Dimension Reduction of Dynamical Systems: Methods, Models, Applications

After presenting some basic introductory ideas concerning dimension reduction and reduced order modelling, an overview of the contents of the papers collected in this Special Issue of Nonlinear Dynamics is given.     

Planktonic Communities and Chaotic Advection in Dynamical Models of Langmuir Circulation

A deterministic mechanism for the production of plankton patches within a typical medium scale oceanic structure is proposed and investigated. By direct numerical simulation of a simple model of Langmuir circulation we quantify the effects of unsteady flows on planktonic communities and demonstrate their importance. Two qualitatively different zones within the flow are identified: chaotic regions that help to spread plankton and locally coherent regions, that do not mix with the chaotic regions and which persist for long periods of time. The relative importance of these regions to both phytoplankton and zooplankton is investigated, taking into account variations in plankton buoyancy. In particular, species-specific retention zone structure is discussed in relation to variations in environmental forcing.     

Sets of Dynamical Systems with Various Limit Shadowing Properties

We study the C ¹-interiors of sets of diffeomorphisms of a closed smooth manifold with various limit shadowing properties. It is shown that, for some natural analogs of the usual limit shadowing property, the corresponding C ¹-interiors coincide with the set of Ω-stable diffeomorphisms. The same problem is considered for two-sided analogs of the limit shadowing property. To Pavol Brunovsky on the occasion of his 70th birthday.     

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.     

Existence of a Singularly Degenerate Heteroclinic Cycle in the Lorenz System and Its Dynamical Consequences: Part I

We prove that the Lorenz system with appropriate choice of parameter values has a specific type of heteroclinic cycle, called a singularly degenerate heteroclinic cycle, that consists of a line of equilibria together with a heteroclinic orbit connecting two of the equilibria. By an arbitrarily small but carefully chosen perturbation to the Lorenz system, we also show that the geometric model of Lorenz attractors formulated by Guckenheimer will bifurcate from it, among other things. Although not proven, one may also expect various other types of chaotic dynamics such as Hénon-like chaotic attractors, Lorenz attractors with hooks which were recently studied by S. Luzzatto and M. Viana [22], and what were observed in the original Lorenz system with large r and small b in the Sparrows book [34]. Our analysis is all done within a family of three dimensional ODEs that contains, as its subfamilies, the Lorenz system, the Rösslers second system and the Shimizu–Morioka system, which are known to exhibit Lorenz-like chaotic dynamics.     

Acoustic Properties of Fontainebleau Samples by Lattice Models

Acoustic properties of dry or saturated porous media can be studied numerically by a combined use of lattice spring models and lattice Boltzmann models. The methodology is briefly described and it is systematically applied to a series of Fontainebleau sandstones whether they are dry or saturated with incompressible or compressible fluids. The velocities of compression and shear waves, as well as their attenuation, are determined. The simulation results are in good agreement compared with the existing methods and experimental data.

     

On Phase-Separation Models: Asymptotics and Qualitative Properties

In this paper we study bound state solutions of a class of two-component nonlinear elliptic systems with a large parameter tending to infinity. The large parameter giving strong intercomponent repulsion induces phase separation and forms segregated nodal domains divided by an interface. To obtain the profile of bound state solutions near the interface, we prove the uniform Lipschitz continuity of bound state solutions when the spatial dimension is N = 1. Furthermore, we show that the limiting nonlinear elliptic system that arises has unbounded solutions with symmetry and monotonicity. These unbounded solutions are useful for rigorously deriving the asymptotic expansion of the minimizing energy which is consistent with the hypothesis of Du and Zhang (Discontin Dynam Sys, 2012). When the spatial dimension is N = 2, we establish the De Giorgi type conjecture for the blow-up nonlinear elliptic system under suitable conditions at infinity on bound state solutions. These results naturally lead us to formulate De Giorgi type conjectures for these types of systems in higher dimensions.     

Dynamical equations for a liquid jet

Equations are described for the three-dimensional motion of a thin jet of viscous liquid. The jet is a liquid body whose transverse dimensions are small compared with the other characteristic dimensions of the problem. The aim in the present paper is to establish a closed system of asymptotic equations for the dynamics of such a thin jet. A more detailed derivation of quasi-one-dimensional asymptotic equations for the dynamics of thin liquid jets and an analysis on the basis of them of the curved decay shape of the jet in the linear and nonlinear stages is contained in [1, 2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 11–18, September–October, 1980.     

Models for the Counter-Gradient-Transport Phenomena

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水分及冻融循环对红砂岩物理力学特性的影响

将干燥红砂岩进行饱水及开放系统下的冻融循环实验,并对干燥、饱水及经历一定冻融次数后的试样进行单轴压缩实验,系统地研究了红砂岩的遇水软化性及冻融损伤劣化特性,分析了岩石的弹性模量、强度、应力-应变关系的变化规律。结果表明：在水分的作用下,红砂岩的弹性模量及强度显著降低,且强度受水分影响引起的损伤较弹性模量大;水分及软弱结构面的存在促进了岩石的冻融损伤劣化,并伴随着岩石延脆性性质的改变,最终在冻融循环的作用下完全崩解;在寒区岩体工程设计和施工过程中,必须考虑水分和温度对工程稳定性的影响,采取加强防排水及隔热保温技术等综合防治措施减少冻融灾害。     

On the Dynamical Rayleigh-Taylor Instability

The classical Rayleigh-Taylor instability is shown to be valid in a nonlinear dynamical sense for smooth density profiles. (Accepted September 23, 2002) Published online February 26, 2003 Communicated by Y. Brenier     

表面纳米化对316L不锈钢低周疲劳性能的影响

超声喷丸(USSP)处理工艺在316L不锈钢表面制备出了纳米表面晶层,对表面纳米化后和未表面纳米化的316L不锈钢试样进行对比拉拉低周疲劳试验,运用数理统计学的方法分析研究了表面纳米化处理对316L不锈钢的低周疲劳性能的影响,并就表面纳米化对疲劳性能的影响机理进行了初步分析探讨.研究结果表明,超声喷丸表面纳米化处理可以有效地提高316L不锈钢的低周疲劳寿命;超声喷丸处理在表面所形成的残余压应力、晶粒细化的纳米强化表层是疲劳寿命提高的主要原因.     

Contribution of Kinematical and Thermal Full-field Measurements for Mechanical Properties Identification: Application to High Cycle Fatigue of Steels

The purpose of this work is to extend the use of unconventional tests and full field measurements (kinematical and thermal) to the identification of the effect of a wide plastic pre-strain range on the high cycle fatigue properties of a dual-phase steel. An unconventional specimen is designed. The geometry of this specimen permits a constant gradient of pre-strain to be obtained after a monotonic tensile test. Then, a self-heating test under cyclic loading is carried out on the pre-strained specimen. During this cyclic test, the thermal field is measured using an infrared camera. Finally, a suitable numerical strategy is proposed to identify a given thermal source model taking into account the influence of plastic pre-strain. The results show that, with the unconventional test and the procedure developed in this work, the influence of a plastic pre-strain range on fatigue properties can be identified by using only one specimen when, for a classical fatigue campaign, a great number of specimens is required.     

Variational Principles for Entropies of Nonautonomous Dynamical Systems

In this paper, we first introduce the measure-theoretic entropy for arbitrary Borel probability measure in nonautonomous case. Then we show that there is certain variational relation for nonautonomous dynamical systems.     

Lyapunov and Dynamical Spectra for Banach State-spaces

We study the Lyapunov exponents and exponential splittings for continuous linear skew-product flows acting on infinite dimensional Banach state-spaces.     

Dynamical behaviors of the periodic parameter-switching system

In this paper, a periodic parameter-switching system about Lorenz oscillators is established. To investigate the bifurcation behavior of this system, Poincaré mapping of the whole system is defined by suitable local sections and local mappings. The location of the fixed point and the parameter values of local bifurcations are calculated by the shooting method and Runge–Kutta method. Then based on the Floquent theory, we conclude that the period-doubling and saddle-node bifurcations play an important role in the generation of various periodic solutions and chaos. Meanwhile, upon the analysis of the equilibrium points of the subsystems, we explore the mechanisms of different periodic switching oscillations.     

Characterization of Measures Satisfying the Pesin Entropy Formula for Random Dynamical Systems

In this paper we first give a formulation of SRB (Sinai–Ruelle–Bowen) property for invariant measures of stationary random dynamical systems and then prove that this property is sufficient and necessary for a formula of Pesin's type relating entropy and Lyapunov exponents of such dynamical systems. This result is a random version of the main result in Part I of Ledrappier and Young's celebrated paper [11].     

Hausdorff Dimension of Invariant Sets for Random Dynamical Systems

Suppose X() is a compact random set, invariant with respect to a continuously differentiable random dynamical system (RDS) on a separable Hilbert space. It is shown that the Hausdorff dimension dim _H (X()) is an invariant random variable, and it is bounded by d, provided the RDS contracts d-dimensional volumes exponentially fast. Both exponential decrease of d-volumes as well as the approximation of the RDS by its linearization are assumed to hold uniformly in . The results are applied to reaction diffusion equations with additive noise and to two-dimensional Navier–Stokes equations with bounded real noise.     

Dynamical models of happiness

A sequence of models for the time evolution of one's happiness in response to external events is described. These models with added nonlinearities can produce stable oscillations and chaos even without external events. Potential implications for psychotherapy and a personal approach to life are discussed.