora of coexisting asymmetric and symmetric attractors (i.e. coexistence of two, three or four different attractors) depending only on the selection of initial states. The magnetization of state space due to the presence of multiple competing solutions is illustrated by means of basins of attraction. More importantly, multistability in the symmetry boundary is discussed by monitoring the bifurcation parameter k. Laboratory experimental results based on the suitably designed electronic analogue of the model confirm the numerical findings.

Dynamical analysis and multistability in autonomous hyperchaotic oscillator with experimental verification

In this contribution, a modified oscillator of Tamasevicius et al. (Electron Lett 33:542–544, 1997) (referred to as the mTCMNL oscillator hereafter) is introduced with antiparallel diodes as nonlinear elements. The model is described by a continuous time of four-dimensional autonomous system with hyperbolic sine nonlinearity based on Shockley diode model. Various methods for characterizing chaos/hyperchaos including bifurcation diagrams, Lyapunov exponents spectrum, frequency spectra, phase portraits, Poincare sections and two parameter Lyapunov exponents diagrams are exploited to point out the rich dynamical behaviors in the model. Numerical results indicate that the system displays extremely rich dynamical behaviors including periodic windows, torus, chaotic and hyperchaotic oscillations. One of the main findings of this work is the presence of a region in the parameter space in which the mTCMNL experiences hysteretic behaviors. This later singularity/phenomenon is marked by the coexistence of multiple attractors (i.e., coexistence of asymmetric pair of periodic, torus and chaotic attractors with symmetric periodic, torus and chaotic attractors), for the same parameters settings. Basins of attractions of various competing attractors depicts symmetric complex basin boundaries, thus suggesting possible jumps between coexisting solutions (i.e., asymmetric pair of attractors with symmetric one) in experiments. A predominant route to chaos/hyperchaos observed in the system for different system parameters is the Afraimovich–Shilnikov scenario with tiny periodic regions. Experimental results from real-time circuit implementation are in good agreement with numerical analysis.

Nonlinear radial breathing vibrations of CFRP laminated cylindrical shell with non-normal boundary conditions subjected to axial pressure and radial line load at two ends

The cylindrical shell structure is an important kind of the primary structures in many engineering community. Under axial excitation and radial line load, the nonlinear radial breathing vibrations of a carbon fiber reinforced polymer (CFRP) laminated cylindrical shell with different temperature is investigated in this paper. Based on von Karman type nonlinear geometric relationship, the first-order shear deformation shell theory is applied to model the kinematics of deformations, and Hamilton’s principle is used to drive the differential governing equations. Galerkin method is utilized to obtain the nonlinear ordinary differential governing equations along the radial displacement of the system under the non-normal boundary conditions that one generatrix of the cylindrical shell is clamped and both ends of the shell are free. The bifurcation diagrams, maximum Lyapunov exponent diagrams, phase portraits, the time history diagrams, three-dimensional phase portraits and poincare maps are obtained by using the fourth-order Runge-Kutta algorithm. The influences of the radial line load, the axial excitation as well as the ratio of length to thickness on the nonlinear radial breath vibrations characteristics of the CFRP laminated cylindrical shell are studied in detail by numerical calculations. The results show that the radial line load, axial excitation and ratio of length to thickness can be used to control the nonlinear dynamic motions.

Memristor-based multi-scroll chaotic system and its pulse synchronization control

The memristor is a nonlinear element and intrinsically possesses memory function. When it works as nonlinear part of a chaotic system, the complexity and the randomness of signal will be enhanced. In this paper memristor is introduced into a three-dimensional chaotic system based on the augmented L system. The interesting and promising behaviors of complex single, double and four-scroll chaotic attractors generated only by varying a parameter have not been reported in memristive chaotic system and thus they deserve to be further investigated. It is also obvious that such a simple change of one parameter could be used to generate a variety of quite complex attractors. Therefore, as a nonlinear device the memristor plays an important role in this system. Firstly, some basic dynamical properties of the memristive chaotic system, including symmetry and in-variance, the existence of attractor, equilibrium, and stability are investigated in detail. By numerically simulating the power spectrum, Lyapunov exponent, Poincare map and bifurcation diagram, in this paper we verify that the proposed system has abundant dynamical behaviors. The sensitivities of system parameters to the chaotic behaviors are further explored by calculating, in detail, its Lyapunov exponent spectrum and bifurcation diagrams. The results of simulation and experiment are in good agreement, thereby proving the veracity of analysis. The memristive chaotic circuit is designed using the memristor, operational amplifier, analog multiplier and other conventional components. The circuit implementation of the memristive system is simulated using SPICE (simulation program with integrated circuit emphasis). The SPICE simulation results and the theoretical analysis are found to be in good agreement, and thus verifying that the system can produce chaos. Pulse synchronization has the following characteristics: low energy consumption, fast synchronization and easy-to-implement single-channel transmission. Therefore, it is more practical in chaotic secure communication. Subsequently the pulse chaos synchronization is realized from the perspective of the maximum Lyapunov exponent, and numerical simulations show the existence of new memristive chaotic system and the feasibility of pulse synchronization control, and also provide an experimental basis for further studying the applications of the memristive chaotic system in voice secure communication and information processing.

Chaotic analysis of fractional Willis delayed aneurysm system

The dynamic system of Willis aneurysm (WAS) has played an important role in theoretical and clinical research of cerebral aneurysms. Fractional differential is an effective mathematical tool that can describe the cerebral aneurysm system accurately and profoundly. However, the existing fractional Willis aneurysm system (FWAS) cannot describe the delayed aneurysm rupture of unknown cause in reality. Therefore, by introducing the time-delay factors into the existing fractional Willis aneurysm system as a rational extension, a new fractional Willis aneurysm system with time-delay (FWASTD) is proposed in this paper.First, FWASTD is introduced in the context, and the comparison of time sequences map between FWAS and FWASTD proves that FWASTD is feasible in the depiction of time-delay situations. The bifurcation diagram and the largest Lyapunov exponent diagram as well as the phase diagram of fractional order also confirm the chaotic characteristics of the FWASTD.Then, the classical analysis methods in chaotic dynamics, such as time series diagram, phase diagram and Poincar section are used to analyze FWASTD in detail. When studying the diagrams of time-delay factors for the important physiological parameters of the system, we find that blood flow resistance coefficient can exert a remarkable effect on the system stability under time-delay. Besides, the experimental results show that the FWASTD becomes stable with the increase of blood flow resistance under a certain condition. Usually, promoting thrombosis is a kind of adjunctive therapy in clinic for cerebral aneurysm. The results of this part can accord with the treatment in clinic and has great significance in clinical diagnosis.Finally, as the chaotic state of the time-delay system indicates that cerebral aneurysm is in a dangerous situation, the primary task of the control for this new system is to achieve stability rather than synchronization. The stability theory of fractional time-delayed system is adopted in a strict proof of the uniqueness of solution for the FWASTD. To make FWASTD stable, a corresponding linear controller is designed based on the stability theory of fractional order delay system. The numerical simulation indicates that the linear controller can control the blood flow velocity and speed up the periodic fluctuation within a small range, which illustrates that it is not easy to rupture the cerebral aneurysm. We also make self-synchronization control between FWASTD and FWAS just in case that the coefficients of the system are not clear.The research results in this paper, to some extent, can serve as theoretical guidance for the clinical diagnosis and the treatment of aneurysm.

Codimension-two bifurcation analysis of a discrete predator–prey model with nonmonotonic functional response

In this paper, we investigate dynamical behaviours of a discrete predator–prey model with nonmonotonic functional response. Codimension-2 bifurcations associated with 1:2, 1:3 and 1:4 resonances are analyzed by using bifurcation theory. Codimension-2 bifurcation diagrams, maximum Lyapunov exponents diagrams and phase portraits, which not only illustrate the validity of the theoretical results but also display some interesting complex dynamical behaviours, are obtained by numerical simulations.

Hyperchaotic Analysis and Adaptive Projective Synchronization of Nonlinear Dynamical System

In this paper, a new nonlinear dynamical system has been studied which is obtained from the 3D chaotic system. The hyperchaotic analysis of the new system is checked in terms of dissipation, equilibrium points and their stability, Lyapunov exponent, time series, phase portraits, Poincare section and bifurcation diagram. Furthermore, the adaptive projective synchronization technique is used to synchronize the novel hyperchaotic system. A brief theoretical analysis and simulation results are presented to prove the behavior of the novel hyperchaotic system.

Mode locking and quasiperiodicity in a discrete-time Chialvo neuron model

The two-dimensional parameter spaces of a discrete-time Chialvo neuron model are investigated. Our studies demonstrate that for all our choice of two parameters (i) the fixed point is destabilized via Neimark–Sacker bifurcation; (ii) there exist mode locking structures like Arnold tongues and shrimps, with periods organized in a Farey tree sequence, embedded in quasiperiodic/chaotic region. We determine analytically the location of the parameter sets where Neimark–Sacker bifurcation occurs, and the location on this curve where Arnold tongues of arbitrary period are born. Properties of the transition that follows the so-called two-torus from quasiperiodicity to chaos are presented clearly and proved strictly by using numerical simulations such as bifurcation diagrams, the largest Lyapunov exponent diagram on MATLAB and C++.

Bifurcations and Chaos in the Duffing Equation with Parametric Excitation and Single External Forcing

We study the Duffing equation with parametric excitation and single external forcing and obtain abundant dynamical behaviors of bifurcations and chaos. The criteria of chaos of the Duffing equation under periodic perturbation are obtained through the Melnikov method. And the existence of chaos of the averaged system of the Duffing equation under the quasi-periodic perturbation [Formula: see text] (where [Formula: see text] is not rational relative to [Formula: see text]) and [Formula: see text] is shown, but the existence of chaos of averaged system of the Duffing equation cannot be proved when [Formula: see text],[Formula: see text]7–15, whereas the occurrence of chaos in the original system can be shown by numerical simulation. Numerical simulations not only show the correctness of the theoretical analysis but also exhibit some new complex dynamical behaviors, including homoclinic or heteroclinic bifurcation surfaces, bifurcation diagrams, Lyapunov exponent diagrams, phase portraits and Poincaré maps. We find a large chaotic region with some solitary period parameter points, a large period and quasi-period region with some solitary chaotic parameter points, period-doubling to chaos and chaos to inverse period-doubling, nondense curvilinear chaotic attractor, nonattracting chaotic motion, nonchaotic attracting set, fragmental chaotic attractors. Almost chaotic motion and almost nonchaotic motion appear through adjusting the parameters of the Duffing system, which can be taken as a strategy of chaotic control or a strategy of chaotic motion to nonchaotic motion.

Coexistence of Multiple Attractors, Metastable Chaos and Bursting Oscillations in a Multiscroll Memristive Chaotic Circuit

In this paper, some complex nonlinear behaviors in a four-dimensional multiscroll autonomous memristor based chaotic system are investigated. This system is derived from the three-dimensional autonomous charge-controlled Muthuswamy–Chua simplest chaotic circuit. The system can generate four different coexisting attractors for a fixed set of parameters and different initial conditions. This phenomenon is relatively rare given that we have four different attractors namely: an equilibrium point, a stable limit cycle, a 16-peak limit cycle and a strange attractor that coexist in the system within a wide range of parameters. The nonlinear phenomenon of transient chaos is studied and revealed numerically in Matlab and Pspice environments. The complex transient dynamics of this memristive system under different initial states shows that the transient time depends strongly on the initial conditions. Moreover, this model displays spiking and bursting oscillations. The bursting behavior is classified according to the dynamics of separated slow and fast subsystems. It is shown to be of the fold-Hopf type. These complex dynamical behaviors of this system are investigated by means of numerical simulations and via Pspice circuit simulations. The use of bifurcation diagrams, Lyapunov exponents diagrams, power spectrums, phase portraits, time series, isospike diagram, basin of attraction, clearly shows these complex phenomena.

Nonlinear Dynamic Analysis of a Simplest Fractional-Order Delayed Memristive Chaotic System

A simplest fractional-order delayed memristive chaotic system is investigated in order to analyze the nonlinear dynamics of the system. The stability and bifurcation behaviors of this system are initially investigated, where time delay is selected as the bifurcation parameter. Some explicit conditions for describing the stability interval and the transversality condition of the emergence for Hopf bifurcation are derived. The period doubling route to chaos behaviors of such a system is discussed by using a bifurcation diagram, a phase diagram, a time-domain diagram, and the largest Lyapunov exponents (LLEs) diagram. Specifically, we study the influence of time delay on the chaotic behavior, and find that when time delay increases, the transitions from one cycle to two cycles, two cycles to four cycles, and four cycles to chaos are observed in this system model. Corresponding critical values of time delay are determined, showing the lowest orders for chaos in the fractional-order delayed memristive system. Finally, numerical simulations are provided to verify the correctness of theoretical analysis using the modified Adams–Bashforth–Moulton method.

Bifurcations and Chaos in the Duffing Equation with One Degenerate Saddle Point and Single External Forcing

In this paper, we study the Duffing equation with one degenerate saddle point and one external forcing and obtain the criteria of chaos of Duffing equation under periodic perturbation through Melnikov method. Numerical simulations not only show the correctness of the theoretical analysis but also exhibit the more new complex dynamical behaviors, including homoclinic bifurcation, bifurcation diagrams, maximum Lyapunov exponents diagrams, phase portraits and Poincaré maps.

Shooting With Deflation Algorithm-Based Nonlinear Response and Neimark-Sacker Bifurcation and Chaos in Floating Ring Bearing Systems

The double-sided fluid film force on the inner and outer ring surfaces of a floating ring bearing (FRB) creates strong nonlinear response characteristics such as coexistence of multiple orbits, Hopf bifurcation, Neimark-Sacker (N-S) bifurcation, and chaos in operations. An improved autonomous shooting with deflation algorithm is applied to a rigid rotor supported by FRBs for numerically analyzing its nonlinear behavior. The method enhances computation efficiency by avoiding previously found solutions in the numerical-based search. The solution manifold for phase state and period is obtained using arc-length continuation. It was determined that the FRB-rotor system has multiple response states near Hopf and N-S bifurcation points, and the bifurcation scenario depends on the ratio of floating ring length and diameter (L/D). Since multiple responses coexist under the same operating conditions, simulation of jumps between two stable limit cycles from potential disturbance such as sudden base excitation is demonstrated. In addition, this paper investigates chaotic motions in the FRB-rotor system, utilizing four different approaches, strange attractor, Lyapunov exponent, frequency spectrum, and bifurcation diagram. A numerical case study for quenching the large amplitude motion by adding unbalance force is provided and the result shows synchronization, i.e., subsynchronous frequency components are suppressed. In this research, the fluid film forces on the FRB are determined by applying the finite element method while prior work has utilized a short bearing approximation. Simulation response comparisons between the short bearing and finite bearing models are discussed.

Hopf bifurcation and chaos in a fractional order delayed memristor-based chaotic circuit system

This paper present Hopf bifurcation and chaos in a fractional order delayed memristor-based chaotic circuit system. Firstly, regarding the time delay τ as a bifurcation parameter, we investigate the stability and bifurcation behaviors of this fractional order delayed memristor-based chaotic circuit system. Some explicit conditions for describing the stability interval and emergence of Hopf bifurcation are derived. Secondly, corresponding to different system parameters, the complex dynamics behaviors of this system are discussed by using the bifurcation diagrams, the Max Lyapunov exponents (MLEs) diagram, the time domain waveforms, the phase portraits and the power spectrums. Thirdly, we study the influence of the two parameters (time delay τ and fractional order q) on the chaotic behavior, and it is found when time delay τ and fractional order q increases, the transitions from period one to period two and period four to chaos are observed in this memristor–based system. Meanwhile, corresponding critical values of time delay τ and fractional order q, the lowest orders q and the minimum time delay τ for generating chaos in the fractional order delayed memristor–based system are determined, respectively. Also, when the system occurs period one, the corresponding frequency is verified theoretically and experimentally. Finally, numerical simulations are provided to demonstrate the validity of theoretical analysis using the improved Adams–Bashforth–Moulton algorithm.