Theoretical and experimental investigation of the resonance responses and chaotic dynamics of a bistable laminated composite shell in the dynamic snap-through mode

The dynamic model of a bistable laminated composite shell simply supported by four corners is further developed to investigate the resonance responses and chaotic behaviors.The existence of the 1:1 resonance relationship between two order vibration modes of the system is verified.The resonance response of this class of bistable structures in the dynamic snap-through mode is investigated,and the four-dimensional(4D)nonlinear modulation equations are derived based on the 1:1 internal resonance relationship by means of the multiple scales method.The Hopf bifurcation and instability interval of the amplitude frequency and force amplitude curves are analyzed.The discussion focuses on investigating the effects of key parameters,e.g.,excitation amplitude,damping coefficient,and detuning parameters,on the resonance responses.The numerical simulations show that the foundation excitation and the degree of coupling between the vibration modes exert a substantial effect on the chaotic dynamics of the system.Furthermore,the significant motions under particular excitation conditions are visualized by bifurcation diagrams,time histories,phase portraits,three-dimensional(3D)phase portraits,and Poincare maps.Finally,the vibration experiment is carried out to study the amplitude frequency responses and bifurcation characteristics for the bistable laminated composite shell,yielding results that are qualitatively consistent with the theoretical results.

Global bifurcations and multi-pulse chaotic dynamics of rectangular thin plate with one-to-one internal resonance

Global bifurcations and multi-pulse chaotic dynamics for a simply supported rectangular thin plate are studied by the extended Melnikov method. The rectangular thin plate is subject to transversal and in-plane excitation. A two-degree-of-freedom nonlinear nonautonomous system governing equations of motion for the rectangular thin plate is derived by the von Karman type equation and the Galerkin approach. A one-to- one internal resonance is considered. An averaged equation is obtained with a multi-scale method. After transforming the averaged equation into a standard form, the extended Melnikov method is used to show the existence of multi-pulse chaotic dynamics, which can be used to explain the mechanism of modal interactions of thin plates. A method for calculating the Melnikov function is given without an explicit analytical expression of homoclinic orbits. Furthermore, restrictions on the damping, excitation, and detuning parameters are obtained, under which the multi-pulse chaotic dynamics is expected. The results of numerical simulations are also given to indicate the existence of small amplitude multi-pulse chaotic responses for the rectangular thin plate.

Revival of Order in the Chaotic Dynamics with Position and Time Dependent Perturbed System

Studying quantum properties of a system has been quite popular in quantum mechanics. One of the most important systems that are very crucial to the framework of quantum mechanics is the system of harmonic oscillator a system whose classical evolution is known to exhibit peculiar chaotic dynamics. We are motivated to investigate the behavior of quantum properties for a system with position and time dependent perturbed. Starting with Hamiltonian, we determined the equation of motion and obtained the wave function. The energy of the whole system using the operator ordering method was found. We show that the quantum mechanical picture alludes to a chaotic dynamics as expected. This is evidenced through the appearance of energy level crossings. An additional signature to this chaotic dynamics is observed in the transition of Eigen values from real to imaginary. We also show numerically that one can give the behavior of the system is Poincare section. By so doing we confirmed that increasing and decreasing the perturbation amplitude of the system becomes chaotic.

Emulation of auditory senses depending on chaotic dynamics of threshold switching memristor

Emulating the auditory sense is a significant challenge in terms of both integration and energy consumption for handling complicated spatiotemporal information.Here,we demonstrate how to utilize the chaotic dynamics of a threshold switching memristor,which usually acts as a leaky integrate and fire neuron in the neuromorphic network,to encode the frequency and amplitude in auditory information.We fabricate a Pd/Nb/NbOx/Nb/Pd memristor domi-nated by the Poole-Frankel conduction mechanism,set its state at the edge of chaos,and stimulate it using periodic perturbations.The memristor's responses to the perturbation frequencies can be categorized into three zones.Two are phase locking with linear phase-frequency rela tionships,and one has a hyper-bolic spike number-frequency relationship.The memristor system also enables intensity coding and tonotopy by modulating the response spikes in either the locked phase or spike number.Based on the emulation of these two features,the memristor system demonstrates sound location and frequency mixing.Our study suggests a novel routine for handling the auditory and visual senses using threshold-switching memristor arrays to enhance the efficiency of neuromorphic networks.

Bifurcation and Chaotic Dynamics of Homoclinic Systems in R^3

We consider perturbations which may or may not depend explicitly on time for the three-dimensional homoclinic systems. We obtain the existence and bifurcation theorems for transversal homoclinic points and homoclinic orbits, and illustrate our results with two examples.

Vibrational chaotic dynamics in triatomic molecules： A comparative study

Within Lie algebraic model, the vibrational chaotic dynamics in triatomic molecules are studied. The molecules of H2S, NO2, and O3 are sampled to explore the dynamical differences between the local and normal mode molecules. The comprehensive effects of the local and normal mode vibrations, resonances and chaos on the dynamical entanglement are studied. The results demonstrate that the resonances as well as chaos can promote the evolution of dynamical entanglement.

Nonlinear dynamics of a classical rotating pendulum system with multiple excitations

We report an attempt to reveal the nonlinear dynamic behavior of a classical rotating pendulum system subjected to combined excitations of constant force and periodic excitation.The unperturbed system characterized by strong irrational nonlinearity bears significant similarities to the coupling of a simple pendulum and a smooth and discontinuous(SD)oscillator,especially the phase trajectory with coexistence of Duffing-type and pendulum-type homoclinic orbits.In order to learn the effect of constant force on this pendulum system,all types of phase portraits are displayed by means of the Hamiltonian function with large constant excitation especially the transitions of complex singular closed orbits.Under sufficiently small perturbations of the viscous damping and constant excitation,the Melnikov method is used to analyze the global structure of the phase space and the feature of trajectories.It is shown,both theoretically and numerically,that this system undergoes a homoclinic bifurcation and then bifurcates a unique attracting rotating limit cycle.Finally,the estimation of the chaotic threshold of the rotating pendulum system with multiple excitations is calculated and the predicted periodic and chaotic motions can be shown by applying numerical simulations.

Chaotic Neural Network Technique for "0-1" Programming Problems

0-1 programming is a special case of the integer programming, which is commonly encountered in many optimization problems. Neural network and its general energy function are presented for 0-1 optimization problem. Then, the 0-1 optimization problems are solved by a neural network model with transient chaotic dynamics (TCNN). Numerical simulations of two typical 0-1 optimization problems show that TCNN can overcome HNN's main drawbacks that it suffers from the local minimum and can search for the global optimal solutions in to solveing 0-1 optimization problems.

Chaos generation by a hybrid integrated chaotic semiconductor laser

We design a hybrid integrated chaotic semiconductor laser with short-cavity optical feedback. It can be assembled in a commercial butterfly shell with just three micro-lenses. One of them is coated by a transflective film to provide the optical feedback for chaos generation while insuring regular laser transmission. We prove the feasibility of the chaos generation in this compact structure and provide critical external parameters for the fabrication by theoretical simulations. Rather than the usual changeless internal parameters used in previous simulation research, we extract the real parameters of the chip by experiment. Moreover, the maps of the largest Lyapunov exponent with varying bias current and feedback intensity Kap demonstrate the dynamic characteristics under different external-cavity conditions. Each laser chip has its own optimal external cavity length（L） and feedback intensity（Kap） to generate chaos because of the different internal parameters. We have acquired two ranges of optimal parameters（L = 4 mm, 0.12 〈 Kap 〈 0.2 and L = 5 mm, 0.07 〈 Kap 〈 0.12） for two different chips.

Study on a new chaotic bitwise dynamical system and its FPGA implementation

In this paper, the structure of a new chaotic bitwise dynamical system （CBDS） is described. Compared to our previous research work, it uses various random bitwise operations instead of only one. The chaotic behavior of CBDS is mathemat- ically proven according to the Devaney＇s definition, and its statistical properties are verified both for uniformity and by a comprehensive, reputed and stringent battery of tests called TestU01. Furthermore, a systematic methodology developing the parallel computations is proposed for FPGA platform-based realization of this CBDS. Experiments finally validate the proposed systematic methodology.

Phase space reconstruction of chaotic dynamical system based on wavelet decomposition

In view of the disadvantages of the traditional phase space reconstruction method, this paper presents the method of phase space reconstruction based on the wavelet decomposition and indicates that the wavelet decomposition of chaotic dynamical system is essentially a projection of chaotic attractor on the axes of space opened by the wavelet filter vectors, which corresponds to the time-delayed embedding method of phase space reconstruction proposed by Packard and Takens. The experimental results show that, the structure of dynamical trajectory of chaotic system on the wavelet space is much similar to the original system, and the nonlinear invariants such as correlation dimension, Lyapunov exponent and Kolmogorov entropy are still reserved. It demonstrates that wavelet decomposition is effective for characterizing chaotic dynamical system.

A novel chaotic stream cipher and its application to palmprint template protection

Based on a coupled nonlinear dynamic filter （NDF）, a novel chaotic stream cipher is presented in this paper and employed to protect palmprint templates. The chaotic pseudorandom bit generator （PRBG） based on a coupled NDF, which is constructed in an inverse flow, can generate multiple bits at one iteration and satisfy the security requirement of cipher design. Then, the stream cipher is employed to generate cancelable competitive code palmprint biometrics for template protection. The proposed cancelable palmprint authentication system depends on two factors： the palmprint biometric and the password/token. Therefore, the system provides high-confidence and also protects the user＇s privacy. The experimental results of verification on the Hong Kong PolyU Palmprint Database show that the proposed approach has a large template re-issuance ability and the equal error rate can achieve 0.02%. The performance of the palmprint template protection scheme proves the good practicability and security of the proposed stream cipher.

A new procedure for exploring chaotic attractors in nonlinear dynamical systems under random excitations

Due to uncertain push-pull action across boundaries between different attractive domains by random excitations, attractors of a dynamical system will drift in the phase space, which readily leads to colliding and mixing with each other, so it is very difficult to identify irregular signals evolving from arbitrary initial states. Here, periodic attractors from the simple cell mapping method are further iterated by a specific Poincare map in order to observe more elaborate structures and drifts as well as possible dynamical bifurcations. The panorama of a chaotic attractor can also be displayed to a great extent by this newly developed procedure. From the positions and the variations of attractors in the phase space, the action mechanism of bounded noise excitation is studied in detail. Several numerical examples are employed to illustrate the present procedure. It is seen that the dynamical identification and the bifurcation analysis can be effectively performed by this procedure.

Off-policy integral reinforcement learning optimal tracking control for continuous-time chaotic systems

This paper estimates an off-policy integral reinforcement learning（IRL） algorithm to obtain the optimal tracking control of unknown chaotic systems. Off-policy IRL can learn the solution of the HJB equation from the system data generated by an arbitrary control. Moreover, off-policy IRL can be regarded as a direct learning method, which avoids the identification of system dynamics. In this paper, the performance index function is first given based on the system tracking error and control error. For solving the Hamilton–Jacobi–Bellman（HJB） equation, an off-policy IRL algorithm is proposed.It is proven that the iterative control makes the tracking error system asymptotically stable, and the iterative performance index function is convergent. Simulation study demonstrates the effectiveness of the developed tracking control method.

Applying Linear Controls to Chaotic Continuous Dynamical Systems

In this case-study, we examine the effects of linear control on continuous dynamical systems that exhibit chaotic behavior using the symbolic computer algebra system Mathematica. Stabilizing (or controlling) higher-dimensional chaotic dynamical systems is generally a difficult problem, Musielak and Musielak, [1]. We numerically illustrate that sometimes elementary approaches can yield the desired numerical results with two different continuous higher order dynamical systems that exhibit chaotic behavior, the Lorenz equations and the Rössler attractor.

A new look at the response surface method for reliability analysis using chaos theory

To overcome excessive computation errors and convergence failures encountered in an iterative calculation of the reliability index using the response surface method （RSM） for some nonlinear limit state functions, this study investigates an essential factor based on chaotic dynamics theory. The bifurcation diagrams of the reliability index are presented for some typical nonlinear limit state functions, and the computation results from the mapping functions due to the RSM iterations show the complicated dynamic phenomena such as the periodic oscillation, as well as bifurcation and chaos. From the numerical examples, it is concluded that the parameter of selection range fplays an important role in the convergence of the RSM iteration, and an improved RSM iterative algorithm is proposed with regard to the incorporation of the iterative sequential function of selection rangef The proposed method is shown to be efficient and to yield accurate results.

Markov transition probability-based network from time series for characterizing experimental two-phase flow

We generate a directed weighted complex network by a method based on Markov transition probability to represent an experimental two-phase flow. We first systematically carry out gas-liquid two-phase flow experiments for measuring the time series of flow signals. Then we construct directed weighted complex networks from various time series in terms of a network generation method based on Markov transition probability. We find that the generated network inherits the main features of the time series in the network structure. In particular, the networks from time series with different dynamics exhibit distinct topological properties. Finally, we construct two-phase flow directed weighted networks from experimental signals and associate the dynamic behavior of gas-liquid two-phase flow with the topological statistics of the generated networks. The results suggest that the topological statistics of two-phase flow networks allow quantitative characterization of the dynamic flow behavior in the transitions among different gas-liquid flow patterns.

Chaos Dynamic Behaviour of Mineralization

Abstract: In this paper a nonlinear dynamic model for the distribution of element content and mineralization in the crust is suggested and the iteration relationship formula of this model coincides with the logistic equation. This shows that mineralization related with migration and enrichment of elements is in chaos, thus resulting in fractal structures of element content and ore reserves and their spatial distribution in the crust.

Modeling the transmission dynamics of a time-delayed epidemic model with saturated treatment rate

In this paper,an attempt has been made to explore a new delayed epidemiological model assuming that the disease is transmitted among the susceptible population and possessing nonlinear incidence function along with a saturated treatment rate.Due attention is paid to the positivity and boundedness of the solutions and the bifurcation of the dynamical system as well.Basic reproduction number is being calculated,and considering the latent period as a bifurcation parameter,it has been examined that a Hopf-bifurcation occurs near the endemic equilibrium point while the parameter attains critical values.We have also discussed the stability and direction of Hopf-bifurcation near the endemic equilibrium point,the global stability analysis and the optimal control theory.We conclude that the system reveals chaotic dynamics through a specific time-delay value.Numerical simulations are being performed in order to explain the accuracy and effectiveness of the acquired theoretical results.