Quasi Non-integrable Hamiltonian Systems Research Articles

Discussion on applicability of the complex fractional moment method in Hamiltonian system

In this paper, the complex fractional moment (CFM) method is extended from the Langevin system to the Hamiltonian system for the first time, the applicability and advantages of the CFM method in Hamiltonian systems are investigated. According to the quasi-periodic property, the stochastic averaging technique is applied to the stochastic quasi-nonintegrable Hamiltonian system, and the one-dimensional Hamiltonian stochastic differential equation is obtained. By Mellin transformation, the associated Fokker Planck Kolmogorov (FPK) equation governing the Hamilton function is transformed into a set of Ordinary Differential Equations (ODEs) of CFM. The semi-analytical solutions of the FPK equation are obtained by solving the ODEs. Furthermore, by a polynomial approximation method for the Hamiltonian case proposed in this paper, the numerical results show the influence of parameters and initial values on the evolution of the transient probability density functions (PDF) of the system. The accuracy of CFM method in the Hamiltonian case is verified by error analysis.

Feedback stabilization of quasi nonintegrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations

A procedure for designing a feedback control to asymptotic Lyapunov stability with probability one of quasi nonintegrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations is proposed. First, a one dimensional partially averaged Itô stochastic differential equation for controlled Hamiltonian is derived from the motion equations of the system by using the stochastic averaging method. Second, the dynamical programming equation for the ergodic control problem of the averaged system with undetermined cost function is set up based on the dynamical programming principle and the jump–diffusion chain stochastic differential rules. The optimal control law is obtained by solving the dynamical programming equation. Third, the analytical expression for the largest Lyapunov exponent of the averaged system is derived. Finally, the asymptotic Lyapunov stability with probability one of the originally controlled system is analyzed approximately by using the largest Lyapunov exponent. The cost function and optimal control forces are determined by the requirements of stabilizing the system. An example is worked out in detail to illustrate the effectiveness of the proposed method for stabilization control, and the control effect of the proposed feedback stabilization varies with the change of parameters is also studied in this paper, such as, the greater the excitation intensity of Gaussian and Poisson white noise, the better the stabilization control effect.

Control of quasi non-integrable Hamiltonian systems for targeting a specified stationary probability density

ABSTRACTAn innovative design procedure for the feedback control of quasi non-integrable Hamiltonian systems to target a specified stationary probability density function (SPDF) is proposed based on the averaged Fokker–Planck–Kolmogorov equation. The control force is split into a conservative part and a dissipative part. The conservative part is designed to make the controlled system and the target SPDF having the same Hamiltonian structure. The dissipative part is then determined to make the target SPDF to be the stationary solution of the controlled system. Then, a Lyapunov function method is adopted for proving that the transient solution of the controlled system approaches to the target SPDF as time t → ∞. The simulation result for an example shows the effectiveness of the proposed control strategy.

Stochastic Stability and Bifurcation of A High Speed Rotor-Bearing System with Random Excitation

In this paper, a high speed rotor-bearing system with stochastic excitation is considered, and study the stability and Hopf bifurcation of the system by using quasi-nonintegrable Hamiltonian system theory. Then, the conditions of local and global stability of system are obtained by largest Lyapunov exponent and boundary category. Finally, the solution of FPK equation can be got, which is stationary probability density function and jointly stationary probability density function, and then, by simulating its graph to illustrate the results.

Asymptotic stability of a class of nonlinear stochastic systems undergoing Markovian jumps

Systems which specifications change abruptly and statistically, referred to as Markovian-jump systems, are considered in this paper. An approximate method is presented to assess the asymptotic stability, with probability one, of nonlinear, multi-degree-of-freedom, Markovian-jump quasi-nonintegrable Hamiltonian systems subjected to stochastic excitations. Using stochastic averaging and linearization, an approximate formula for the largest Lyapunov exponent of the Hamiltonian equations is derived, from which necessary and sufficient conditions for asymptotic stability are obtained for different jump rules. In a Markovian-jump system with unstable operating forms, the stability conditions prescribe limitations on time spent in each unstable form so as to render the entire system asymptotically stable. The validity and utility of this approximate technique are demonstrated by a nonlinear two-degree-of-freedom oscillator that is stochastically driven and capable of Markovian jumps.

Transient stochastic response of quasi non-integerable Hamiltonian system

The approximate transient response of multi-degree-of-freedom (MDOF) quasi non-integrable Hamiltonian system under Gaussian white noise excitation is investigated. First, the averaged Itô equation for Hamiltonian and the associated Fokker–Planck–Kolmogorov (FPK) equation governing the transient probability density of Hamiltonian of the system are derived by applying the stochastic averaging method for quasi non-integrable Hamiltonian systems. Then, the approximate solution of the transient probability density of Hamiltonian is obtained by applying the Galerkin method to solve the FPK equation. The approximate transient solution is expressed as a series in terms of properly selected basis functions with time-dependent coefficients. The transient probability densities of displacements and velocities can be derived from that of Hamiltonian. One example is given to illustrate the application of the proposed procedure. It is shown that the results for the example obtained by using the proposed procedure agree well with those from Monte Carlo simulation of the original system.

Stationary Response of a Class of Nonlinear Stochastic Systems Undergoing Markovian Jumps

Systems whose specifications change abruptly and statistically, referred to as Markovian-jump systems, are considered in this paper. An approximate method is presented to assess the stationary response of multidegree, nonlinear, Markovian-jump, quasi-nonintegrable Hamiltonian systems subjected to stochastic excitation. Using stochastic averaging, the quasi-nonintegrable Hamiltonian equations are first reduced to a one-dimensional Itô equation governing the energy envelope. The associated Fokker–Planck–Kolmogorov equation is then set up, from which approximate stationary probabilities of the original system are obtained for different jump rules. The validity of this technique is demonstrated by using a nonlinear two-degree oscillator that is stochastically driven and capable of Markovian jumps.

Asymptotic Analytical Solutions of First-Passage Rate to Quasi-Nonintegrable Hamiltonian Systems

The first-passage problem of quasi-nonintegrable Hamiltonian systems subject to light linear/nonlinear dampings and weak external/parametric random excitations is investigated here. The motivation is to acquire asymptotic analytical solution of the first-passage rate or the mean first-passage time based on the averaged Itô stochastic differential equation for quasi-nonintegrable Hamiltonian systems. By using the probability current equation and the Laplace integral method, a new method is proposed to obtain the asymptotic analytical expressions for the first-passage rate in the case of high passage threshold. The associated functions such as the reliability function and the probability density function of first-passage time can then be obtained from the first-passage rate. High passage threshold is the crucial condition for the validity of the proposed method. The random bistable oscillator is studied as an illustrative example using the method. The analytical result obtained from the asymptotic analysis shows its consistency with the Kramers formula. A coupled two-degree-of-freedom (2DOF) nonlinear oscillator subjected to stochastic excitations is studied to illustrate the procedure of acquiring the asymptotic analytical solution. The results obtained from the analytical solution agree well with those from numerical simulation, which verifies the accuracy of the proposed method.

Optimal bounded control of quasi-nonintegrable Hamiltonian systems using stochastic maximum principle

A new procedure for designing optimal bounded control of quasi-nonintegrable Hamiltonian systems with actuator saturation is proposed based on the stochastic averaging method for quasi-nonintegrable Hamiltonian systems and the stochastic maximum principle. First, the stochastic averaging method for controlled quasi-nonintegrable Hamiltonian systems is introduced. The original control problem is converted into one for a partially averaged equation of system energy together with a partially averaged performance index. Then, the adjoint equation and the maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The bounded optimal control forces are obtained from the maximum condition and solving the forward–backward stochastic differential equations (FBSDE). For infinite time-interval ergodic control, the adjoint variable is stationary process, and the FBSDE is reduced to an ordinary differential equation. Finally, the stationary probability density of the Hamiltonian and other response statistics of optimally controlled system are obtained by solving the Fokker–Plank–Kolmogorov equation associated with the fully averaged Ito equation of the controlled system. For comparison, the bang–bang control is also presented. An example of two degree-of-freedom quasi-nonintegrable Hamiltonian system is worked out to illustrate the proposed procedure and its effectiveness. Numerical results show that the proposed control strategy has higher control efficiency and less discontinuous control force than the corresponding bang–bang control at the price of slightly less control effectiveness.

Nonlinear Dynamic Characteristics of Giant Magnetostrictive Nanofilm-SMA Composite Beam in Axial Stochastic Excitation

In this paper, nonlinear dynamic characteristics of giant magnetostructive nanofilm-shape memory alloy (SMA) composite beam in axial stochastic excitation were studied. Von del Pol nonlinear difference item was introduced to interpret the hysteresis phenomenon of the strain-stress curve of SMA, and the hysteretic nonlinear dynamic model of giant magnetostructive nanofilm-SMA composite beam in axial stochastic excitation was developed. The steady-state probability density function and the joint probability density function of the system were obtained in quasi-nonintegrable Hamiltonian system theory. The result of simulation shows that the stability of the trivial solution varies with bifurcation parameter, and stochastic Hopf bifurcation appears in the process. The result is helpful to stochastic bifurcation control to giant magnetostructive nanofilm-SMA composite beam.

Stochastic stability of quasi non-integrable Hamiltonian systems under parametric excitations of Gaussian and Poisson white noises

The asymptotic Lyapunov stability with probability one of n-degree-of-freedom (n-DOF) quasi non-integrable Hamiltonian systems subject to weakly parametric excitations of combined Gaussian and Poisson white noises is studied by using the largest Lyapunov exponent. First, an n-DOF quasi non-integrable Hamiltonian system subject to weakly parametric excitations of combined Gaussian and Poisson white noises is reduced to a one-dimensional averaged Itô stochastic differential equation (SDE) for Hamiltonian by using the stochastic averaging method for quasi non-integrable Hamiltonian systems. Then, the expression for the Lyapunov exponent of the averaged Itô SDE is derived and the approximately necessary and sufficient condition for the asymptotic Lyapunov stability with probability one of the trivial solution of the original system is obtained. Finally, one example is worked out to illustrate the proposed procedure and its effectiveness is confirmed by comparing with Monte Carlo simulation. It is found that analytical and simulation results agree well.

Research on Stochastic Stability of Wheelsets with Primary Suspension

A new model for elastic constraint wheelset system of rail vehicle is proposed. Assuming the stochastic excitation as Gauss white noise, a stochastic model is built for elastic constraint wheelset system. Here two kinds of stochastic excitations are considered: one is the internal multiplicative excitation inherited in the internal system such as the spring and wheelset/rail contact geometric relationship, the other is the external excitation induced by track random irregularities. The model defined here is considered as a weak damping, weak excitation quasi non-integrable Hamiltonian system. The maximal Lyapunov exponent is calculated by quasi non-integrable Hamiltonian theory and oseledec multiplicative ergodic theory, and the stochastic local stability conditions are obtained. Meanwhile, the stochastic global stability conditions are derived by considering the modality of the singular boundary condition.

Stochastic optimal control of quasi non-integrable Hamiltonian systems with stochastic maximum principle

A new procedure for designing optimal control of quasi non-integrable Hamiltonian systems under stochastic excitations is proposed based on the stochastic averaging method for quasi non-integrable Hamiltonian systems and the stochastic maximum principle. First, the control problem consisting of 2n-dimensional equations governing the controlled quasi non-integrable system and performance index is converted into a partially averaged one consisting of one-dimensional equation of the controlled system and performance index by using the stochastic averaging method. Then, the adjoint equation and the maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The optimal control forces are determined from the maximum condition and solving the forward–backward stochastic differential equations (FBSDE). For infinite time-interval ergodic control, the adjoint variable is a stationary process and the FBSDE is reduced to a partial differential equation. Finally, the response statistics of optimally controlled system is predicted by solving the Fokker–Plank equation (FPE) associated with the fully averaged Ito equation of the controlled system. An example of two degree-of-freedom (DOF) quasi non-integrable Hamiltonian system is worked out to illustrate the proposed procedure and its effectiveness.

Stochastic optimal control of first-passage failure for rectangular thin plate vibration model under gaussian white-noise excitations

A rectangular thin plate vibration model subjected to inplane stochastic excitation is simplified to a quasi-nonintegrable Hamiltonian system with two degrees of freedom. Subsequently a one-dimensional Ito stochastic differential equation for the system is obtained by applying the stochastic averaging method for quasi-nonintegrable Hamiltonian systems. The conditional reliability function and conditional probability density are both gained by solving the backward Kolmogorov equation numerically. Finally, a stochastic optimal control model is proposed and solved. The numerical results show the effectiveness of this method.

Robustness of feedback stabilization of quasi non-integrable Hamiltonian systems with parametric uncertainty

The robustness of feedback stabilization for quasi non-integrable Hamiltonian systems with uncertain parameters is investigated. The uncertain parameters are modeled as bounded random variables with λ-probability density function. Based on the independence of uncertain parameters and stochastic excitations, a feedback control to asymptotically stabilize, with probability one, the nominal system (with average parameter values) is firstly obtained by applying the stochastic averaging method, the expression for the Lyapunov exponent of quasi non-integrable Hamiltonian systems and the stochastic dynamical programming principle. Then, the mean and standard deviations of the Lyapunov exponent of the uncertain quasi non-integrable Hamiltonian system are calculated by using the stochastic averaging method and probabilistic analysis. Finally, the robustness of the feedback stabilization for quasi non-integrable Hamiltonian systems with parametric uncertainty is evaluated in terms of the sensitivity of the variation coefficient of the Lyapunov exponent of a controlled system to the variation coefficients of uncertain parameters. An example is worked out to illustrate the robustness of the feedback stabilization.

BIFURCATIONS IN A STOCHASTIC BUSINESS CYCLE MODEL

We introduce a stochastic business cycle model and study the underlying stochastic Hopf bifurcations with respect to probability densities at different parameter values. Our analysis is based on the calculation of the largest Lyapunov exponent via multiplicative ergodic theorem and the theory of boundary analysis for quasi-nonintegrable Hamiltonian systems. Some numerical simulations of the model are performed.

Stochastic optimal bounded control of MDOF quasi nonintegrable-Hamiltonian systems with actuator saturation

A new bounded optimal control strategy for multi-degree-of-freedom (MDOF) quasi nonintegrable-Hamiltonian systems with actuator saturation is proposed. First, an n-degree-of-freedom (n-DOF) controlled quasi nonintegrable-Hamiltonian system is reduced to a partially averaged Ito stochastic differential equation by using the stochastic averaging method for quasi nonintegrable-Hamiltonian systems. Then, a dynamical programming equation is established by using the stochastic dynamical programming principle, from which the optimal control law consisting of optimal unbounded control and bang–bang control is derived. Finally, the response of the optimally controlled system is predicted by solving the Fokker–Planck–Kolmogorov (FPK) equation associated with the fully averaged Ito equation. An example of two controlled nonlinearly coupled Duffing oscillators is worked out in detail. Numerical results show that the proposed control strategy has high control effectiveness and efficiency and that chattering is reduced significantly compared with the bang–bang control strategy.

Nonlinear stochastic optimal bounded control of hysteretic systems with actuator saturation

A modified nonlinear stochastic optimal bounded control strategy for random excited hysteretic systems with actuator saturation is proposed. First, a controlled hysteretic system is converted into an equivalent nonlinear nonhysteretic stochastic system. Then, the partially averaged Ito stochastic differential equation and dynamical programming equation are established, respectively, by using the stochastic averaging method for quasi non-integrable Hamiltonian systems and stochastic dynamical programming principle, from which the optimal control law consisting of optimal unbounded control and bang-bang control is derived. Finally, the response of optimally controlled system is predicted by solving the Fokker-Planck-Kolmogorov (FPK) equation associated with the fully averaged Ito equation. Numerical results show that the proposed control strategy has high control effectiveness and efficiency.

Stochastic dynamics and denaturation of thermalized DNA

In the first part of the paper, the stochastic dynamics of the Peyrard-Bishop-Dauxois (PBD) DNA model is studied. A one-dimensional averaged Itô stochastic differential equation governing the total energy of the system and the associated Fokker-Planck equation governing the transition probability density function of the total energy are derived from the Langevin equations for the base-pair (bp) separation of the PBD DNA model by using the stochastic averaging method for quasinonintegrable Hamiltonian systems. The stationary probability density function of the average energy and the mean square of the bp separation are obtained by solving the reduced Fokker-Planck equation. In the second part of the paper, the local denaturation of the thermalized PBD DNA model is studied as a first-passage-time problem in the energy. A backward Kolmogorov equation and a Pontryagin equation are derived from the averaged Itô equation and solved to yield the waiting-time distribution and the mean bp opening time. All the analytical results are confirmed with those from Monte Carlo simulation. It is pointed out that the proposed method may yield a reasonable mean bp opening time if the friction coefficient is fixed using experimental results.

Stochastic stability and bifurcation in a macroeconomic model

Abstract On the basis of the work of Goodwin and Puu, a new business cycle model subject to a stochastically parametric excitation is derived in this paper. At first, we reduce the model to a one-dimensional diffusion process by applying the stochastic averaging method of quasi-nonintegrable Hamiltonian system. Secondly, we utilize the methods of Lyapunov exponent and boundary classification associated with diffusion process respectively to analyze the stochastic stability of the trivial solution of system. The numerical results obtained illustrate that the trivial solution of system must be globally stable if it is locally stable in the state space. Thirdly, we explore the stochastic Hopf bifurcation of the business cycle model according to the qualitative changes in stationary probability density of system response. It is concluded that the stochastic Hopf bifurcation occurs at two critical parametric values. Finally, some explanations are given in a simply way on the potential applications of stochastic stability and bifurcation analysis.