A model-reduction method for nonlinear discrete-time skew-product dynamical systems in the presence of model uncertainty

The present research work proposes a new systematic approach to the problem of model reduction for nonlinear discrete-time skew-product dynamical systems in the presence of model uncertainty. The problem of interest is addressed within the context of functional equation theory, and in particular, through a system of invariance functional equations for which a general set of conditions for solvability is provided. Within the class of analytic solutions, this set of conditions guarantees the existence and uniqueness of a locally analytic solution which represents the system’s slow invariant manifold attracting all dynamic trajectories in the absence of model uncertainty. An exact reduced-order model is then obtained through the restriction of the original discrete-time system dynamics on the slow manifold. The analyticity property of the solution to the invariance functional equations enables the development of a series solution method that can be easily implemented using MAPLE leading to polynomial approximations up to the desired degree of accuracy. Furthermore, the aforementioned attractivity property and the system’s transition towards the above manifold is analyzed and characterized in the presence of model uncertainty. Finally, the proposed method is evaluated through an illustrative biological reactor example.     

Global bifurcations and chaotic motions of a flexible multi-beam structure

Global bifurcations and multi-pulse chaotic motions of flexible multi-beam structures derived from an L-shaped beam resting on a vibrating base are investigated considering one to two internal resonance and principal resonance. Base on the exact modal functions and the orthogonality conditions of global modes, the PDEs of the structure including both nonlinear coupling and nonlinear inertia are discretized into a set of coupled autoparametric ODEs by using Galerkin’s technique. The method of multiple scales is applied to yield a set of autonomous equations of the first order approximations to the response of the dynamical system. A generalized Melnikov method is used to study global dynamics for the “resonance case”. The present analysis indicates multi-pulse chaotic motions result from the existence of Šilnikov’s type of homoclinic orbits and the critical parameter surface under which the system may exhibit chaos in the sense of Smale horseshoes are obtained. The global results are finally interpreted in terms of the physical motion of such flexible multi-beam structure and the dynamical mechanism on chaotic pattern conversion between the localized mode and the coupled mode are revealed.     

Existence of traveling wave solutions for a nonlinear dissipative-dispersive equation

In this paper, we consider a dissipative-dispersive nonlinear equation appliable to many physical phenomena. Using the geometric singular perturbation method based on the theory of dynamical systems, we investigate the existence of its traveling wave solutions with the dissipative terms having sufficiently small coefficients. The results show that the traveling waves exist on a two-dimensional slow manifold in a three-dimensional system of ordinary differential equations （ODEs）. Then, we use the Melnikov method to establish the existence of a homoclinic orbit in this manifold corresponding to a solitary wave solution of the equation. Furthermore, we present some numerical computations to show the approximations of such wave orbits.     

Invariant Manifolds for Steady Boltzmann Flows and Applications

We develop a theory of invariant manifolds for the steady Boltzmann equation and apply it to the study of boundary layers and nonlinear waves. The steady Boltzmann equation is an infinite dimensional differential equation, so the standard center manifold theory for differential equations based on spectral information does not apply here. Instead, we employ a time-asymptotic approach using the pointwise information of Green’s function for the construction of the linear invariant manifolds. At the resonance cases when the Mach number at the far field is around one of the critical values of ?1, 0 or 1, the truly nonlinear theory arises. In such a case, there are wave patterns combining the fast decaying Knudsen-type and slow varying fluid-like waves. The key Knudsen manifolds consisting of only Knudsentype layers are constructed through delicate analysis of identifying the singular behavior around the critical Mach numbers. Around Mach number ± 1, the fluidlike waves are compressive and expansive waves; and around the Mach number 0, they are linear thermal layers. The quantitative analysis of the fluid-like waves is done using the reduction of dimensions to the center manifolds.Two-scale nonlinear dynamics based on those on the Knudsen and center manifolds are formulated for the study of the global dynamics of the combined wave patterns. There are striking bifurcations in the transition of evaporation to condensation and in the transition of the Milne’s problem with a subsonic far field to one with a supersonic far field. The analysis of these wave patterns allows us to understand the Sone Diagram for the study of the complete condensation boundary value problem. The monotonicity of the Boltzmann shock profiles, a problem that initially motivated the present study, is shown as a consequence of the quantitative analysis of the nonlinear fluid-like waves.     

Invariant Manifolds,Nonclassical Normal Modes,and Proper Orthogonal Modes in the Dynamics of the Flexible Spherical Pendulum

It is shown that the flexible spherical pendulum undergoes purely slow motions with master and slaved components. The family of slow motions is realized as a three-dimensional invariant manifold in phase space. This manifold is computed analytically by applying the method of geometric singular perturbations. This manifold is nonlinear and for all energy and angular momentum levels is characterized precisely by three PO (proper orthogonal) modes. Its submanifold of zero angular momentum is a two-dimensional invariant manifold; it is the geometric realization of a nonclassical nonlinear normal mode. This normal mode is characterized precisely by two PO modes. The slaved slow dynamics are characterized precisely by a single PO mode. The stability of the slow invariant manifold as well as interactions between fast and slow dynamics are considered.     

Heteroclinic Orbits in Slow–Fast Hamiltonian Systems with Slow Manifold Bifurcations

Motivated by a problem in which a heteroclinic orbit represents a moving interface between ordered and disordered crystalline states, we consider a class of slow–fast Hamiltonian systems in which the slow manifold loses normal hyperbolicity due to a transcritical or pitchfork bifurcation as a slow variable changes. We show that under assumptions appropriate to the motivating problem, a singular heteroclinic solution gives rise to a true heteroclinic solution. In contrast to previous approaches to such problems, our approach uses blow-up of the bifurcation manifold, which allows geometric matching of inner and outer solutions.     

Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations

In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets and its behavior under perturbation. We prove that a bounded hyperbolic global solutions persists under singular perturbations and that their nonlinear unstable manifold behave continuously. To accomplish this, we need to establish results on roughness of exponential dichotomies under these singular perturbations. Our results imply that, if the limiting pullback attractor of a nonautonomous dynamical system is the closure of a countable union of unstable manifolds of global bounded hyperbolic solutions, then it behaves continuously (upper and lower) under singular perturbations.     

An Optimization Approach to Kinetic Model Reduction for Combustion Chemistry

Model reduction methods are relevant when the computation time of a full convection–diffusion–reaction simulation based on detailed chemical reaction mechanisms is too large. In this article, we consider a model reduction approach based on optimization of trajectories and its applicability to realistic combustion models. As many model reduction methods, it identifies points on a slow invariant manifold based on time scale separation in the dynamics of the reaction system. The numerical approximation of points on the manifold is achieved by solving a semi-infinite optimization problem, where the dynamics enter the problem as constraints. The proof of existence of a solution for an arbitrarily chosen dimension of the reduced model (slow manifold) is extended to the case of realistic combustion models including thermochemistry by considering the properties of proper maps. The model reduction approach is finally applied to two models based on realistic reaction mechanisms: ozone decomposition as a small test case and syngas combustion as a test case including all features of a detailed combustion mechanism.     

Analysis on global and chaotic dynamics of nonlinear wave equations for truss core sandwich plate

This paper presents the study on the chaotic wave and chaotic dynamics of the nonlinear wave equations for a simply supported truss core sandwich plate combined with the transverse and in-plane excitations. Based on the governing equation of motion for the simply supported sandwich plate with truss core, the reductive perturbation method is used to simplify the partial differential equation. According to the exact solution of the unperturbed equation, two different kinds of the topological structures are derived, which one structure is the resonant torus and another structure is the heteroclinic orbit. The characteristic of the singular points in the neighborhood of the resonant torus for the nonlinear wave equation is investigated. It is found that there exists the homoclinic orbit on the unperturbed slow manifold. The saddle-focus type of the singular point appears when the homoclinic orbit is broken under the perturbation. Additionally, the saddle-focus type of the singular point occurs when the resonant torus on the fast manifold is broken under the perturbation. It is known that the dynamic characteristics are well consistent on the fast and slow manifolds under the condition of the perturbation. The Melnikov method, which is called the first measure, is applied to study the persistence of the heteroclinic orbit in the perturbed equation. The geometric analysis, which is named the second measure, is used to guarantee that the heteroclinic orbit on the fast manifold comes back to the stable manifold of the saddle on the slow manifold under the perturbation. The theoretical analysis suggests that there is the chaos for the Smale horseshoe sense in the truss core sandwich plate. Numerical simulations are performed to further verify the existence of the chaotic wave and chaotic motions in the nonlinear wave equation. The damping coefficient is considered as the controlling parameter to study the effect on the propagation property of the nonlinear wave in the sandwich plate with truss core. The numerical results confirm the validity of the theoretical study.     

Dimensional Reduction for Nonlinear Time-Delayed Systems Composed of Stiff and Soft Substructures

This paper presents a new approach, based on the center manifoldtheorem, to reducing the dimension of nonlinear time-delay systemscomposed of both stiff and soft substructures. To complete the reductionprocess, the dynamic equation of a delayed system is first formulated asa set of singular perturbed equations that exhibit dynamic behaviorevolving in two different time scales. In terms of the fast time scale,the dynamic equation of system can be converted into the standard formof a functional differential equation in critical cases, namely, to aform that can be treated by means of the center manifold theorem. Then,the approximated center manifold is determined by solving a series ofboundary-value problems. The center manifold theorem ensures that thedominant dynamics of the system is described by a set of ordinarydifferential equations of low order, the dimension of which is identicalto that of the phase space of slowly variable states. As an applicationof the proposed approach, a detailed stability analysis is made for aquarter car model equipped with an active suspension with a time delaycaused by a hydraulic actuator. The analysis shows that the dimensionalreduction is surprisingly effective within a wide range of the systemparameters.     

On the Global Geometric Structure of the Dynamics of the Elastic Pendulum

We approach the planar elastic pendulum as a singular perturbation of the pendulum to show that its dynamics are governed by global two-dimensional invariant manifolds of motion. One of the manifolds is nonlinear and carries purely slow periodic oscillations. The other one, on the other hand, is linear and carries purely fast radial oscillations. For sufficiently small coupling between the angular and radial degrees of freedom, both manifolds are global and orbitally stable up to energy levels exceeding that of the unstable equilibrium of the system. For fixed value of coupling, the fast invariant manifold bifurcates transversely to create unstable radial oscillations exhibiting energy transfer. Poincaré sections of iso-energetic manifolds reveal that only motions on and near a separatrix emanating from the unstable region of the fast invariant manifold exhibit energy transfer.     

Multi-step transversal and tangential linearization methods applied to a class of nonlinear beam equations

A novel and continuously parameterized form of multi-step transversal linearization (MTrL) method is developed and numerically explored for solving nonlinear ordinary differential equations governing a class of boundary value problems (BVPs) of relevance in structural mechanics. A similar family of multi-step tangential linearization (MTnL) methods is also developed and applied to such BVP-s. Within the framework of MTrL and MTnL, a BVP is treated as a constrained dynamical system, i.e. a constrained initial value problem (IVP). While the MTrL requires the linearized solution manifold to transversally intersect the nonlinear solution manifold at a chosen set of points across the axis of the independent variable, the essential difference of the present MTrL method from its previous version [Roy, D., Kumar, R., 2005. A multistep transversal linearization (MTL) method in nonlinear structural dynamics. J. Sound Vib. 17, 829–852.] is that it has the flexibility of treating nonlinear damping and stiffness terms as time-variant damping and stiffness terms in the linearized system. The resulting time-variant linearized system is then solved using Magnus’ characterization [Magnus, W., 1954. On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math., 7, 649–673.]. Towards numerical illustrations, response of a tip loaded cantilever beam (Elastica) is first obtained. Next, the response of a simply supported nonlinear Timoshenko beam is obtained using a variationally correct (VC) model for the beam [Marur, S., Prathap, G., 2005. Nonlinear beam vibration problems and simplification in finite element model. Comput. Mech. 35(5), 352–360.]. The new model does not involve any simplifications commonly employed in the finite element formulations in order to ease the computation of nonlinear stiffness terms from nonlinear strain energy terms. A comparison of results through MTrL and MTnL techniques consistently indicate a superior quality of approximations via the transversal linearization technique. While the usage of tangential system matrices is common in nonlinear finite element practices, it is demonstrated that the transversal version of linearization offers an easier and more general implementation, requires no computations of directional derivatives and leads to a consistently higher level of numerical accuracy. It is also observed that higher order versions of MTrL/MTnL with Lagrangian interpolations may not work satisfactorily and hence spline interpolations are suggested to overcome this problem.     

On application of Newton’s law to mechanical systems with motion constraints

This work is devoted to deriving and investigating conditions for the correct application of Newton’s law to mechanical systems subjected to motion constraints. It utilizes some fundamental concepts of differential geometry and treats both holonomic and nonholonomic constraints. This approach is convenient since it permits one to view the motion of any dynamical system as a path of a point on a manifold. In particular, the main focus is on the establishment of appropriate conditions, so that the form of Newton’s law of motion remains invariant when imposing an additional set of motion constraints on a mechanical system. Based on this requirement, two conditions are derived, specifying the metric and the form of the connection on the new manifold, which results after enforcing the additional constraints. The latter is weaker than a similar condition obtained by imposing a metric compatibility condition holding on Riemannian manifolds and employed frequently in the literature. This is shown to have several practical implications. First, it provides a valuable freedom for selecting the connection on the manifold describing large rigid body rotation, so that the group properties of this manifold are preserved. Moreover, it is used to state clearly the conditions for expressing Newton’s law on the tangent space and not on the dual space of a manifold, which is the natural geometrical space for this. Finally, the Euler–Lagrange operator is examined and issues related to equations of motion for anholonomic and vakonomic systems are investigated and clarified further.     

Dynamic instabilities in coupled oscillators induced by geometrically nonlinear damping

The dynamics of a system of coupled oscillators possessing strongly nonlinear stiffness and damping is examined. The system consists of a linear oscillator coupled to a strongly nonlinear, light attachment, where the nonlinear terms of the system are realized due to geometric effects. We show that the effects of nonlinear damping are far from being purely parasitic and introduce new dynamics when compared to the corresponding systems with linear damping. The dynamics is analyzed by performing a slow/fast decomposition leading to slow flows, which in turn are used to study transient instability caused by a bifurcation to 1:3 resonance capture. In addition, a new dynamical phenomenon of continuous resonance scattering is observed that is both persistent and prevalent for the case of the nonlinearly damped system: For certain moderate excitations, the transient dynamics “tracks” a manifold of impulsive orbits, in effect transitioning between multiple resonance captures over definitive frequency and energy ranges. Eventual bifurcation to 1:3 resonance capture generates the dynamic instability, which is manifested as a sudden burst of the response of the light attachment. Such instabilities that result in strong energy transfer indicate potential for various applications of nonlinear damping such as in vibration suppression and energy harvesting.     

A differential flatness theory approach to observer-based adaptive fuzzy control of MIMO nonlinear dynamical systems

The paper proposes a solution to the problem of observer-based adaptive fuzzy control for MIMO nonlinear dynamical systems (e.g. robotic manipulators). An adaptive fuzzy controller is designed for a class of nonlinear systems, under the constraint that only the system’s output is measured and that the system’s model is unknown. The control algorithm aims at satisfying the $H_\infty $ tracking performance criterion, which means that the influence of the modeling errors and the external disturbances on the tracking error is attenuated to an arbitrary desirable level. After transforming the MIMO system into the canonical form, the resulting control inputs are shown to contain nonlinear elements which depend on the system’s parameters. The nonlinear terms which appear in the control inputs are approximated with the use of neuro-fuzzy networks. Moreover, since only the system’s output is measurable the complete state vector has to be reconstructed with the use of a state observer. It is shown that a suitable learning law can be defined for the aforementioned neuro-fuzzy approximators so as to preserve the closed-loop system stability. With the use of Lyapunov stability analysis, it is proven that the proposed observer-based adaptive fuzzy control scheme results in $H_{\infty }$ tracking performance.     

The Geometry of the Solution Set of Nonlinear Optimal Control Problems

In an optimal control problem one seeks a time-varying input to a dynamical systems in order to stabilize a given target trajectory, such that a particular cost function is minimized. That is, for any initial condition, one tries to find a control that drives the point to this target trajectory in the cheapest way. We consider the inverted pendulum on a moving cart as an ideal example to investigate the solution structure of a nonlinear optimal control problem. Since the dimension of the pendulum system is small, it is possible to use illustrations that enhance the understanding of the geometry of the solution set. We are interested in the value function, that is, the optimal cost associated with each initial condition, as well as the control input that achieves this optimum. We consider different representations of the value function by including both globally and locally optimal solutions. Via Pontryagin’s maximum principle, we can relate the optimal control inputs to trajectories on the smooth stable manifold of a Hamiltonian system. By combining the results we can make some firm statements regarding the existence and smoothness of the solution set.     

A reduced‐order approach for optimal control of fluids using proper orthogonal decomposition

In this article, a reduced‐order modeling approach, suitable for active control of fluid dynamical systems, based on proper orthogonal decomposition (POD) is presented. The rationale behind the reduced‐order modeling is that numerical simulation of Navier–Stokes equations is still too costly for the purpose of optimization and control of unsteady flows. The possibility of obtaining reduced‐order models that reduce the computational complexity associated with the Navier–Stokes equations is examined while capturing the essential dynamics by using the POD. The POD allows the extraction of a reduced set of basis functions, perhaps just a few, from a computational or experimental database through an eigenvalue analysis. The solution is then obtained as a linear combination of this reduced set of basis functions by means of Galerkin projection. This makes it attractive for optimal control and estimation of systems governed by partial differential equations (PDEs). It is used here in active control of fluid flows governed by the Navier–Stokes equations. In particular, flow over a backward‐facing step is considered. Reduced‐order models/low‐dimensional dynamical models for this system are obtained using POD basis functions (global) from the finite element discretizations of the Navier–Stokes equations. Their effectiveness in flow control applications is shown on a recirculation control problem using blowing on the channel boundary. Implementational issues are discussed and numerical experiments are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered in state space and in direct second order (structural) form. In state space order reduction methods, the equations of motion are expressed as a set of first order equations and transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of new set of equations are time invariant. At this stage, four order reduction methodologies, namely linear, nonlinear projection via singular perturbation, post-processing approach and invariant manifold technique, are suggested. The invariant manifold technique yields a unique ‘reducibility condition’ that provides the conditions under which an accurate nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An alternate approach of deriving reduced order models in direct second order form is also presented. Here the system is converted into an equivalent second order nonlinear system with time invariant linear system matrices and periodically modulated nonlinearities via the L–F and other canonical transformations. Then a master-slave separation of degrees of freedom is used and a nonlinear relation between the slave coordinates and the master coordinates is constructed. This method yields the same ‘reducibility conditions’ obtained by invariant manifold approach in state space. Some examples are given to show potential applications to real problems using above mentioned methodologies. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘true combination resonances’ are discussed. A generalization of these ideas to periodic-quasiperiodic systems is included and demonstrated by means of an example.     

扁球面网壳的混沌运动研究

在圆形三向网架非线性动力学基本方程的基础上,用拟壳法给出了圆底扁球面三向网壳的非线性动力学基本方程．在固定边界条件下,引入了异于等厚度壳的无量纲量,对基本方程和边界条件进行无量纲化,通过Galerkin作用得到了一个含二次、三次的非线性动力学方程．为求Melnikov函数,对一类非线性动力系统的自由振动方程进行了求解,得到了此类问题的准确解．在无激励情况下,讨论了稳定性问题．在外激励情况下,通过求Melnikov函数,给出了可能发生混沌运动的条件．通过数字仿真绘出了平面相图,证实了混沌运动的存在．     

Chaotic vibrations of spherical and conical axially symmetric shells

Summary Chaotic vibrations of deterministic, geometrically nonlinear, elastic, spherical and conical axially summetric shells, subject to sign-changing transversal load using the variational principle, are analysed. The paper is motivated by an observation that variational equations of the hybrid type are suitableto solve many dynamical problems of the shells theory. It is assumed that the shell material is isotropic, and the Hook's principle holds. Intertial forces in directions tangent to mean shell surface and rotation inertia of a normal shell cross section are neglected. A transition form PDEs to ODEs (the Cauchy problem) is realized through the Ritz procedure. Next, the Cauchy problem is solved using the fourth-order Runge-Kutta method. Qualitative and quantitative analysis is carried out in the frame of both nonlinear dynamics and quantitative theory of differential equations. New scenarios from harmonic to chaotic dynamics are detected. Various vibration forms development versus control parameters (rise of arc; amplitude and frequency of the exciting force and number of vibrational modes accounted) are illustrated and discussed.