Lyapunov Exponent Spectrum Research Articles

Extreme Multistability and Extreme Events in a Novel Chaotic Circuit with Hidden Attractors

Extreme multistable systems can show vibrant dynamical properties and infinitely many coexisting attractors generated by changing the initial conditions while the system and its parameters remain unchanged. On the other hand, the frequency of extreme events in society is increasing which could have a catastrophic influence on human life worldwide. Thus, complex systems that can model such behaviors are very significant in order to avoid or control various extreme events. Also, hidden attractors are a crucial issue in nonlinear dynamics since they cannot be located and recognized with conventional methods. Hence, finding such systems is a vital task. This paper proposes a novel five-dimensional autonomous chaotic system with a line of equilibria, which generates hidden attractors. Furthermore, this system can exhibit extreme multistability and extreme events simultaneously. The fascinating features of this system are examined by dynamical analysis tools such as Poincaré sections, connecting curves, bifurcation diagrams, Lyapunov exponents spectra, and attraction basins. Moreover, the reliability of the introduced system is confirmed through analog electrical circuit design so that this chaotic circuit can be employed in many engineering fields.

Dynamical Analysis of the Incommensurate Fractional-Order Hopfield Neural Network System and Its Digital Circuit Realization

Dynamical analysis of the incommensurate fractional-order neural network is a novel topic in the field of chaos research. This article investigates a Hopfield neural network (HNN) system in view of incommensurate fractional orders. Using the Adomian decomposition method (ADM) algorithm, the solution of the incommensurate fractional-order Hopfield neural network (FOHNN) system is solved. The equilibrium point of the system is discussed, and the dissipative characteristics are verified and discussed. By varying the order values of the proposed system, different dynamical behaviors of the incommensurate FOHNN system are explored and discussed via bifurcation diagrams, the Lyapunov exponent spectrum, complexity, etc. Finally, using the DSP platform to implement the system, the results are in good agreement with those of the simulation. The actual results indicate that the system shows many complex and interesting phenomena, such as attractor coexistence and an inversion property, with dynamic changes of the order of q0, q1, and q2. These phenomena provide important insights for simulating complex neural system states in pathological conditions and provide the theoretical basis for the later study of incommensurate fractional-order neural network systems.

Modeling and complexity analysis of a fractional-order memristor conservative chaotic system

Memristors are often utilized in circuit model analysis as one of the fundamental circuit components. In this paper, a five-dimensional conservative memristor chaotic system is built after the introduction of the memristor into a four-dimensional conservative chaotic system. The dynamic changes of the system are examined using phase diagram, mean value, and Lyapunov exponent spectrum. A line equilibrium point, symmetry and multi-stability are characteristics of the system; the phase trajectory can also produce shrinking and structure transformation behavior with the change of parameters. Furthermore, the system has initial offset boosting behaviors, conservative flows of it can be altered in position by changing two initial values, respectively. Most notably, we discover that the complexity of the system rises with the inclusion of memristor and again with the addition of fractional differential operators. It is shown that the complexity of chaotic systems may increase with the addition of memristors and fractional-order differential operators. At last, the NIST is used to test the randomness of the sequence, and the system's physical realizability is confirmed by the DSP platform.

Optical Chaos in Saturated Nonlinear Media

In the study of the evolution of Gaussian beam in saturated nonlinear media, it is found that the probability of optical rogue waves changes with the change of nonlinearity. The light intensity distribution on the exit surface of nonlinear medium can be characterized by scintillation index, and the change of rogue wave corresponds to the evolution of scintillation index. The rogue wave probability shows a complex trend with the evolution of nonlinearity. The Lyapunov exponent and power spectrum method are used to determine that the probability of rogue wave is chaotic with nonlinear evolution.

Fingerprint construction of optical transmitters based on the characteristic of electro-optic chaos for secure authentication.

In this paper, an optical transmitter authentication method using hardware fingerprints based on the characteristic of electro-optic chaos is proposed. By means of phase space reconstruction of chaotic time series generated by an electro-optic feedback loop, the largest Lyapunov exponent spectrum (LLES) is defined and used as the hardware fingerprint for secure authentication. The time division multiplexing (TDM) module and the optical temporal encryption (OTE) module are introduced to combine chaotic signal and the message to ensure the security of the fingerprint. Support vector machine (SVM) models are trained to recognize legal and illegal optical transmitters at the receiver. Simulation results show that LLES of chaos has the fingerprint characteristic and is highly sensitive to the time delay of the electro-optic feedback loop. The trained SVM models can distinguish electro-optic chaos generated by different feedback loops with a time delay difference of only 0.03ns and have a good anti-noise ability. Experimental results show that the recognition accuracy of the authentication module based on LLES can reach 98.20% for both legal and illegal transmitters. Our strategy can improve the defense ability of optical networks against active injection attacks and has high flexibility.

Modelling and dynamic analysis of a novel seven-dimensional Hamilton conservative hyperchaotic systems with wide range of parameter

The study of chaotic attractors has been a hot issue in complex science research in recent years. However, most of the current research has focused on low-dimensional dissipative systems. High-dimensional conservative systems have both conservative and hyperchaotic properties, the phase space is integer dimensional and does not have attractors, and the trajectories expand in multiple directions, thus having higher complexity and spatial ergodicity. In addition, the high dimensional conservative system with wide parameter range not only has better dynamic characteristics, but also has a good application prospect in the field of information security. In this paper, a novel seven-dimensional Hamiltonian conservative hyperchaotic system (7D-HCHCS) is constructed. The dynamical properties of this system are described by analyzing the rate of change of phase space volume, phase trajectory diagram, Poincaré map, Lyapunov exponential spectrum (LEs), bifurcation diagram, equilibrium point, and system complexity. A new pseudo-random number generator (PRNG) is designed on this basis, and the key stream generated by this PRNG passes the NIST test. Besides, the phase diagrams and Poincaré map under a wide range of parameters are compared. The results show that the proposed system satisfies the Hamilton energy conservation and can generate hyperchaotic flow. It also has good pseudorandom characteristics, ergodicity under a large range of control parameters, which also has good prospects in the field of information security.

Complex Dynamic Analysis, Circuit Design and Simplified Predefined Time Synchronization for a Jerk Absolute Memristor Chaotic System

In this parper, a 4D absolute memristor Jerk chaotic system is proposed. Firstly, complex dynamics are studied by phase diagram, Poincaré section, power spectrum, bifurcation diagram, 0-1 test, and Lyapunov exponent spectrum. Then, the period doubling bifurcation, degradation, and offset boosting are revealed. For the feasibility of practical application, the analog circuit and FPGA digital circuit are designed. Finally, a simplified predefined time synchronization scheme is proposed; comparing with the full control input synchronization scheme, the simplified predefined time synchronization scheme can not only reduce the controller inputs but also predefine the synchronization time.

Discrete fracmemristor-based chaotic map by Grunwald–Letnikov difference and its circuit implementation

The topic of discrete memristor has attracted more and more attention in recent years, but there is not enough discussion about discrete memristor under the fractional-order difference definition, especially for the system model defined by Grunwald–Letnikov. In this paper, a discrete fractional-order memristor model based on Grunwald–Letnikov definition is derived, and two new chaotic maps are designed based on this fractional-order model. Dynamic behaviors are investigated by the volt–ampere curve, bifurcation diagram, Lyapunov exponent spectrum, attractor phase diagram and fuzzy entropy complexity algorithm. Simulation results show that the two fractional-order systems can produce rich dynamic behaviors with the change of fractional order value, and both of them can present hyperchaos. Finally, the proposed systems are implemented in FPGA digital circuit and analog circuit respectively, and the phenomena in the numerical simulation are verified.

The dynamics of a duopoly Stackelberg game with marginal costs among heterogeneous players.

One of the famous economic models in game theory is the duopoly Stackelberg model, in which a leader and a follower firm manufacture a single product in the market. Their goal is to obtain the maximum profit while competing with each other. The desired dynamics for a firm in a market is the convergence to its Nash equilibrium, but the dynamics of real-world markets are not always steady and can result in unpredictable market changes that exhibit chaotic behaviors. On the other hand, to approach reality more, the two firms in the market can be considered heterogeneous. The leader firm is bounded rationale, and the follower firm is adaptable. Modifying the cost function that affects the firms' profit by adding the marginal cost term is another step toward reality. We propose a Stackelberg model with heterogeneous players and marginal costs, which exhibits chaotic behavior. This model's equilibrium points, including the Nash equilibrium, are calculated by the backward induction method, and their stability analyses are obtained. The influence of changing each model parameter on the consequent dynamics is investigated through one-dimensional and two-dimensional bifurcation diagrams, Lyapunov exponents spectra, and Kaplan-Yorke dimension. Eventually, using a combination of state feedback and parameter adjustment methods, the chaotic solutions of the model are successfully tamed, and the model converges to its Nash equilibrium.

Fractional-Order sliding mode control of a 4D memristive chaotic system

Chaotic systems depict complex dynamics, thanks to their nonlinear behaviors. With recent studies on fractional-order nonlinear systems, it is deduced that fractional-order analysis of a chaotic system enriches its dynamic behavior. Therefore, the investigation of the chaotic behavior of a 4D memristive Chen system is aimed in this study by taking the order of the system as fractional. The nonlinear behavior of the system is observed numerically by comparing the fractional-order bifurcation diagrams and Lyapunov Exponents Spectra with 2D phase portraits. Based on these analyses, two different fractional orders (i.e., q = 0.948 and q = 0.97) are determined where the 4D memristive system shows chaotic behavior. Furthermore, a single state fractional-order sliding mode controller (FOSMC) is designed to maintain the states of the fractional-order memristive chaotic system on the equilibrium points. Then, control method results are obtained by both numerical simulations and different illustrative experiments of microcontroller-based realization. As expected, voltage outputs of the microcontroller-based realization are in good agreement with the time series of numerical simulations.

Coexisting Behavior and Status Transition of the Hodgkin-Huxley Cardiac Purkinje Fiber Model Under External AC Injection

The membrane current I_m of the Hodgkin-Huxley (HH) cardiac Purkinje fiber (CPF) model is usually calculated as direct current (DC). In this paper, a conventional alternating current (AC) is used, namely I_{AC} = textrm{Asin}(2 pi ft), as the impressed injection. Dynamic characters of the model with different AC parameters and various initial conditions are observed through phase plane orbits, waveforms, bifurcation diagrams, and Lyapunov exponent spectra, which reveal multiple coexisting membrane action potential patterns, corresponding to the coexisting attractors of the same or different periods in the phase diagram. Meanwhile, the model undergoes period, quasi-period and local forward or inverse period-doubling bifurcations with changes in amplitude A or frequency f, which further proves the complex nonlinear property of the AC-injected model. In addition, by changing the external current I_m, the sodium-ion and potassium-ion equilibrium potentials, i.e. E_{Na} and E_{K}, respectively, the regularity of the CPF heartbeat frequency is observed. The state transformations of CPF are found between normal, abnormal and sudden cardiac arrest, and the method adjusting from the dangerous state to the normal heartbeat frequency range is investigated. This study may provide a reference for exploring the evolution of nonlinear dynamics in HH CPF model and protecting the health of life and heart.

Generation mechanisms of strange nonchaotic attractors and multistable dynamics in a class of nonlinear economic systems

In this paper, we study strange nonchaotic attractors (SNAs) and multistable dynamics in a class of nonlinear economic systems. For quasiperiodically forced case, the generation and evolution mechanisms of SNAs are discussed. The fractal, Heagy–Hammel, torus doubling, and intermittency routes to SNAs are identified. The Lyapunov exponent, phase-sensitive function and power spectrum are used to characterize the dynamical and geometrical properties of SNAs. Moreover, when multistable phenomenon occur in the system, the boundaries of the basin of attraction may become intertwined, which leads to the economic unpredictability in the long run.

A novel image encryption scheme based on Logistic cosine cascade maps

A two-dimensional Logistic cosine cascade map (2D-LCCM) and a three-dimensional Logistic cosine cascade map (3D-LCCM) are proposed based on a logistic map and infinite collapse cascade model. We analyze the chaotic properties of 2D-LCCM and 3D-LCCM, such as Lyapunov exponential spectrum, attractors, and complexity, which show that the 2D-LCCM and 3D-LCCM are hyperchaotic and have large chaotic regions. We also proposed a DNA encryption algorithm based on double chaotic maps. 2D-LCCM was used for scrambling and 3D-LCCM for diffusion. First, the sequence no-duplicate zigzag transform (SNZT) is used to scramble the plaintext. Z-transform is applied to the permutation sequence to make the sequence more random. Then a repetition-free permutation associated with the plaintext is applied to each block, and the pixel position is effectively scrambled. Finally, DNA diffusion is performed using DNA coding to diffuse pixel values. Simulation results show that the algorithm has a good encryption effect, high efficiency and anti-interference performance. Therefore, the proposed new hyperchaotic map and DNA encryption algorithm can be effectively applied to the process of image encryption.

RC Bridge Oscillation Memristor Chaotic Circuit for Electrical and Electronic Technology Extended Simulation Experiment.

The fourth basic circuit component, the memristor, has been proposed for a long time, but it is not mentioned in the experiment teaching system of Electrical and Electronic Technology. In this paper, an RC bridge oscillation chaotic circuit based on memristor is designed to solve this problem. The dynamical behavior of the circuit system is analyzed using Lyapunov exponents spectrum, bifurcation diagram, phase portrait and Poincaré map. A series of complex dynamical behaviors such as symmetric single-scroll coexistence, asymmetrical single-scroll coexistence, symmetric double-scroll coexistence and asymmetrical limit-cycle coexistence exist in the circuit system. This research plays a critical role in enriching students' knowledge and improving the experiment teaching system of Electrical and Electronic Technology.

How close are integrable and nonintegrable models: A parametric case study based on the Salerno model.

In the present work we revisit the Salerno model as a prototypical system that interpolates between a well-known integrable system (the Ablowitz-Ladik lattice) and an experimentally tractable, nonintegrable one (the discrete nonlinear Schrödinger model). The question we ask is, for "generic" initial data, how close are the integrable to the nonintegrable models? Our more precise formulation of this question is, How well is the constancy of formerly conserved quantities preserved in the nonintegrable case? Upon examining this, we find that even slight deviations from integrability can be sensitively felt by measuring these formerly conserved quantities in the case of the Salerno model. However, given that the knowledge of these quantities requires a deep physical and mathematical analysis of the system, we seek a more "generic" diagnostic towards a manifestation of integrability breaking. We argue, based on our Salerno model computations, that the full spectrum of Lyapunov exponents could be a sensitive diagnostic to that effect.

Dynamical behaviors, circuit design, and synchronization of a novel symmetric chaotic system with coexisting attractors

In this paper, we introduce a novel three-dimension chaotic system with strange characteristic by applying construction of a 3D chaotic circuit method. Multiple equilibria and abundant coexisting attractors exist in this system. A mathematical model is developed and detailed stability analyses for equilibrium points are executed with obtaining significant results of the period-doubling bifurcation patterns confirmed by phase plane plots and Lyapunov exponent spectra. By varying the initial value and unique controlled parameter, the double-scroll chaotic attractor is broken up into a pair of symmetric singular attractors. Then, the local basins of attraction are investigated concerning the initial condition. Next, the circuit synthesis results generated by Multisim simulation tool validate the self-excitation characteristics of this system. Finally, the feedback control technique is used to study difference synchronization of this system. Main conclusions prove the validity and reliability of difference synchronization.

Neural network method for Lyapunov exponents spectrum calculation in analysis of classic systems nonlinear dynamics

Neural network method for Lyapunov exponents spectrum calculation in analysis of classic systems nonlinear dynamics

From Chaos to Pseudorandomness: A Case Study on the 2-D Coupled Map Lattice.

Applying the chaos theory for secure digital communications is promising and it is well acknowledged that in such applications the underlying chaotic systems should be carefully chosen. However, the requirements imposed on the chaotic systems are usually heuristic, without theoretic guarantee for the resultant communication scheme. Among all the primitives for secure communications, it is well accepted that (pseudo) random numbers are most essential. Taking the well-studied 2-D coupled map lattice (2D CML) as an example, this article performs a theoretical study toward pseudorandom number generation with the 2D CML. In so doing, an analytical expression of the Lyapunov exponent (LE) spectrum of the 2D CML is first derived. Using the LEs, one can configure system parameters to ensure the 2D CML only exhibits complex dynamic behavior, and then collect pseudorandom numbers from the system orbits. Moreover, based on the observation that least significant bit distributes more evenly in the (pseudo) random distribution, an extraction algorithm E is developed with the property that when applied to the orbits of the 2D CML, it can squeeze uniform bits. In implementation, if fixed-point arithmetic is used in binary format with a precision of z bits after the radix point, E can ensure that the deviation of the squeezed bits is bounded by 2-z . Further simulation results demonstrate that the new method not only guides the 2D CML model to exhibit complex dynamic behavior but also generates uniformly distributed independent bits with good efficiency. In particular, the squeezed pseudorandom bits can pass both NIST 800-22 and TestU01 test suites in various settings. This study thereby provides a theoretical basis for effectively applying the 2D CML to secure communications.

Chaos and complexity in the dynamics of nonlinear Alfvén waves in a magnetoplasma.

The nonlinear dynamics of circularly polarized dispersive Alfvén wave (AW) envelopes coupled to the driven ion-sound waves of plasma slow response is studied in a uniform magnetoplasma. By restricting the wave dynamics to a few number of harmonic modes, a low-dimensional dynamical model is proposed to describe the nonlinear wave-wave interactions. It is found that two subintervals of the wave number of modulation k of AW envelope exist, namely, (3/4)kc<k<kc and 0<k<(3/4)kc, where kc is the critical value of k below which the modulational instability (MI) occurs. In the former, where the MI growth rate is low, the periodic and/or quasi-periodic states are shown to occur, whereas the latter, where the MI growth is high, brings about the chaotic states. The existence of these states is established by the analyses of Lyapunov exponent spectra together with the bifurcation diagram and phase-space portraits of dynamical variables. Furthermore, the complexities of chaotic phase spaces in the nonlinear motion are measured by the estimations of the correlation dimension as well as the approximate entropy and compared with those for the known Hénon map and the Lorenz system in which a good qualitative agreement is noted. The chaotic motion, thus, predicted in a low-dimensional model can be a prerequisite for the onset of Alfvénic wave turbulence to be observed in a higher dimensional model that is relevant in the Earth's ionosphere and magnetosphere.

Determining Lyapunov exponents of fractional-order systems: A general method based on memory principle

Lyapunov exponents provide quantitative evidence for determining the stability and classifying the limit set of dynamical systems. There are several well-established techniques to compute Lyapunov exponent of integer-order systems, however, these techniques failed to generalize to fractional-order systems due to the nonlocality of fractional-order derivatives. In this paper, a method for determining the Lyapunov exponent spectrum of fractional-order systems is proposed. The proposed method is rigorously derived based on the memory principle of Grünwald–Letnikov derivative so that it is generally applicable and even well compatible with integer-order systems. Three classical examples, which are the fractional-order Lorenz system, fractional-order Duffing oscillator, and 4-dimensional fractional-order Chen system, are respectively employed to demonstrate the effectiveness of the proposed method for incommensurate, nonautonomous and low effective order systems as well as hyperchaotic systems. The simulation results suggest that the proposed method is indeed superior to the existing methods in accuracy and correctness.