Nonlinear Dynamic System Identification of ARX Model for Speech Signal Identification

System Identification becomes very crucial in the field of nonlinear and dynamic systems or practical systems.As most practical systems don’t have prior information about the system behaviour thus,mathematical modelling is required.The authors have proposed a stacked Bidirectional Long-Short Term Memory(Bi-LSTM)model to handle the problem of nonlinear dynamic system identification in this paper.The proposed model has the ability of faster learning and accurate modelling as it can be trained in both forward and backward directions.The main advantage of Bi-LSTM over other algorithms is that it processes inputs in two ways:one from the past to the future,and the other from the future to the past.In this proposed model a backward-running Long-Short Term Memory(LSTM)can store information from the future along with application of two hidden states together allows for storing information from the past and future at any moment in time.The proposed model is tested with a recorded speech signal to prove its superiority with the performance being evaluated through Mean Square Error(MSE)and Root Means Square Error(RMSE).The RMSE and MSE performances obtained by the proposed model are found to be 0.0218 and 0.0162 respectively for 500 Epochs.The comparison of results and further analysis illustrates that the proposed model achieves better performance over other models and can obtain higher prediction accuracy along with faster convergence speed.

A Fuzzy-Neural Network Control of Nonlinear Dynamic Systems

In this paper, an adaptive dynamic control scheme based on a fuzzy neural network is presented, that presents utilizes both feed-forward and feedback controller elements. The former of the two elements comprises a neural network with both identification and control role, and the latter is a fuzzy neural algorithm, which is introduced to provide additional control enhancement. The feedforward controller provides only coarse control, whereas the feedback controller can generate on-line conditional proposition rule automatically to improve the overall control action. These properties make the design very versatile and applicable to a range of industrial applications.

IDENTIFICATION OF NONLINEAR DYNAMIC SYSTEMS:TIME-FREQUENCY FILTERING AND SKELETON CURVES

The nonlinear behavior varying with the instantaneous response was analyzed through the joint time-frequency analysis method for a class of S. D. O. F nonlinear system. A masking operator an definite regions is defined and two theorems are presented. Based on these, the nonlinear system is modeled with a special time-varying linear one, called the generalized skeleton linear system (GSLS). The frequency skeleton curve and the damping skeleton curve are defined to describe the main feature of the non-linearity as well. Moreover, an identification method is proposed through the skeleton curves and the time-frequency filtering technique.

TWO-DIMENSIONAL NONLINEAR DYNAMIC SYSTEM MODEL OF INTERSPECIFIC INTERACTION AND NUMERICAL SIMULATION RESEARCH ON IT

The mechanism and the course of two_dimensional nonlinear dynamic system of interspecific interaction were dealt with systematically. By extending the Lotka_Volterra model from the viewpoint of biomechanics, it developed new models of two_dimensional nonlinear autonomous and nonautonomous dynamic systems, with its equilibrium point's stability and the existence and stability of its periodical solutions analyzed, and did numerical simulation experiments on its dynamics course. The results show that efficiency of interaction between two populations, time_varying effort, and change direction of action coefficient and reaction coefficient have important influences on the stability of dynamic system, that too large or too small interspecific interaction efficiency and contrary change direction of action coefficient and reaction coefficient may result in the nonstability of the system, and thus it is difficult for two populations to coexist, and that time_varying active force contributes to system stability.

A METHOD FOR FOLLOWING THE UNSTABLE PATH BETWEEN TWO SADDLE-NODE BIFURCATION POINTS IN NONLINEAR DYNAMIC SYSTEM

A computation algorithm based on the Poincaré Mapping in combination with Pseudo_Arc Length Continuation Method is presented for calculating the unstable response with saddle_node bifurcation, and the singularity, which occurs using the general continuation method combined with Poincaré Mapping to follow the path, is also proved. A normalization equation can be introduced to avoid the singularity in the process of iteration, and a new iteration algorithm will be presented too. There will be two directions in which the path can be continued at each point, but only one can be used. The method of determining the direction will be presented in the paper. It can be concluded that this method is effective in analysis of nonlinear dynamic system with saddle_node bifurcations.

Advanced Studies for Resolving Nonlinear Dynamic Systems Involving Complexity Based on Systems Thinking for Non-Physicists

There are many examples of nonlinear dynamics, such as food chains or thermodynamic systems within closed-loop systems. In modern physics, these problems have been resolved based on logical thinking by using the chaos theory in statistical physics, which was arranged by classical physicists in the 17th century. However, this is a significantly erroneous problem because, in engineering science, nonlinear dynamics must be resolved and cleared using systems analysis theory based on systems thinking. It is the main concept in a new solution that is studied through interdisciplinary research;it is needed to introduce control theory into physics. In 2015, on the behalf of physicists, the author successfully resolved and achieved a new solution, which is a significant achievement in modern science. Unfortunately, physicists have not welcomed it because it is disadvantageous to them, similar to the Copernican theory. So, they themselves became outsiders. If so, non-physicists need to follow their chaos theory in physics unless they do not clarify their solution;moreover, non-physicists on behalf of physicists can use their own new solution without risk. Hence, all scientists need to learn the systems analytic method in engineering.

THE ACCOMPANIED SLOWLY-VARIANT-SYSTEM OF NONLINEAR DYNAMIC SYSTEMS

The slowly-variant-system is defined and analyzed in this paper and the nonlinear relationship between its instantaneous parameters and the instantaneous amplitude and frequency of its free vibration response is established. By defining the band-pass mapping, a slowly-variant-system which we call the accompanied slowly-variant-system is extracted from the nonlinear system; and the relationship between the two systems is discussed. Also, the skeleton curves that can illustrate the nonlinearity and the main properties of the nonlinear system directly and concisely are defined. Work done in this paper opens a new way for nonlinearity detection and identification for nonlinear systems.

Nonlinear dynamics of a circular curved cantilevered pipe conveying pulsating fluid based on the geometrically exact model

Due to the novel applications of flexible pipes conveying fluid in the field of soft robotics and biomedicine,the investigations on the mechanical responses of the pipes have attracted considerable attention.The fluid-structure interaction(FSI)between the pipe with a curved shape and the time-varying internal fluid flow brings a great challenge to the revelation of the dynamical behaviors of flexible pipes,especially when the pipe is highly flexible and usually undergoes large deformations.In this work,the geometrically exact model(GEM)for a curved cantilevered pipe conveying pulsating fluid is developed based on the extended Hamilton's principle.The stability of the curved pipe with three different subtended angles is examined with the consideration of steady fluid flow.Specific attention is concentrated on the large-deformation resonance of circular pipes conveying pulsating fluid,which is often encountered in practical engineering.By constructing bifurcation diagrams,oscillating shapes,phase portraits,time traces,and Poincarémaps,the dynamic responses of the curved pipe under various system parameters are revealed.The mean flow velocity of the pulsating fluid is chosen to be either subcritical or supercritical.The numerical results show that the curved pipe conveying pulsating fluid can exhibit rich dynamical behaviors,including periodic and quasi-periodic motions.It is also found that the preferred instability type of a cantilevered curved pipe conveying steady fluid is mainly in the flutter of the second mode.For a moderate value of the mass ratio,however,a third-mode flutter may occur,which is quite different from that of a straight pipe system.

THE ASYMPTOTIC BEHAVIOR AND OSCILLATION FOR A CLASS OF THIRD-ORDER NONLINEAR DELAY DYNAMIC EQUATIONS

In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero.Third,we obtain new oscillation criteria by employing the Potzsche chain rule.Then,using the generalized Riccati transformation technique and averaging method,we establish the Philos-type oscillation criteria.Surprisingly,the integral value of the Philos-type oscillation criteria,which guarantees that all unbounded solutions oscillate,is greater than θ_(4)(t_(1),T).The results of Theorem 3.5 and Remark 3.6 are novel.Finally,we offer four examples to illustrate our results.

Target layer state estimation in multi-layer complex dynamical networks considering nonlinear node dynamics

In many engineering networks, only a part of target state variables are required to be estimated.On the other hand,multi-layer complex network exists widely in practical situations.In this paper, the state estimation of target state variables in multi-layer complex dynamical networks with nonlinear node dynamics is studied.A suitable functional state observer is constructed with the limited measurement.The parameters of the designed functional observer are obtained from the algebraic method and the stability of the functional observer is proven by the Lyapunov theorem.Some necessary conditions that need to be satisfied for the design of the functional state observer are obtained.Different from previous studies, in the multi-layer complex dynamical network with nonlinear node dynamics, the proposed method can estimate the state of target variables on some layers directly instead of estimating all the individual states.Thus, it can greatly reduce the placement of observers and computational cost.Numerical simulations with the three-layer complex dynamical network composed of three-dimensional nonlinear dynamical nodes are developed to verify the effectiveness of the method.

Finite Deformation, Finite Strain Nonlinear Dynamics and Dynamic Bifurcation in TVE Solids with Rheology

This paper presents a mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and ordered rate constitutive theories in Lagrangian description derived using entropy inequality and the representation theorem for thermoviscoelastic solids (TVES) with rheology. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics and are based on contravariant deviatoric second Piola-Kirchhoff stress tensor and its work conjugate covariant Green’s strain tensor and their material derivatives of up to order m and n respectively. All published works on nonlinear dynamics of TVES with rheology are mostly based on phenomenological mathematical models. In rare instances, some aspects of CBL are used but are incorrectly altered to obtain mass, stiffness and damping matrices using space-time decoupled approaches. In the work presented in this paper, we show that this is not possible using CBL of CCM for TVES with rheology. Thus, the mathematical models used currently in the published works are not the correct description of the physics of nonlinear dynamics of TVES with rheology. The mathematical model used in the present work is strictly based on the CBL of CCM and is thermodynamically and mathematically consistent and the space-time coupled finite element methodology used in this work is unconditionally stable and provides solutions with desired accuracy and is ideally suited for nonlinear dynamics of TVES with memory. The work in this paper is the first presentation of a mathematical model strictly based on CBL of CCM and the solution of the mathematical model is obtained using unconditionally stable space-time coupled computational methodology that provides control over the errors in the evolution. Both space-time coupled and space-time decoupled finite element formulations are considered for obtaining solutions of the IVPs described by the mathematical model and are presented in the paper. Factors or the physics influencing dynamic response and dynamic bifurcation for TVES with rheology are identified and are also demonstrated through model problem studies. A simple model problem consisting of a rod (1D) of TVES material with memory fixed at one end and subjected to harmonic excitation at the other end is considered to study nonlinear dynamics of TVES with rheology, frequency response as well as dynamic bifurcation phenomenon.

An operator methodology for the global dynamic analysis of stochastic nonlinear systems

In a global dynamic analysis,the coexisting attractors and their basins are the main tools to understand the system behavior and safety.However,both basins and attractors can be drastically influenced by uncertainties.The aim of this work is to illustrate a methodology for the global dynamic analysis of nondeterministic dynamical systems with competing attractors.Accordingly,analytical and numerical tools for calculation of nondeterministic global structures,namely attractors and basins,are proposed.First,based on the definition of the Perron-Frobenius,Koopman and Foias linear operators,a global dynamic description through phase-space operators is presented for both deterministic and nondeterministic cases.In this context,the stochastic basins of attraction and attractors’distributions replace the usual basin and attractor concepts.Then,numerical implementation of these concepts is accomplished via an adaptative phase-space discretization strategy based on the classical Ulam method.Sample results of the methodology are presented for a canonical dynamical system.

Posterior Cramér-Rao Bounds for Nonlinear Dynamic System with Colored Noises

A mean squared error lower bound for the discrete-time nonlinear filtering with colored noises is derived based on the posterior version of the Cramer-Rao inequality.The colored noises are characterized by the auto-regressive model including the auto-correlated process noise and autocorrelated measurement noise simultaneously.Moreover,the proposed lower bound is also suitable for a general model of nonlinear high order auto-regressive systems.Finally,the lower bound is evaluated by a typical example in target tracking.It shows that the new lower bound can assess the achievable performance of suboptimal filtering techniques,and the colored noise has a significantly effect on the lower bound and the performance of filters.

NONLINEAR DYNAMIC CHARACTERISTICS OF HYDRODYNAMIC JOURNAL BEARING-FLEXIBLE ROTOR SYSTEM

The nonlinear dynamic behaviors of flexible rotor system with hydrodynamicbearing supports are analyzed. The shaft is modeled by using the finite element method that takesthe effect of inertia and shear into consideration. According to the nonlinearity of thehydrodynamic journal bearing-flexible rotor system, a modified modal synthesis technique withfree-interface is represented to reduce degrees-of-freedom of model of the flexible rotor system.According to physical character of oil film, variational constrain approach is introduced tocontinuously revise the variational form of Reynolds equation at every step of dynamic integrationand iteration. Fluid lubrication problem with Reynolds boundary is solved by the isoparametricfinite element method without the increasing of computing efforts. Nonlinear oil film forces andtheir Jacobians are simultaneously calculated and their compatible accuracy is obtained. Theperiodic motions are obtained by using the Poincare -Newton-Floquet (PNF) method. A method,combining the predictor-corrector mechanism to the PNF method, is presented to calculate thebifurcation point of periodic motions to be subject to change of system parameters. The localstability and bifurcation behaviors of periodic motions are obtained by Floquet theory. The chaoticmotions of the bearing-rotor system are investigated by power spectrum. The numerical examples showthat the scheme of this study saves computing efforts but also is of good precision.

Nonlinear dynamics of flexible tethered satellite system subject to space environment

The paper studies the nonlinear dynamics of a flexible tethered satellite system subject to space environments, such as the J2 perturbation, the air drag force, the solar pressure, the heating effect, and the orbital eccentricity. The flexible tether is modeled as a series of lumped masses and viscoelastic dampers so that a finite multi- degree-of-freedom nonlinear system is obtained. The stability of equilibrium positions of the nonlinear system is then analyzed via a simplified two-degree-freedom model in an orbital reference frame. In-plane motions of the tethered satellite system are studied numerically, taking the space environments into account. A large number of numerical simulations show that the flexible tethered satellite system displays nonlinear dynamic characteristics, such as bifurcations, quasi-periodic oscillations, and chaotic motions.

Dynamics and response reshaping of nonlinear predator-prey system undergoing random abrupt disturbances

An actual ecological predator-prey system often undergoes random environmental mutations owing to the impact of natural disasters and man-made destruction, which may destroy the balance between the species. In this paper,the stochastic dynamics of the nonlinear predator-prey system considering random environmental mutations is investigated, and a feedback control strategy is proposed to reshape the response of the predator-prey system against random abrupt environmental mutations. A delayed Markov jump system(MJS) is established to model such a predator-prey system. A novel first integral is constructed which leads to better approximation solutions of the ecosystem. Then, by applying the stochastic averaging method based on this novel first integral, the stochastic response of the predator-prey system is investigated, and an analytical feedback control is designed to reshape the response of the ecosystem from the disturbed state back to the undisturbed one.Numerical simulations finally illustrate the accuracy and effectiveness of the proposed procedure.

Complex nonlinear behaviors of a rotor dynamical system with non-analytical journal bearing supports

Nonlinear dynamic behaviors of a rotor dynamical system with finite hydrodynamic bearing supports were investigated. In order to increase the numerical accuracy and decrease computing costs, the isoparametric finite element method based on variational constraint approach is introduced because analytical bearing forces are not available. This method calculates the oil film forces and their Jacobians simultaneously while it can ensure that they have compatible accuracy. Nonlinear motion of the bearing-rotor system is caused by strong nonlinearity of oil film forces with respect to the displacements and velocities of the center of the rotor. A method consisting of a predictor-corrector mechanism and Newton-Raphson method is presented to calculate equilibrium position and critical speed corresponding to Hopf bifurcation point of the bearing-rotor system. Meanwhile the dynamic coefficients of bearing are obtained. The nonlinear unbalance periodic responses of the system are obtained by using Poincaré-Newton-Floquet method and a combination of predic- tor-corrector mechanism and Poincaré-Newton-Floquet method. The local stability and bifuration behaviors of periodic motions are analyzed by the Floquet theory. Chaotic motion of long term dynamic behaviors of the system is analyzed with power spectrum. The numerical results reveal such complex nonlinear behaviors as periodic, quasi-periodic, chaotic, jumped and coexistent solutions.

Chaos and chaotic control in a relative rotation nonlinear dynamical system under parametric excitation

This paper studies the chaotic behaviours of a relative rotation nonlinear dynamical system under parametric excitation and its control. The dynamical equation of relative rotation nonlinear dynamical system under parametric excitation is deduced by using the dissipation Lagrange equation. The. criterion of existence of chaos under parametric excitation is given by using the Melnikov theory. The chaotic behaviours are detected by numerical simulations including bifurcation diagrams, Poincare map and maximal Lyapunov exponent. Furthermore, it implements chaotic control using nomfeedback method. It obtains the parameter condition of chaotic control by the Melnikov theory. Numerical simulation results show the consistence with the theoretical analysis. The chaotic motions can be controlled to periodmotions by adding an excitation term.

Nonlinear fast–slow dynamics of a coupled fractional order hydropower generation system

Internal effects of the dynamic behaviors and nonlinear characteristics of a coupled fractional order hydropower generation system(HGS) are analyzed. A mathematical model of hydro-turbine governing system(HTGS) with rigid water hammer and hydro-turbine generator unit(HTGU) with fractional order damping forces are proposed. Based on Lagrange equations, a coupled fractional order HGS is established. Considering the dynamic transfer coefficient eis variational during the operation, introduced e as a periodic excitation into the HGS. The internal relationship of the dynamic behaviors between HTGS and HTGU is analyzed under different parameter values and fractional order. The results show obvious fast–slow dynamic behaviors in the HGS, causing corresponding vibration of the system, and some remarkable evolution phenomena take place with the changing of the periodic excitation parameter values.

Reconstructing the Nonlinear Dynamical Systems by Evolutionary Computation Techniques

We introduce a new dynamical evolutionary algorithm（DEA） based on the theory of statistical mechanics and investigate the reconstruction problem for the nonlinear dynamical systems using observation data. The convergence of the algorithm is discussed. We make the numerical experiments and test our model using the two famous chaotic systems （mainly the Lorenz and Chen systems）. The results show the relatively accurate reconstruction of these chaotic systems based on observational data can be obtained. Therefore we may conclude that there are broad prospects using our method to model the nonlinear dynamical systems.