Showing papers on "Bifurcation diagram published in 2019"

Entropy Analysis and Neural Network-Based Adaptive Control of a Non-Equilibrium Four-Dimensional Chaotic System with Hidden Attractors

Abstract: Today, four-dimensional chaotic systems are attracting considerable attention because of their special characteristics. This paper presents a non-equilibrium four-dimensional chaotic system with hidden attractors and investigates its dynamical behavior using a bifurcation diagram, as well as three well-known entropy measures, such as approximate entropy, sample entropy, and Fuzzy entropy. In order to stabilize the proposed chaotic system, an adaptive radial-basis function neural network (RBF-NN)-based control method is proposed to represent the model of the uncertain nonlinear dynamics of the system. The Lyapunov direct method-based stability analysis of the proposed approach guarantees that all of the closed-loop signals are semi-globally uniformly ultimately bounded. Also, adaptive learning laws are proposed to tune the weight coefficients of the RBF-NN. The proposed adaptive control approach requires neither the prior information about the uncertain dynamics nor the parameters value of the considered system. Results of simulation validate the performance of the proposed control method.

A simple Jerk-like system without equilibrium: Asymmetric coexisting hidden attractors, bursting oscillation and double full Feigenbaum remerging trees

Abstract: The topic associated with hidden attractor and multistability has been received considerable attention recently. In this paper, a novel no-equilibrium Jerk-like chaotic system is constructed and explored. Particularly, owing to the absence of the equilibria, such a new system can be categorized as a system with hidden attractors. More interestingly, this system holds three conspicuous characteristics. The first one is that various asymmetric coexisting hidden attractors and complicated transient chaos behaviors are obtained. The second one is the new finding of the periodic bursting oscillation and unusual phenomenon of transient periodic bursting oscillation in the system. The third one is the observation of the amazing and rare phenomenon of one to two full Feigenbaum remerging trees, namely, antimonotonicity. To the best knowledge of us, the last two special features are first discovered and have never been reported, especially in such no-equilibrium chaotic system that exhibits hidden attractors. With the help of phase portraits, time series, bifurcation diagram, Lyapunov exponents, chaotic dynamical diagram, basin of attraction and so forth, the rich hidden dynamical properties of this system are systematically analyzed and investigated. Additionally, a hardware electronic circuit on a breadboard is carried out. A very good similarity between the hardware experimental results and the theoretical analysis testifies the feasibility and practicality of this original system.

Multi-scroll hidden attractors with two stable equilibrium points.

Abstract: Multiscroll hidden attractors have attracted extensive research interest in recent years. However, the previously reported multiscroll hidden attractors belong to only one category of hidden attractors, namely, the hidden attractors without equilibrium points. Up to now, multiscroll hidden attractors with stable equilibrium points have not been reported. This paper proposes a multiscroll chaotic system with two equilibrium points. The number of scrolls can be increased by adding breakpoints of a nonlinear function. Moreover, the two equilibrium points are stable node-foci equilibrium points. According to the classification of hidden attractors, the multiscroll attractors generated by a novel system are the hidden attractors with stable equilibrium points. The dynamical characteristics of the novel system are studied using the spectrum of Lyapunov exponents, a bifurcation diagram, and a Poincare map. Furthermore, the novel system is implemented by electronic circuits. The hardware experiment results are consistent with the numerical simulations.

A Novel Chaotic System with Two Circles of Equilibrium Points: Multistability, Electronic Circuit and FPGA Realization

Abstract: This paper introduces a new chaotic system with two circles of equilibrium points. The dynamical properties of the proposed dynamical system are investigated through evaluating Lyapunov exponents, bifurcation diagram and multistability. The qualitative study shows that the new system exhibits coexisting periodic and chaotic attractors for different values of parameters. The new chaotic system is implemented in both analog and digital electronics. In the former case, we introduce the analog circuit of the proposed chaotic system with two circles of equilibrium points using amplifiers, which is simulated in MultiSIM software, version 13.0 and the results of which are in good agreement with the numerical simulations using MATLAB. In addition, we perform the digital implementation of the new chaotic system using field-programmable gate arrays (FPGA), the experimental observations of the attractors of which confirm its suitability to generate chaotic behavior.

A Novel Chaotic System With Boomerang-Shaped Equilibrium, Its Circuit Implementation and Application to Sound Encryption

Abstract: In the chaos literature, there has been much attention paid to chaotic systems with uncountable equilibrium points such as systems with line equilibrium, curve equilibrium. This paper reports a 3-D chaotic system with a closed curve of equilibrium points, which has the shape of a boomerang. Dynamics of the chaotic system with the boomerang equilibrium has been studied by using phase portraits, bifurcation diagram, Lyapunov exponents and Lyapunov dimension. Also, we design an electronic circuit implementation of the theoretical system to check its feasibility. As an application of the new chaotic system, we have derived new results for sound encryption with the new chaotic system.

Entropy Analysis and Image Encryption Application Based on a New Chaotic System Crossing a Cylinder

Abstract: Designing chaotic systems with specific features is a hot topic in nonlinear dynamics. In this study, a novel chaotic system is presented with a unique feature of crossing inside and outside of a cylinder repeatedly. This new system is thoroughly analyzed by the help of the bifurcation diagram, Lyapunov exponents’ spectrum, and entropy measurement. Bifurcation analysis of the proposed system with two initiation methods reveals its multistability. As an engineering application, the system’s efficiency is tested in image encryption. The complexity of the chaotic attractor of the proposed system makes it a proper choice for encryption. States of the chaotic attractor are used to shuffle the rows and columns of the image, and then the shuffled image is XORed with the states of chaotic attractor. The unpredictability of the chaotic attractor makes the encryption method very safe. The performance of the encryption method is analyzed using the histogram, correlation coefficient, Shannon entropy, and encryption quality. The results show that the encryption method using the proposed chaotic system has reliable performance.

Bifurcation diagram and dynamic response of a MEMS resonator with a 1:3 internal resonance

Abstract: The dynamic response of a nonlinear resonator in the presence of resonant mode coupling is studied experimentally and theoretically. For the case of a clamped-clamped beam resonator in the presence of a 1:3 internal resonance, we show that at the onset of internal resonance, steady state oscillations cannot be sustained. At higher drive levels, stable oscillations can be maintained but the resonator amplitude undergoes amplitude modulated responses. We use these dynamic responses to build a bifurcation diagram that can be described remarkably well with a simple model consisting of a Duffing resonator coupled to a linear one.The dynamic response of a nonlinear resonator in the presence of resonant mode coupling is studied experimentally and theoretically. For the case of a clamped-clamped beam resonator in the presence of a 1:3 internal resonance, we show that at the onset of internal resonance, steady state oscillations cannot be sustained. At higher drive levels, stable oscillations can be maintained but the resonator amplitude undergoes amplitude modulated responses. We use these dynamic responses to build a bifurcation diagram that can be described remarkably well with a simple model consisting of a Duffing resonator coupled to a linear one.

On the Stability and Multistability in a Duopoly Game with R&D Spillover and Price Competition

Abstract: In this paper, a dynamical two-stage game with R&D competition and joint profit maximization is built. The stability of all the equilibrium points is discussed through Jury condition, and the stability region of the Nash equilibrium point is then given. The influence of the parameters on the system is discussed, and we find that the firm can even benefit from chaos, when it has higher innovation efficiency and higher adjusting speed. And then the coexistence of multiple attractors is studied using basin of attraction. Our research result shows that the coexisting attractors can be observed in the two-parameter bifurcation diagram. At last, the boundary of feasible region, global bifurcations, and formation mechanism of fractal structure of attracting basin are analyzed through critical curves and noninvertible map theory.

A class of higher-dimensional hyperchaotic maps

Abstract: Based on the hybrid modulation coupling (HMC) pattern, a class of higher-dimensional (HD) hyperchaotic maps is proposed using three one-dimensional (1D) seed maps. The seed maps are chaotic maps or the combination of chaotic maps and non-chaotic maps. Taking the HMC of iterative chaotic map with infinite collapse (ICMIC), Sine map and a linear map (ISL-HMC) as an example, the equilibrium points are mathematically analyzed. The dynamical performance of the 3D ISL-HMC map is evaluated by phase diagram, Lyapunov exponents (LEs), bifurcation diagram and chaos diagram. Furthermore, compared with existing chaotic maps, complexity and distribution characteristic are analyzed. As application of the ISL-HMC map, a pseudorandom number generator (PRNG) is designed and tested by NIST SP 800-22 and TestU01. Experimental results show that the ISL-HMC map has rich dynamical behaviors and good randomness. So this class of HD hyperchaotic maps is a potential model for cryptography and other applications.

Darcy-Bénard convection of Newtonian liquids and Newtonian nanoliquids in cylindrical enclosures and cylindrical annuli

Abstract: An analytical study of linear and nonlinear Darcy-Benard convection of Newtonian liquids and Newtonian nanoliquids confined in a cylindrical porous enclosure is made. The effect of concentric insertion of a solid cylinder into the hollow circular cylinder on onset and heat transport is also investigated. An axisymmetric mode is considered, and the Bessel functions are the eigenfunctions for the problem. The two-phase model is used in the case of nanoliquids. Weakly nonlinear stability analysis is performed by considering the double Fourier-Bessel series expansion for velocity, temperature, and nanoparticle concentration fields. Water well-dispersed with copper nanoparticles of very high thermal conductivity, and one of the five different shapes is chosen as the working medium. The thermophysical properties of nanoliquids are calculated using the phenomenological laws and the mixture theory. It is found that the effect of concentric insertion of a solid cylinder into the hollow cylinder is to enhance the heat transport. The results of rectangular enclosures are obtained as limiting cases of the present study. In general, curvature enhances the heat transport and hence the heat transport is maximum in the case of a cylindrical annulus followed by that in cylindrical and rectangular enclosures. Among the five different shapes of nanoparticles, blade-shaped nanoparticles help transport maximum heat. An analytical expression is obtained for the Hopf bifurcation point in the cases of the fifth-order and the third-order Lorenz models. Regular, chaotic, mildly chaotic, and periodic behaviors of the Lorenz system are discussed using plots of the maximum Lyapunov exponent and the bifurcation diagram.

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

Abstract: 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.

Imperfect Bifurcation for the Quasi-Geostrophic Shallow-Water Equations

Abstract: We study analytical and numerical aspects of the bifurcation diagram of simply-connected rotating vortex patch equilibria for the quasi-geostrophic shallow-water (QGSW) equations. The QGSW equations are a generalisation of the Euler equations and contain an additional parameter, the Rossby deformation length e −1 , which enters in the relation between streamfunction and (potential) vorticity. The Euler equations are recovered in the limit e → 0. We prove, close to circular (Rankine) vortices, the persistence of the bifurcation diagram for arbitrary Rossby deformation length. However we show that the twofold branch, corresponding to Kirchhoff ellipses for the Euler equations, is never connected even for small values e, and indeed is split into a countable set of disjoint connected branches. Accurate numerical calculations of the global structure of the bifurcation diagram and of the limiting equilibrium states are also presented to complement the mathematical analysis.

A New Chaotic System with Line of Equilibria: Dynamics, Passive Control and Circuit Design

Abstract: A new chaotic system with line equilibrium is introduced in this paper. This system consists of five terms with two transcendental nonlinearities and two quadratic nonlinearities. Various tools of dynamical system such as phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, bifurcation diagram and Poincare map are used. It is interesting that this system has a line of fixed points and can display chaotic attractors. Next, this paper discusses control using passive control method. One example is given to insure the theoretical analysis. Finally, for the new chaotic system, An electronic circuit for realizing the chaotic system has been implemented. The numerical simulation by using MATLAB 2010 and implementation of circuit simulations by using MultiSIM 10.0 have been performed in this study.

Reduced Basis Approaches for Parametrized Bifurcation Problems held by Non-linear Von Kármán Equations

Abstract: This work focuses on the computationally efficient detection of the buckling phenomena and bifurcation analysis of the parametric Von Karman plate equations based on reduced order methods and spectral analysis. The computational complexity—due to the fourth order derivative terms, the non-linearity and the parameter dependence—provides an interesting benchmark to test the importance of the reduction strategies, during the construction of the bifurcation diagram by varying the parameter(s). To this end, together the state equations, we carry out also an analysis of the linearized eigenvalue problem, that allows us to better understand the physical behaviour near the bifurcation points, where we lose the uniqueness of solution. We test this automatic methodology also in the two parameter case, understanding the evolution of the first buckling mode.

Chaotic Map with No Fixed Points: Entropy, Implementation and Control

Abstract: A map without equilibrium has been proposed and studied in this paper. The proposed map has no fixed point and exhibits chaos. We have investigated its dynamics and shown its chaotic behavior using tools such as return map, bifurcation diagram and Lyapunov exponents’ diagram. Entropy of this new map has been calculated. Using an open micro-controller platform, the map is implemented, and experimental observation is presented. In addition, two control schemes have been proposed to stabilize and synchronize the chaotic map.

Complex Two-Parameter Bifurcation Diagrams of a Simple Oscillating Circuit

Abstract: It is shown that a simple active oscillating circuit with memristor yields quite complicated two-parameter bifurcation diagrams When two (out of three) parameters of the circuit change simultaneously, various periodic, chaotic and unstable solutions are obtained Those solutions result from parallel computing, as a single two-parameter bifurcation diagram requires solving the underlying model hundred of thousands (or even a few million) times Such diagrams when complemented with one-parameter ones show surprisingly complex bifurcation patterns for the simple circuit having a very weak nonlinearity present in only one of the three differential equations Color two-parameter bifurcation diagrams are included in this brief

Complexity in the muscular blood vessel model with variable fractional derivative and external disturbances

Abstract: In this paper, a variable fractional-order muscular blood vessel system under the influence of different external disturbances is proposed. Specifically, those external disturbances includes periodic forces, stochastic signal and time delay force. Its dynamical analysis is then carried out by employing the bifurcation diagram and phase portraits. Meanwhile, multiscale C 0 complexity measure and multiscale 0–1 test are applied to analyze complexity and to detect chaos. It shows that the multiscale coarse graining process deduces better analysis results. Rich dynamics and high complexity are found in the system. Specifically, the noise and non-constant derivative order functions make the system more complex such that the risk of vascular disease is increased. It provides a reference for understanding the vascular disease with complex external disturbances.

Evolution of the geometric structure of strange attractors of a quasi-zero stiffness vibration isolator

Abstract: The aim of this work is to evaluate the dynamics of the quasi-zero stiffness vibration isolator, with particular emphasis on the density distribution of Poincare cross-section points for different conditions of the system's excitation. The research was mainly focused on model tests carried out to examine the structures representing the geometry of strange attractors of a quasi-zero stiffness vibration isolator. The analysed mechanical system consists of one main spring and two compensation springs. Energy losses are caused by friction at the connection points of the compensation springs and vibroisolated mass. On the basis of a formulated non-linear mathematical model, the ranges of variation of physical parameters of an external dynamic input for which the system motion is chaotic are identified. Taking into account three selected values of the dynamic input amplitude, a bifurcation diagram and a diagram defining the number of phase stream intersections (NPSI) with the abscissa of the phase plane are generated. Based on the diagrams, a solution is selected. The diagram proposed by us (NPSI) is an alternative method of identifying areas in which the system moves chaotically. The classic Poincare cross-section combined by us with information about the density of point distribution on the trajectory intersection with the control plane serve as a basis for the assessment of evolution of the geometric structure of strange attractors. The evaluation of the density distribution of Poincare cross-section points provides important information regarding the evolution of geometrical structures of strange attractors. It has been shown that in relation to large ranges of changes in the control parameter, the geometric structure of the strange attractor is stretched and curved. However, in the area of small changes in the control parameter, the evolution of the attractor’ geometric structure can only be observed by analysing the density of point distribution on the Poincare cross-section by the probability density function (PDF). The areas with the highest density of the Poincare cross-section are usually located in places where the strange attractor is curved. Taking into account the practical application of research results, operational guidelines are formulated.

Novel Nonlinear Function Shift Method for Generating Multiscroll Attractors Using Memristor-Based Control Circuit

Abstract: In this paper, a novel nonlinear function shift method for generating multiscroll attractors is proposed, and a memristor-based control circuit is used to realize the shift controller. Three types of shift modes, namely, horizontal shift, vertical shift, and combined shift, are added in a Jerk system. The dynamic behavior is analyzed through equilibria distribution, bifurcation diagram, Lyapunov exponent spectrum, and phase portraits. Research shows that various equilibria distributions and bifurcation phenomena can be obtained by adding different shifts, thereby producing diverse attractors including periodic orbits, single-scroll, double-scroll, and multiscroll attractors. Furthermore, symmetrical and asymmetrical attractors that are unusual dynamic behaviors can also be found. The circuit construction based on CMOS technology is given, and a memristor-based control circuit is designed to implement the proposed shift method. Different multiscroll attractors can be obtained by regulating the applied control signals instead of redesigning the nonlinear circuit, which simplifies the circuit design of multiscroll system. Our theoretical analysis, numerical simulations, and PSpice simulations together demonstrate the simpleness and effectiveness of the proposed methodology.

Dynamical Analysis of a Novel Complex Chaotic System and Application in Image Diffusion

Abstract: In this paper, a novel complex chaotic system is constructed. The characteristics of new complex chaotic system are analyzed by symmetric, dissipative and stability, then dynamical performances studied using bifurcation diagram, Lyapunov exponent spectrum and complexity. On the basis of this, an image diffusion algorithm is proposed based on the model of law of gravity. Security performances of the proposed algorithm are researched through the key space, statistics, information entropy, noise attack. The experimental results illustrates that the novel complex chaotic system has abundant dynamical behaviors and large parameters range, the security analysis shows that the proposed image encryption algorithm possess higher security features. Therefore, the research will provide theoretical guidance and experimental basis for chaotic secure communication and information security.

Dynamics and Entropy Analysis for a New 4-D Hyperchaotic System with Coexisting Hidden Attractors

Abstract: This paper presents a new no-equilibrium 4-D hyperchaotic multistable system with coexisting hidden attractors. One prominent feature is that by varying the system parameter or initial value, the system can generate several nonlinear complex attractors: periodic, quasiperiodic, multiple topology chaotic, and hyperchaotic. The dynamics and complexity of the proposed system were investigated through Lyapunov exponents (LEs), a bifurcation diagram, a Poincare map, and spectral entropy (SE). The simulation and calculation results show that the proposed multistable system has very rich and complex hidden dynamic characteristics. Additionally, the circuit of the chaotic system is designed to verify the physical realizability of the system. This study provides new insights into uncovering the dynamic characteristics of the coexisting hidden attractors system and provides a new choice for nonlinear control or chaotic secure communication technology.

A new supply chain model and its synchronization behaviour

Abstract: Marketing researchers and practitioners have noticed that the demand of many retail items are proportional to the amount of inventory displayed. In this paper, a new supply chain model is proposed assuming that demand of a product does not increase monotonically with the increase of inventory. In this model it is assumed that the demand has a saturation level and it does not increase monotonically with the inventory. The stationary solutions (fixed points) of the model are determined and their stability natures are determined using Routh–Hurwitz criteria. The variation of the model dynamics for different parameter values are presented numerically by drawing time evolution diagrams and phase diagrams. Bifurcation diagram of the model with respect to the demand saturation parameter is presented to note the importance of the parameter. Next, synchronization behavior of two coupled identical supply chain models under both unidirectional and bidirectional coupling are discussed. Sufficient conditions for synchronization in case of bidirectional coupling are derived. Numerical simulation results are presented to validate the analytical findings.

A new chaotic oscillator containing generalised memristor, single op-amp and RLC with chaos suppression and an application for the random number generation

Abstract: In this paper, a new chaotic oscillator consists of a single op-amp, two capacitors, one resistor, one inductor, and memristive diode bridge cascaded with an inductor is proposed. The proposed chaotic oscillator has a line of equilibria. In the new oscillator circuit, negative feedback, i.e. inverting terminal of the op-amp is used, and the non-inverting terminal is grounded. The new oscillator has chaotic, periodic, quasi-periodic behaviours, as seen from the Lyapunov spectrum plots. Some more theoretical and numerical tools are used to present the dynamical behaviours of the new oscillator like bifurcation diagram, phase plot. Further, a non-singular terminal sliding mode control (N-TSMC) is designed for the suppression of the chaotic states of the new oscillator. An application of the new oscillator is shown by designing a chaos-based random number generator. Raspberry Pi 3 is used for the realisation of the random number generator.