Lyapunov Exponents Research Articles

Design and synchronization control application of a new five-dimensional memristor CNN conservative hyperchaotic system

Abstract Incorporating memristors into a cellular neural network (CNN) and introducing chaotic characteristics can generate in highly complex and unpredictable dynamic behaviors. To advance this research area, this paper proposes a new five-dimensional memristor CNN conservative hyperchaotic system and systematically analyzes its dynamic properties. The analysis content includes equilibrium point analysis, Poincaré sections, Lyapunov exponent spectra, bifurcation diagrams, two-parameter Lyapunov exponent spectra, complexity assessment, homogeneous and heterogeneous extreme multistability, etc. In addition, the simulation circuit for the new system is designed and constructed. The digital circuit of the new system is implemented using a microcontroller (MCU). After running simulations, the experimental results from the analog circuit, digital circuit, and numerical simulation are consistent with each other, demonstrating the feasibility of the circuit implementation. Finally, two different synchronization control strategies are employed to achieve synchronization control within a finite time.

Optical solitons, qualitative analysis, and chaotic behaviors to the highly dispersive nonlinear perturbation Schrödinger equation

In this paper, we study the highly dispersive nonlinear perturbation Schrödinger equation, which has arbitrary form of Kudryashov's with sextic‐power law refractive index and generalized nonlocal laws. For the equation has highly dispersive nonlinear terms and higher order derivatives, it cannot be integrated directly, so we build an integrable factor equation for the approximated equation and apply the trial equation method and the complete discrimination system for polynomial method to create new soliton solutions. On the other hand, we use the bifurcation theory to qualitatively analyze the equation and find the model has periodic solutions, bell‐shaped soliton solutions, and solitary wave solutions via phase diagrams. The topological stability of the solutions with respect to the parameters is explored in order to better understand the effect of parameters perturbations on the stability of the model's solutions. Furthermore, we analyze the modulation instability and give the corresponding linear criterion. After accounting for external perturbation terms, we analyze the chaotic behaviors of the equation through the largest Lyapunov exponents and phase diagrams.

Stability, uniqueness and existence of solutions to McKean–Vlasov SDEs: a multidimensional Yamada–Watanabe approach

We establish stability and pathwise uniqueness of solutions to Wiener noise driven McKean–Vlasov equations with random non-Lipschitz continuous coefficients. In the deterministic case, we also obtain the existence of unique strong solutions. By using our approach, which is based on an extension of the Yamada–Watanabe ansatz to the multidimensional setting and which does not rely on the construction of Lyapunov functions, we prove first moment and pathwise exponential stability. Furthermore, Lyapunov exponents are computed explicitly.

Stability and bifurcation analysis of a 2DOF dynamical system with piezoelectric device and feedback control

This study aims to demonstrate the behaviors of a two degree-of-freedom (DOF) dynamical system consisting of attached mass to a nonlinear damped harmonic spring pendulum with a piezoelectric device. Such a system is influenced by a parametric excitation force on the direction of the spring’s elongation and an operating moment at the supported point. A negative-velocity-feedback (NVF) controller is inserted into the main system to reduce the undesired vibrations that affect the system’s efficiency, especially at the resonance state. The equations of motion (EOM) are derived by using Lagrangian equations. Through the use of the multiple-scales-strategy (MSS), approximate solutions (AS) are investigated up to the third order. The accuracy of the AS is verified by comparing them to the obtained numerical solutions (NS) through the fourth-order Runge-Kutta Method (RK-4). The study delves into resonance cases and solvability conditions to provide the modulation equations (ME). Graphical representations showing the time histories of the obtained solutions and frequency responses are presented utilizing Wolfram Mathematica 13.2 in addition to MATLAB software. Additionally, discusses the bifurcation diagrams, Poincaré maps, and Lyapunov exponent spectrums to show the various behavior patterns of the system. To convert vibrating motion into electrical power, a piezoelectric sensor is connected to the dynamical model, which is just one of the energy harvesting (EH) technologies with extensive applications in the commercial, industrial, aerospace, automotive, and medical industries. Moreover, the time histories of the obtained solutions with and without control are analyzed graphically. Finally, resonance curves are used to discuss stability analysis and steady-state solutions.

Nonlinear dynamic of flexible rotor supported by turbulent bearings with quadratic damping and temperature dependent viscosity

A nonlinear dynamic model of a rotor supported by the turbulent journal bearing with quadratic damping under nonlinear suspension is presented in this study, and the lubricating oil is assumed to be temperature-dependent. The bifurcation diagrams, dynamic trajectories, power spectra, Poincaré map, Lyapunov exponent, and fractal dimension are used to verify the dynamic characteristics of the rotor-bearing system, and abundant subharmonic, and even chaotic motions are found. Temperature is one of the key parameters affecting the dynamic responses of the system, and the dynamic responses will perform non-periodic or even chaotic motions with higher working temperatures of lubricant fluid, as proved in this study. Nonlinear dynamic simulation analysis results will help engineers avoid unnecessary trial and error when designing and applying rotating machinery systems.

Impact of Horizontal Model Resolution on Mixing and Dispersion in the Northeastern Gulf of Mexico

AbstractIn this paper, the importance of model horizontal resolution in mixing and dispersion is investigated by comparing two data‐assimilative high‐resolution simulations (4 and 1 km), one of which is submesoscale‐permitting. By employing both Eulerian and Lagrangian metrics, upper‐ocean differences between the mesoscale‐resolving and submesoscale‐permitting simulations are examined in the northeastern Gulf of Mexico, a region of high mesoscale and submesoscale activity. Mixing in both simulations is explored by conducting Lagrangian experiments to track the generation of Lagrangian coherent structures (LCSs) and their associated transport barriers. Finite‐time Lyapunov exponent (FTLE) fields show higher separation rates of fluid particles in the submesoscale‐permitting case, which indicate more vigorous mixing with differences being more pronounced in the shelf regions (depths ≤ 500 m). The extent of the mixing homogeneity is examined using probability density functions (PDFs) of FTLEs with results suggesting that mixing is heterogeneous in both simulations, but some homogeneity is exhibited in the submesoscale‐permitting case. The FTLE fields also indicate that chaotic advection dominates turbulent mixing in both simulations regardless of the horizontal resolution. In the submesoscale‐permitting experiment, however, smaller scale LCSs emerge as noise‐like filaments that suggest a larger turbulent mixing component than in the mesoscale‐resolving experiment. The impact of resolution is then explored by investigating the spread of oil particles at the location of the Deepwater Horizon oil spill.

An enhanced key expansion module based on 2D hyper chaotic map and Galois field

Key expansion is an essential component in block cryptography, which serves the round function. By analyzing key expansion module of AES, it is found that the round-keys are highly correlated, and current round-key can be derived from its previous or the next round-key. To address these weaknesses, first, a 2D exponential chaotic map (2D-ECM) that exhibits ergodicity and superior randomness was constructed. Then, the Lyapunov exponents (LEs) were calculated based on the singular value decomposition (SVD) method. In addition, TestU01 test results showed that the sequences generated by 2D-ECM have better randomness. Further, an enhanced key expansion module was designed utilizing 2D-ECM and primitive polynomial over GF(2n), which has irreversibility and parallelism, and the round-keys are independent of each other. Simulation results and performance analysis demonstrated the effectiveness of the proposed enhanced key expansion module.

Orbits of a system of three point vortices and the associated chaotic mixing.

We study the general periodic motion of a set of three point vortices in the plane, as well as the potentially chaotic motion of one or more tracer particles. While the motion of three vortices is simple in that it can only be periodic, the actual orbits can be surprisingly complex and varied. This rich behavior arises from the existence of both co-linear and equilateral relative equilibria (steady motion in a rotating frame of reference). Here, we start from a general (unsteady) co-linear array with arbitrary vortex circulations. The subsequent motion may take the vortices close to a distinct co-linear relative equilibrium or to an equilateral one. Both equilibrium states are necessarily unstable, as we demonstrate by a linear stability analysis. We go on to study mixing by examining Poincaré sections and finite-time Lyapunov exponents. Both indicate widespread chaotic motion in general, implying that the motion of three vortices efficiently mixes the nearby surrounding fluid outside of small regions surrounding each vortex.

Chaotic dynamics in a class of generalized memristive maps.

The memory effects of the memristors in nonlinear systems make the systems generate complicated dynamics, which inspires the development of the applications of memristors. In this article, the model of the discrete memristive systems with the generalized Ohm's law is introduced, where the classical Ohm's law is a linear relationship between voltage and current, and a generalized Ohm's law is a nonlinear relationship. To illustrate the rich dynamics of this model, the complicated dynamical behavior of three types of maps with three types of discrete memristances is investigated, where a cubic function representing a kind of generalized Ohm's law is used, and this cubic function is a simplified characteristic of the famous tunnel diode. The existence of attractors with one or two positive Lyapunov exponents (corresponding to chaotic or hyperchaotic dynamics) is obtained, and the coexistence of (infinitely) many attractors is observable. A hardware device is constructed to implement these maps and the analog voltage signals are experimentally acquired.

About the chaos influence on a system with multi-frequency quasi-periodicity and the Landau-Hopf scenario

The interaction of system demonstrating multi-frequency quasi-periodic oscillations and several steps of the Landau-Hopf scenario with chaotic Rössler system is considered. The quasi-periodic subsystem is a network of five non-identical van der Pol oscillators. It is shown that as the coupling parameter between the subsystems decreases, successive quasi-periodic Hopf bifurcations and doublings of high-dimensional invariant tori are observed. The chaos arising in this system can have several (in our case up to five) additional zero Lyapunov exponents. In case of weak coupling parameter between chaotic and quasi-periodic subsystems, when the coupling parameter of van der Pol oscillators changes, the points at which the attractor transformation occurs are observed. This is a new type of bifurcations that are responsible for a consistent increase in the number of additional zero Lyapunov exponents. As the coupling parameter between chaotic and quasi-periodic subsystems increases, the observed stages of the Landau-Hopf scenario turns out to be resistant to interaction with the chaotic system.

Effect of a perforation on the flow characteristics of corrugated wall

The flow characteristics on a corrugated wall and the variations caused by a perforation are investigated experimentally based on two-dimensional particle image velocimetry (PIV), and the passive control mechanism of the perforation on the corrugated wall is studied. The corrugated wall has an amplitude-to-wavelength ratio 2a/λ = 0.1, with the wavelength Reynolds number Reλ = 14400 and bulk Reynolds number Reb = 17500. The perforation is located on the eleventh cycle of the corrugated wall. The results show that perforation increases the area of the recirculation zone, reduces the effect of frictional resistance, weakens the turbulence intensity and the Reynolds normal stress on the corrugated wall, but enhances the Reynolds shear stress. The POD and the Finite-Time Lyapunov Exponent(FTLE) are used to analyze the vortex structures. From the FTLE result, it can be seen that the perforation disturbs the original shear layer and redistributes the vortex structure of the flow field. The instantaneous fluctuating flow field of the first 50 % and the last 50 % of the energy content after POD mode decomposition is reconstructed to study the effect of perforation on on large and small scale structures in the flow field. It is found that the impact of perforation on small-scale structures is greater than that on large-scale structures.

Mixed-mode oscillations and chaos in a complex chemical reaction network involving heterogeneous catalysis.

The nonlinear dynamical behavior in a complex isothermal reaction network involving heterogeneous catalysis is studied. The method first determines the multiple steady states in the reaction network. This is followed by an analysis of bifurcation continuations to identify several kinds of bifurcations, including limit point, Bogdanov-Takens, generalized Hopf, period doubling, and generalized period doubling. Numerical simulations are performed around the period doubling and generalized period doubling bifurcations. Rich nonlinear behaviors are observed, including simple sustained oscillations, mixed-mode oscillations, non-mixed-mode chaotic oscillations, and mixed-mode chaotic oscillations. Concentration-time plots, 2D phase portraits, Poincaré maps, maximum Lyapunov exponents, frequency spectra, and cascade of bifurcations are reported. Period-doubling and period-adding routes leading to chaos are observed. Maximum Lyapunov exponents are positive for all the chaotic cases, but they are also positive for some non-chaotic orbits. This result diminishes the reliability of using maximum Lyapunov exponents as a tool for determining chaos in the network under study.

Utilizing Inertial Measurement Units for Detecting Dynamic Stability Variations in a Multi-Condition Gait Experiment.

This study proposes a wearable gait assessment method using inertial measurement units (IMUs) to evaluate gait ability in daily environments. By focusing on the estimation of the margin of stability (MoS), a key kinematic stability parameter, a method using a convolutional neural network, was developed to estimate the MoS from IMU acceleration time-series data. The relationship between MoS and other stability indices, such as the Lyapunov exponent and the multi-site time-series (MSTS) index, using data from five IMU sensors placed on various body parts was also examined. To simulate diverse gait conditions, treadmill speed was varied, and a knee-ankle-foot orthosis was used to restrict left knee extension, inducing gait asymmetry. The model achieved over 90% accuracy in classifying MoS in both forward and lateral directions using three-axis acceleration data from the IMUs. However, the correlation between MoS and the Lyapunov exponent or MSTS index was weak, suggesting that these indices may capture different aspects of gait stability.

Analytic mappings of the unit disk which almost preserve hyperbolic area

AbstractIn this paper, we study analytic self‐maps of the unit disk which distort hyperbolic areas of large hyperbolic disks by a bounded amount. We give a number of characterizations involving angular derivatives, Lipschitz extensions, Möbius distortion, the distribution of critical points and Aleksandrov–Clark measures. We also examine the Lyapunov exponents of their Aleksandrov–Clark measures.

In-phase collective unconventional photon blockade and its stability in asymmetrical cavity containing N bosonic atoms

Abstract We present a work of cavity-driven QED system combined an asymmetrical Fabry-Perot cavity and N two-level atoms (TLAs) and show the convenience of simplifying from distinguishable atoms to undistinguishable bosons when atoms are prepared in the same initial state. Such simplification is valid even when the atoms are not prepared in the in-phase condition, since any partial in-phase initial state will evolve into the ground state through a relaxation process. Thus, we get a reduced group of differential equations by introducing the Dicke states, and the under-zero Lyapunov exponents verify its stability. We also work out the collective unconventional photon blockade (UCPB) and get two kinds of giant nonreciprocal UCPBs (NUCPBs) in the weak-driving approximation. Results show that we can employ N non-interacting bosonic atoms to generate a collective UCPB instead of a monoatomic UCPB as UCPB conditions do not vary with the number of atoms. Furthermore, the forward giant NUCPB only occurring for N larger than a certain number, as well as the backward giant NUCPB, are controllable by the cavity asymmetry and by the number of atoms. Our findings suggest a prospective approach to the generation of quantum nonreciprocity by N identical atoms.

A New Class of Circulant Chaotic Systems with Unidirectional or Mutual Coupling: Theoretical Investigation and Arduino-Based Electronic Implementation

The study of chaotic systems with special properties addresses one of the ongoing research topics. In this paper, we introduce a new class of third-order chaotic systems possessing a cyclic symmetry obeying unidirectional or mutual coupling. These new models are built by considering nonlinearities with three, four, or five segments and are distinguished from each other by the topological structure of the associated chaotic attractor. The projection of the novel chaotic attractors on certain planes reveals their multiscroll or multiwing character. For illustration, one of these models is studied in detail using analytical and numerical methods. The mechanism giving rise to the establishment of the chaotic regime starting from the state of equilibrium under the variation of a parameter is described using bifurcation diagrams, phase portraits, basins of attraction as well as the maximum Lyapunov exponent. Several zones of parameters are identified where the model experiences two or more attractors (i.e. multistability). In order to demonstrate the practical feasibility of the proposed circulant models, a series of experimental measurements are carried out using a physical implementation with an Arduino microcontroller.

A Comparison between four Chaotic Indicators in Systems with Hidden Attractors

Abstract A non-regular oscillation is not enough to define a system as chaotic. A more in-depth investigation is required to prove the existence of chaotic behavior, which is challenging. Although many scientists use the Lyapunov Characteristic Exponents to detect chaos, it is not the only method. Several scientists have introduced different methods that utilize various properties of dynamical systems. The paper presents four different chaotic indicators based on the evolution of the deviation vectors: the maximal Lyapunov Exponent, the Lyapunov Characteristic Exponents, the Fast Lyapunov Index, and the Small Alignment Index. It includes their properties and the advantages and disadvantages of each method. Also, it includes the algorithms to calculate them and their implementation in Python. The paper closes with a comparison between the four indices applied to a system with hidden attractors.

FPGA Implementation for a Class of Generalized Hamiltonian Conservative Chaotic Systems Based on Integrated 4D Euler Equations with Multistability

In this paper, two newly proposed generalized Hamiltonian Conservative Chaotic Systems (CCSs) based on integrated 4D Euler equations are first discussed. Second, another generalized Hamiltonian CCS is also proposed. Furthermore, a class of generalized Hamiltonian CCSs based on integrated 4D Euler equations is given and studied. Subsequently, numerical analysis for the class of generalized Hamiltonian CCSs is further investigated to show their advantages over most of the existing CCSs, where the corresponding Lyapunov exponents’ diagrams, bifurcation diagrams and phase portraits are all given to show their multistability and complex dynamics. Finally, the class of generalized Hamiltonian CCSs is implemented by using FPGA technology, and the results observed in FPGA implementation are consistent with those observed in the numerical analysis. All these results not only show multistability from a physical point of view, but also provide new physical models for chaos applications.

Chaos and attraction domain of fractional Φ6‐van der Pol with time delay velocity

This article investigates the chaotic analysis and attractive domain of a fractional‐order ‐van der Pol with time delay velocity under harmonic excitation. Firstly, eight different types of bifurcation states of the system under different parameters are calculated by using the undisturbed system. Secondly, the Melnikov method is used to explore the effect of time delay velocity on the threshold of chaos in the Smale horseshoe sense under the double‐well potential and three‐well potential of the system. Finally, through numerical analysis of the phase diagram, bifurcation diagram, and maximum Lyapunov exponent, the influence of time delay velocity on system chaos is studied. The results indicate that an increase in the delay velocity coefficient will lead to the system transitioning from a chaotic state to a periodic state, while an increase in the delay velocity term will lead to the system transitioning from a periodic state to a chaotic state. In the study of system bifurcation, it is found that the position of the equilibrium points of the system changes during periodic motion. Therefore, cell mapping is used to draw the attractive domain of the system is studying the influence of initial conditions on the equilibrium point of the system and the results show that there is a close relationship between the attraction domain and the process of chaos occurrence.

Transforming chaotic and stiff systems to improve numerical accuracy

AbstractSystems of differential equations can exhibit chaotic or stiff behavior under specific conditions, posing challenges for time-stepping numerical methods. For explicit methods, the required time step resolution significantly exceeds the resolution associated with the smoothness of the exact solution for specified accuracy. In order to improve efficiency, the question arises whether transformation to asymptotically stable solutions can be performed, for which neighbouring solutions converge towards each other at a controlled rate. Employing the concept of local Lyapunov exponents, it is shown that chaotic differential equations can indeed be transformed to obtain high accuracy, whereas stiff equations cannot. For instance, the accuracy of explicit fourth order Runge–Kutta solution of the Lorenz chaotic equations can be increased by several orders of magnitude. Alternatively, the time step can be significantly extended with retained accuracy. The transform method applies broadly to chaotic systems, including weather prediction and turbulence.