On null controllability of linear systems with recurrent coefficients and constrained controls

Given a family of time-dependent linear control processes, we study conditions under which local null controllability implies global null controllability. This is done by employing methods of dynamical systems and the Sacker-Sell spectral theory. We show that the above implication holds almost surely for recurrent families provided the spectrum of the associated linear system is contained in (–,0].     

A constant-contour-length reptation model without independent alignment or consistent averaging approximations for chain retraction

A reptation model for the primitive chain that does not assume independent alignment or consistent-averaging for the retraction process, or equilibrium relaxation for the reptation process is proposed and compared to the analytical expressions of Doi and Edwards in single-step, double-step strains and steady-state shear flow. The Doi and Edwards model with independent alignment approximation underpredicts the magnitude of the relaxation modulus by 25%, and consistently overpredicts the magnitude of the damping function; for steady shear flow, it predicts the correct shape for the steady-state viscosity and the first normal stress difference coefficient, although the magnitude is incorrect. The analytical expressions of Doi and Edwards without independent alignment approximation are excellent approximations to the damping function. In double-step strains, the expressions of Doi assuming consistent averaging, but no independent alignment, predict well the stress decay following the second strain. Linear response theory is found to be invalid for describing the stress relaxation following single-step strain for the models considered. Similar to the Doi and Edwards model, no overshoot for the first normal stress difference is observed for the simulation model. Unlike the Doi equation derived without the independent alignment approximation but restricted to double-step strains, the simulation model proposed here can be easily generalized to complex flow fields. No contour length fluctuation or constraint release is considered in this model, and chain retraction is assumed to be instantaneous.     

Spectral Properties of Dynamical Systems, Model Reduction and Decompositions

In this paper we discuss two issues related to model reduction of deterministic or stochastic processes. The first is the relationship of the spectral properties of the dynamics on the attractor of the original, high-dimensional dynamical system with the properties and possibilities for model reduction. We review some elements of the spectral theory of dynamical systems. We apply this theory to obtain a decomposition of the process that utilizes spectral properties of the linear Koopman operator associated with the asymptotic dynamics on the attractor. This allows us to extract the almost periodic part of the evolving process. The remainder of the process has continuous spectrum. The second topic we discuss is that of model validation, where the original, possibly high-dimensional dynamics and the dynamics of the reduced model – that can be deterministic or stochastic – are compared in some norm. Using the “statistical Takens theorem” proven in (Mezić, I. and Banaszuk, A. Physica D, 2004) we argue that comparison of average energy contained in the finite-dimensional projection is one in the hierarchy of functionals of the field that need to be checked in order to assess the accuracy of the projection.     

Generalizations of the Lotka-Volterra Population Ecology Model: Theory,Simulation, and Applications

Social scientists have attempted in vain to explain and predict the social phenomenon and particularly the behavior of the social system, with the unsatisfactory result that they were not so successful in terms of the accuracy of the prediction that they started to look into chaos theory. Several authors presented the ability of even the most simple predator-prey models to yield damped and explosive oscillations as well as stable limit cycles. Lotka and Volterra suggested models of population dynamics incorporating interpopulation competition. In this paper, biological population ecology model, especially Lotka-Volterra model is applied to organizations and social systems at large. This paper demonstrates the power of merging system dynamics with population ecology models to assess the sensitivity to initial conditions. The dynamical properties of the generalized Lotka-Volterra model were made by simulations using the Ithink software. The implications of using simulation in the analysis of chaotic behavior are presented.     

Synchronization of N different coupled chaotic systems with ring and chain connections

Synchronization of N different coupled chaotic systems with ring and chain connections is investigated. The New system, the Chen system, the Lii system, the Lorenz system, and the Rossler system are used as examples in verifying effectiveness of the method. Based on the Lyapunov stability theory, the form of the controller is designed and the area of the coupling coefficients is determined. Simulations indicate that global synchronization of the N different chaotic systems can be realized by choosing appropriate coupling coefficients by using the controller.     

The adjoining cell mapping and its recursive unraveling,part I: Description of adaptive and recursive algorithms

A new type of cell mapping, referred to as an adjoining cell mapping, is developed in this paper for autonomous dynamical systems employing the cellular state space. It is based on an adaptive time integration employed to compute an associated cell mapping for the system. This technique overcomes the problem of determining an appropriate duration of integration time for the simple cell mapping method. Employing the adjoining mapping principle, the first type of algorithm developed here is an adaptive mapping unraveling algorithm to determine equilibria and limit cycles of the dynamical system in a way similar to that of the simple cell mapping. In addition, it is capable of providing useful information regarding the behavior of dynamical systems possessing pathological dynamics and of systems with rapidly changing vector field. The adjoining property inherent in the adjoining cell mapping method, in general, permits development of new recursive algorithms for unraveling dynamics. The required computer memory for a practical implementation of such algorithms is considerably less than that required by the simple cell mapping algorithm since they allow for a recursive partitioning of state space for trajectory analysis. The second type of algorithm developed in this paper is a recursive unraveling algorithm based on adaptive integration and recursive partitioning of state space into blocks of cells with a view toward its practical implementation. It can find equilibria of the system in the same manner as the simple cell mapping method but is more efficient in locating periodic solutions.     

The adjoining cell mapping and its recursive unraveling,part II: Application to selected problems

Several applications of the adjoining cell mapping technique are provided here by employing the adaptive mapping unraveling algorithm to analyze smooth and pathological autonomous dynamical systems. The performance of an implementation of recursive unraveling algorithm is also illustrated regarding its low memory requirements for computational purposes when compared with the simple cell mapping method. The applications considered here illustrate the effectiveness of the adjoining cell mapping technique in its ability to determine limit cycles and to unravel nonstandard dynamics. The advantages of this new technique of global analysis over the simple cell mapping method are discussed.     

Synchronization of N different coupled chaotic systems with ring and chain connections

Synchronization of N different coupled chaotic systems with ring and chain Lorenz system, and the R(o)ssler system are used as examples in verifying effectiveness of the method. Based on the Lyapunov stability theory, the form of the controller is designed and the area of the coupling coefficients is determined. Simulations indicate that global synchronization of the N different chaotic systems can be realized by choosing appropriate coupling coefficients by using the controller.     

Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations

A vector bundle morphism of a vector bundle with strongly ordered Banach spaces as fibers is studied. It is assumed that the fiber maps of this morphism are compact and strongly positive. The existence of two complementary, dimension-one and codimension-one, continuous subbundles invariant under the morphism is established. Each fiber of the first bundle is spanned by a positive vector (that is, a nonzero vector lying in the order cone), while the fibers of the other bundle do not contain a positive vector. Moreover, the ratio between the norms of the components (given by the splitting of the bundle) of iterated images of any vector in the bundle approaches zero exponentially (if the positive component is in the denominator). This is an extension of the Krein-Rutman theorem which deals with one compact strongly positive map only. The existence of invariant bundles with the above properties appears to be very useful in the investigation of asymptotic behavior of trajectories of strongly monotone discrete-time dynamical systems, as demonstrated by Poláik and Tereák (Arch. Ration. Math. Anal. 116, 339–360, 1991) and Hess and Poláik (preprint). The present paper also contains some new results on typical asymptotic behavior in scalar periodic parabolic equations.     

Simulation of acoustic fields in fluid-, solid- and porous layers by the combined transfer matrix/angular spectrum approach with applications in bioacoustics

A highly accurate semi-analytical method was developed to predict the acoustic field generated by a real transducer in an axisymmetric sonobioreactor consisting of multiple fluid-, linear elastic solid-, and/or poroelastic-layers. The accuracy of the method is independent of the spacing of the grid-points and computational costs are not proportional to the ratio of the system’s characteristic dimensions to the acoustic wavelength, both improvements over the use of full numerical methods. Contrary to similar semi-analytical approaches, the method is not limited to the prediction of freely propagating waves. Acoustic reflection and perfect absorption are readily implemented. The method was numerically validated and matched the analytical function describing the pressure amplitude along the axis of a cylindrical transducer with a root-mean-square error of less than 2%. The method was also experimentally validated, but it was shown that the method is not applicable when certain components of the system have a diameter smaller than that of the acoustic beam. The method was used to model an ultrasonic bioreactor as an example problem, where its accuracy and computational efficiency were shown to be instrumental in bioreactor design.