Finite-dimensional Ordinary Differential Equation Research Articles

Periodic Center Manifolds for Nonhyperbolic Limit Cycles in ODEs

In this paper, we deal with a classical object, namely, a nonhyperbolic limit cycle in a system of smooth autonomous ordinary differential equations. While the existence of a center manifold near such a cycle was assumed in several studies on cycle bifurcations based on periodic normal forms, no proofs were available in the literature until recently. The main goal of this paper is to give an elementary proof of the existence of a periodic smooth locally invariant center manifold near a nonhyperbolic cycle in finite-dimensional ordinary differential equations by using the Lyapunov–Perron method. In addition, we provide several explicit examples of analytic vector fields admitting (non)-unique, (non)-[Formula: see text]-smooth and (non)-analytic periodic center manifolds.

Calculation of microwave heating temperature distribution based on SVD truncation

In mathematical models of microwave heating with infinite-dimensional characteristics, it is difficult to use traditional numerical methods to improve computational efficiency. In this work, we propose a fast and accurate method to calculate the temperature distribution of materials under microwave heating. First, we analysed the relationship between the choice of model order and the solution accuracy by downscaling the infinite-dimensional heat conduction partial differential equation (PDE) model into a finite-dimensional ordinary differential equation (ODE) model. Additionally, the effect of different boundary conditions on the global temperature distribution was analysed. Second, the equilibrium conversion matrix was calculated using the singular value decomposition (SVD) truncation method under homogeneous boundary conditions. Using this matrix, a lower-dimensional microwave heating ODE model was further obtained. Finally, the numerical simulation results showed that the root mean square error (RMSE) was only 0.07 and the maximum relative error was only −0.85%. The computation time of the equilibrium conversion matrix was 2.12 ∼ 3.00 ms, and the model calculation time was reduced by 97.78%. We compared the calculated temperature rise curves with those obtained using the conventional COMSOL model. The SVD truncation method achieved an efficient and accurate solution for the microwave heating model.

Random vortex dynamics via functional stochastic differential equations

In this paper, we present a novel, closed, three-dimensional random vortex dynamics system, which is equivalent to the Navier–Stokes equations for incompressible viscous fluid flows. The new random vortex dynamics system consists of a stochastic differential equation which is, in contrast with the two-dimensional random vortex dynamics equations, coupled to a finite-dimensional ordinary functional differential equation. This new random vortex system paves the way for devising new numerical schemes (random vortex methods) for solving three-dimensional incompressible fluid flow equations by Monte Carlo simulations. In order to derive the three-dimensional random vortex dynamics equations, we have developed two powerful tools: the first is the duality of the conditional distributions of a couple of Taylor diffusions, which provides a path space version of integration by parts; the second is a forward type Feynman–Kac formula representing solutions to nonlinear parabolic equations in terms of functional integration. These technical tools and the underlying ideas are likely to be useful in treating other nonlinear problems.

Investigation on the Nonlinear Vibration Characteristics of Current-Carrying Crescent Iced Conductors under Aerodynamic Forces, Ampere’s Forces, and Forced Excitation Conditions

Aiming at the problem of nonlinear vibration of current-carrying iced conductors, the aerodynamic forces are introduced into the previous vibration equation of current-carrying conductors that only considered Ampere’s forces. At the same time, on this basis, a forced excitation load is further introduced to study the influence of dynamic wind on the nonlinear vibration characteristics of current-carrying iced conductors, and a new current-carrying iced conductors system under the combined action of Ampere’s forces, forced excitation, and aerodynamic forces has been established, and the improved theoretical modeling of current-carrying iced transmission lines made the model more in line with practical engineering. Firstly, the model of current-carrying iced conductors was established, and then the vibration equation of the model was derived. And the vibration equation was transformed into a finite dimensional ordinary differential equation by using the Galerkin method. The amplitude-frequency response functions of the nonlinear forced primary resonances and super-harmonic and subharmonic resonances of the system are derived by using the multiscale method. Through numerical calculation, the influence of current-carrying, spacing, wind velocity, tension, and excitation amplitude on the response amplitude when the primary resonance of the system appears is analyzed, and the difference between the two working conditions (considering the aerodynamic forces and without considering aerodynamic forces) is compared. The influence of the variation of current-carrying i on the response amplitude of super-harmonic and subharmonic resonances and the stability of the steady-state solution of forced primary resonance was analyzed. The results show that the response amplitude and the nonlinearilty of system under the action of aerodynamic forces are smaller and weaker than without the action of aerodynamic forces; the variation of line parameters has a certain influence on the response amplitude of conductor and the nonlinearity of system; the response amplitudes of the primary resonance, super-harmonic resonance, and subharmonic resonance increase with the increase in the excitation amplitudes, and the resonance peak is offset towards the negative value of the tuning parameter σ, showing the characteristics of soft spring, and the response amplitudes are accompanied by complex nonlinear dynamic behaviors such as the multivalue and jump phenomenon. The change of current-carrying i has an obvious effect on the nonlinearity of the system. The nonlinear and response amplitudes of the system are also enhanced with the increase in wind velocity. The stability of the system is judged when the primary resonance occurs, and it is found that the response amplitude shows synchronization and the out-of-step phenomenon with the change of tuning parameters. The research results obtained in this paper would help to further improve the theoretical modeling about current-carrying iced lines, and the research of line parameters can give a certain reference value to practical engineering, and it will have a positive effect on the safe operation of high-voltage transmission lines.

Reservoir Computing with an Inertial Form

We propose a framework of a reservoir with an inertial manifold, which is a finite-dimensional object that attracts all trajectories. In addition, based on the theoretical results concerning infinite-dimensional dynamical systems, we demonstrate that this reservoir should have some desirable properties called the echo state property and the separation property. By considering the projection of the system onto the inertial manifold, we describe the behavior of the reservoir as a system of finite-dimensional ordinary differential equations called an inertial form. Next, we present a finite-dimensional model that approximates the inertial form. Finally, we propose a temporally discretized model, which is more suitable for computer computation. We give the theoretical proofs of our theorems herein.

Fault Detection of a Class of Nonlinear Uncertain Parabolic PDE Systems

This letter investigates the fault detection (FD) problem of a class of uncertain distributed parameter systems modeled by nonlinear parabolic partial differential equations (PDEs). A novel FD scheme is proposed with a neural network-based adaptive dynamics learning approach. Specifically, based on the Galerkin method, a finite dimensional ordinary differential equation (ODE) system is first derived to capture the dominant dynamics of the PDE system. An adaptive dynamics learning approach using radial basis function neural networks (RBF NN) is then developed to achieve locally accurate identification of the associated uncertain system dynamics. The learned knowledge can be obtained and stored in a bank of constant RBF NN models, which can be used to further construct a bank of FD estimators. A threshold is finally designed for real-time FD decision making. One important feature of the proposed FD scheme is that the spatio-temporal uncertain dominant dynamics of the parabolic PDE system can be locally accurately identified, such that occurring faults can be effectively distinguished from system uncertainties for accurate FD. Extensive simulation studies are conducted to verify the effectiveness and advantages of the proposed FD scheme.

Nonlinear Dynamic Behavior of Winding Hoisting Rope under Head Sheave Axial Wobbles

Axial wobbles of the head sheave due to manufacturing accuracy and installation technology can exert periodic lateral excitations to the winding hoisting rope. A hoisting rope consists of a constant catenary rope from the drum to the head sheave and a variable vertical rope from the head sheave to the conveyance. The longitudinal and lateral vibration responses of the hoisting rope are coupled to each other. In this paper, the coupled lateral-longitudinal governing equations of the winding hoisting system under the axial wobble excitations of the head sheave are established by the Hamilton principle. The governing equations are nonlinear infinite-dimensional partial differential equations, which are discretized into the finite-dimensional ordinary differential equations through the Galerkin method. The dynamic responses of the hoisting rope under the head sheave axial wobbles are given by MATLAB simulation when the winding hoist is lifting or lowering. The results show that lateral vibration displacements of the vertical rope under the head sheave axial wobbles are larger than those of the catenary rope. The axial wobble amplitude range of the head sheave is given to limit the lateral vibration displacements of the hoisting rope.

Reduction approach to the dynamics of interacting front solutions in a bistable reaction–diffusion system and its application to heterogeneous media

The dynamics of pulse solutions in a bistable reaction-diffusion system are studied analytically by reducing partial differential equations (PDEs) to finite-dimensional ordinary differential equations (ODEs). For the reduction, we apply the multiple-scales method to the mixed ODE-PDE system obtained by taking a singular limit of the PDEs. The reduced equations describe the interface motion of a pulse solution formed by two interacting front solutions. This motion is in qualitatively good agreement with that observed for the original PDE system. Furthermore, it is found that the reduction not only facilitates the analytical study of the pulse solution, especially the specification of the onset of local bifurcations, but also allows us to elucidate the global bifurcation structure behind the pulse behavior. As an application, the pulse dynamics in a heterogeneous bump-type medium are explored numerically and analytically. The reduced ODEs clarify the transition mechanisms between four pulse behaviors that occur at different parameter values.

Improved receding horizon H∞ temperature spectrum tracking control for Debye media in microwave heating process

This paper considers the problem of tracking control for the coupled thermal-electromagnetic model with temperature-dependent permittivity in microwave heating process, where the incident electrical field is subject to the asymmetrical saturation. Due to the nonhomogeneous Neumann boundary condition and inherent infinite-dimensional characteristics for microwave heating process, it is difficult for us to readily design and implement tracking controller. With the help of improved spectral Galerkin method, the traditional partial differential equation (PDE) model is transformed into the linear finite-dimensional ordinary differential equation (ODE) model which can be subsequently used for controller design. Then, the receding horizon H∞ control strategy is improved to stabilize the derived uncertain ODE error model with asymmetrical input saturation and bounded external disturbance. The outcome of improved algorithm is formulated as a linear matrix inequalities (LMIs) optimization problem, which can be on-line updated by applying the finite-different time-domain (FDTD) method. Finally, the improved control strategy is successfully applied in one-dimensional cavity model for tracking temperature spectrum.

Numerical solution of the unsteady diffusion-convection-reaction equation based on improved spectral Galerkin method

The aim of this paper is to present an explicit numerical algorithm based on improved spectral Galerkin method for solving the unsteady diffusion-convection-reaction equation. The principal characteristics of this approach give the explicit eigenvalues and eigenvectors based on the time-space separation method and boundary condition analysis. With the help of Fourier series and Galerkin truncation, we can obtain the finite-dimensional ordinary differential equations which facilitate the system analysis and controller design. By comparing with the finite element method, the numerical solutions are demonstrated via two examples. It is shown that the proposed method is effective.

Receding horizon H∞ guaranteed cost tracking control for microwave heating medium with temperature-dependent permittivity

This paper considers the temperature spectrum tracking control of microwave heating model, in the presence of asymmetrical input saturation, nonhomogeneous Neumann boundary condition and temperature-dependent permittivity. The sufficient condition for the existence of receding horizon H∞ guaranteed cost control is proposed based on the derived finite-dimensional ordinary differential equation (ODE) error model. Furthermore, by on-line updating and solving linear matrix inequalities (LMIs) optimization problem, the constrained tracking controller can be obtained in the sense of minimizing H∞ norm and satisfying the quadratic cost performance. The proposed control strategy is implemented on a one-dimensional cavity heating model and its performance is evaluated through the simulation.

Method of evolving junctions: A new approach to optimal path-planning in 2D environments with moving obstacles

We propose a novel algorithm to find the global optimal path in 2D environments with moving obstacles, where the optimality is understood relative to a general convex continuous running cost. By leveraging the geometric structures of optimal solutions and using gradient flows, we convert the path-planning problem into a system of finite dimensional ordinary differential equations, whose dimensions change dynamically. Then a stochastic differential equation based optimization method, called intermittent diffusion, is employed to obtain the global optimal solution. We demonstrate, via numerical examples, that the new algorithm can solve the problem efficiently.

Finite dimensional disturbance observer based control for nonlinear parabolic PDE systems via output feedback

This paper considers the problem of finite dimensional disturbance observer based control (DOBC) via output feedback for a class of nonlinear parabolic partial differential equation (PDE) systems. The external disturbance is generated by an exosystem modeled by ordinary differential equations (ODEs), which enters into the PDE system through the control channel. Motivated by the fact that the dominant dynamic behavior of parabolic PDE systems can be characterized by a finite number of degrees of freedom, the modal decomposition technique is initially applied to the PDE system to derive a slow subsystem of finite dimensional ODEs. Subsequently, based on the slow subsystem and the exosystem, a disturbance observer (DO) and a slow mode observer (SMO) are constructed to estimate the disturbance and the slow modes. Moreover, an observation spillover observer (OSO) is also constructed to cancel approximately the effect of the observation spillover. Then, a finite dimensional DOBC design via output feedback is developed to estimate and compensate the disturbance, such that the closed-loop PDE system is exponentially stable in the presence of the disturbance. The condition for the existence of the proposed controller is given in terms of bilinear matrix inequality. Two algorithms based on the linear matrix inequality (LMI) technique are provided for solving control and observer gain matrices of the proposed controller. Finally, the developed design method is applied to the control of a one-dimensional diffusion-reaction process to illustrate its effectiveness.

A neural network-based distributed parameter model identification approach for microcantilever

The microcantilever used in micro–nanomanipulator is a spatially distributed and flexible mechanical system. An accurate model of the microcantilever is essential for the accurate tip positioning and force sensing. Traditional lumped parameter model will lose the spatial dynamics. Though the nominal Euler–Bernoulli model is a distributed parameter model, in practice there are still some unknown nonlinear dynamics. In this study, a neural network-based distributed parameter model identification approach is proposed for modelling the microcantilever. First, a nominal Euler–Bernoulli beam model is derived. To compensate unknown nonlinear dynamics, a nonlinear term that needs to be estimated is added in the nominal model. For finite-dimensional implementation, the infinite-dimensional partial differential equation model is reduced into a finite-dimensional ordinary differential equation model using the Galerkin method. Next, a neural network-based intelligent learning approach is developed to learn the unknown nonlinearities from the input–output data. A radial basis function recurrent neural network observer is designed to estimate the finite-dimensional states from a few sensors of measurements. After that, a general regression neural network model is identified to establish the nonlinear spatiotemporal dynamic model between the inputs and outputs. The effectiveness of the proposed neural network-based distributed parameter modelling approach is verified by the simulations on a typical microcantilever.

Coupled Electromagnetic and Heat Transfer ODE Model for Microwave Heating With Temperature-Dependent Permittivity

In the traditional microwave heating partial differential equation (PDE) model, one of the main characteristics is the infinite-dimensional nature, which does not allow to readily design and implement a controller. Motivated by this obstruction, this paper proposes a microwave heating finite-dimensional ordinary differential equation (ODE) model, which can not only describe the thermodynamics field with nonhomogeneous boundary conditions but also be coupled with the variation of electromagnetic field in temperature-dependent dielectric media. Initially, the equivalent PDE model with a homogeneous boundary condition is derived by constructing an auxiliary function in order to directly derive the eigenspectrum of the spatial differential operator. With the help of model-reduction techniques, the dominant dynamics of temperature distribution are subsequently captured with a reasonable Galerkin truncation. The simulation results on microwave heating a water prototype show that the temporal and the spatial evolution of the temperature profile can be described by solving the temperature-dependent electromagnetic field and the finite-dimensional ODE model. Moreover, the effectiveness of the model is verified by comparing with the numerical results from the traditional COMSOL model. A further development of this ODE model may provide a useful numerical tool for the design and synthesis of microwave heaters to avoid thermal runaway phenomena.

Approximate finite-dimensional ODE temperature model for microwave heating

In this paper, a finite-dimensional ordinary differential equation (ODE) model is proposed for predicting the temperature profile with microwave heating to accomplish lower computing complexity. The traditional parabolic partial different equation (PDE) model with integrating Maxwell's equation and heat transport equation is not suitable for designing the on-line controller. Based on the obstruction, using an auxiliary function derives an intermediate model, which is analyzed and discussed for model reduction by employing the parameter separation method and Galerkin's method. The simulation experiments on one-dimensional waveguide and cavity demonstrate that the proposed approximate model is effective.

Dynamic optimization of open-loop input signals for ramp-up current profiles in tokamak plasmas

Establishing a good current spatial profile in tokamak fusion reactors is crucial to effective steady-state operation. The evolution of the current spatial profile is related to the evolution of the poloidal magnetic flux, which can be modeled in the normalized cylindrical coordinates using a parabolic partial differential equation (PDE) called the magnetic diffusion equation. In this paper, we consider the dynamic optimization problem of attaining the best possible current spatial profile during the ramp-up phase of the tokamak. We first use the Galerkin method to obtain a finite-dimensional ordinary differential equation (ODE) model based on the original magnetic diffusion PDE. Then, we combine the control parameterization method with a novel time-scaling transformation to obtain an approximate optimal parameter selection problem, which can be solved using gradient-based optimization techniques such as sequential quadratic programming (SQP). This control parameterization approach involves approximating the tokamak input signals by piecewise-linear functions whose slopes and break-points are decision variables to be optimized. We show that the gradient of the objective function with respect to the decision variables can be computed by solving an auxiliary dynamic system governing the state sensitivity matrix. Finally, we conclude the paper with simulation results for an example problem based on experimental data from the DIII-D tokamak in San Diego, California.

Distributed Consensus Observers-Based <inline-formula> <tex-math notation="TeX">$H_{\infty} $</tex-math></inline-formula> Control of Dissipative PDE Systems Using Sensor Networks

This paper considers the problem of finite dimensional output feedback H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> control for a class of nonlinear spatially distributed processes described by highly dissipative partial differential equations (PDEs), whose state is observed by a sensor network (SN) with a given topology. This class of systems typically involves a spatial differential operator whose eigenspectrum can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement. Motivated by this fact, the modal decomposition and singular perturbation techniques are initially applied to the PDE system to derive a finite-dimensional ordinary differential equation model, which accurately captures the dominant dynamics of the PDE system. Subsequently, based on the slow system and the SN topology, a set of finite-dimensional distributed consensus observers is constructed to estimate the state of the slow system. Then, a centralized control scheme, which only uses the available estimates from a specified group of SN nodes, is proposed for the PDE system. An H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> control design is developed in terms of a bilinear matrix inequality (BMI), such that the closed-loop PDE system is exponentially stable and a prescribed level of disturbance attenuation is satisfied for the slow system. Furthermore, a suboptimal H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> controller is also provided to make the attenuation level as small as possible, which can be obtained via a local optimization algorithm that treats the BMI as a double linear matrix inequality. Finally, the proposed method is applied to the control of the 1-D Kuramoto-Sivashinsky equation system.

A Grey-Box Distributed Parameter Modeling Approach for a Flexible Manipulator with Nonlinear Dynamics

The flexible manipulator is a spatially distributed mechanical system. An accurate model of the flexible manipulator is essential for the positioning control of the end effector. In this study, a greybox distributed parameter modeling approach is proposed for the flexible manipulator with unknown nonlinear dynamics. First, a nominal Euler-Bernoulli beam model is derived to describe the linear dynamics. To compensate unknown nonlinear dynamics, a nonlinear term is added in the nominal model.The Galerkin method is used to reduce the infinite-dimensional partial differential equation (PDE) model into a finite-dimensional ordinary differential equation (ODE) model. A neural network is designed to estimate the unknown nonlinearities from the input-output data. The effectiveness of the proposed greybox distributed parameter modeling approach is verified by the simulations on a flexible manipulator.

Modulated traveling fronts for a nonlocal Fisher-KPP equation: A dynamical systems approach

We consider a nonlocal generalization of the Fisher-KPP equation in one spatial dimension. As a parameter is varied, the system undergoes a Turing bifurcation. We study the dynamics near this Turing bifurcation. Our results are two-fold. First, we prove the existence of a two-parameter family of bifurcating stationary periodic solutions and derive a rigorous asymptotic approximation of these solutions. We also study the spectral stability of the bifurcating stationary periodic solutions with respect to almost co-periodic perturbations. Second, we restrict to a specific class of exponential kernels for which the nonlocal problem is transformed into a higher order partial differential equation. In this context, we prove the existence of modulated traveling fronts near the Turing bifurcation that describe the invasion of the Turing unstable homogeneous state by the periodic pattern established in the first part. Both results rely on a center manifold reduction to a finite dimensional ordinary differential equation.