Homoclinic Saddle-Node Bifurcations in Singularly Perturbed Systems

In this paper we study the creation of homoclinic orbits by saddle-node bifurcations. Inspired on similar phenomena appearing in the analysis of so-called localized structures in modulation or amplitude equations, we consider a family of nearly integrable, singularly perturbed three dimensional vector fields with two bifurcation parameters a and b. The O() perturbation destroys a manifold consisting of a family of integrable homoclinic orbits: it breaks open into two manifolds, W ^s() and W ^u(), the stable and unstable manifolds of a slow manifold . Homoclinic orbits to correspond to intersections W ^s()W ^u(); W ^s()W ^u()= for a<a*, a pair of 1-pulse homoclinic orbits emerges as first intersection of W ^s() and W ^u() as a>a*. The bifurcation at a=a* is followed by a sequence of nearby, O( ²(log)²) close, homoclinic saddle-node bifurcations at which pairs of N-pulse homoclinic orbits are created (these orbits make N circuits through the fast field). The second parameter b distinguishes between two significantly different cases: in the cooperating (respectively counteracting) case the averaged effect of the fast field is in the same (respectively opposite) direction as the slow flow on . The structure of W ^s()W ^u() becomes highly complicated in the counteracting case: we show the existence of many new types of sometimes exponentially close homoclinic saddle-node bifurcations. The analysis in this paper is mainly of a geometrical nature.     

On the Start-Up Behavior and Steady-State Oscillation of Singularly Perturbed Harmonic Oscillators

In this paper we present an analytic methodology for the analysis of a class of electrical harmonic oscillators. We combine geometric methods with the theory of singularly perturbed systems, which we use as tool for reduced order modeling. So we are able to define an easy to use formula in order to reduce an oscillatory system to center manifold. Thus we get a model for the start-up behavior as well as for the steady-state oscillation of sinusoidal oscillators. Furthermore, we demonstrate our technique by means of the Clapp oscillator which is an important member of the Colpitts, Clapp and Pierce oscillator family.     

Stability of Singularly Perturbed Systems with Structural Perturbations. Some Applications

Singularly perturbed systems with structural perturbations are analyzed for stability on the basis of matrix-valued Lyapunov functions. Sufficient conditions of stability and uniform asymptotic stability for automatic control and stabilization systems of an orbital observatory are established     

Numerical Continuation of Homoclinic Orbits to Non-Hyperbolic Equilibria in Planar Systems

In this paper we develop numerical algorithms for thecontinuation of degenerate homoclinic orbits to non-hyperbolicequilibria in planar systems. The first situation corresponds to asaddle-node equilibrium (a zero eigenvalue) and the second one is theso-called cuspidal loop (double-zero eigenvalue). The methods proposedmay deal with codimension-two and -three homoclinic connections.Application of the algorithms to several examples supports its validityand demonstrates its usefulness.     

On the Homoclinic Orbits in a Class of Two-Degree-of-Freedom Systems under the Resonance Conditions

ＩｎｔｒｏｄｕｃｔｉｏｎＴｗｏ＿ｄｅｇｒｅｅ＿ｏｆ＿ｆｒｅｅｄｏｍｓｙｓｔｅｍｓｈａｖｉｎｇｃｕｂｉｃｎｏｎｌｉｎｅａｒｉｔｉｅｓａｒｅｅｘｔｅｎｓｉｖｅｌｙｕｓｅｄｉｎｐｈｙｓｉｃｓ,ｍｅｃｈａｎｉｃｓ.Ｆｏｒｅｘａｍｐｌｅ :ｔｈｅｌａｒｇｅ＿ａｍｐｌｉｔｕｄｅｖｉｂｒａｔｉｏｎｓｏｆｓｔｒｉｎｇｓ,ｂｅａｍｓ,ｍｅｍｂｒａｎｅｓａｎｄｐｌａｔｅｓ ,ｄｙｎａｍｉｃｖｉｂｒａｔｉｏｎ＿ｉｓｏｌａｔｉｏｎｓｙｓｔｅｍｓ ,ｄｙｎａｍｉｃｖｉｂｒａｔｉｏｎａｂｓｏｒｂｅｒｓ,ｔｈｅｍｏｔｉｏｎｏｆｓｐｈｅ…     

Determination of the Chaos Onset in Mechanical Systems with Several Equilibrium Positions

Determination of the chaos onset in some mechanical systems with several equilibrium positions are analyzed. Namely, the snap-through truss and the oscillator with a nonlinear dissipation force, under the external periodical excitation, are considered. Two approaches are used for the chaos onset determination. First, Padé and quasi-Padé approximants are used to construct closed homoclinic trajectories for a case of small dissipation. Convergence condition used earlier in the theory of nonlinear normal vibration modes as well conditions at infinity make possible to evaluate initial amplitude values for the trajectories with admissible precision. Mutual instability of phase trajectories is used as criterion of chaotic behavior in nonlinear systems for a case of not small dissipation. The numerical realization of the Lyapunov stability definition gives us a possibility to observe a process of appearance and fast enlargement of the chaotic behavior regions if some selected parameters of the dynamical systems under consideration are changing.     

A New Model for the Dynamics of Dispersions in a Batch Reactor: Numerical Approach

In this paper, we develop some considerations concerning the structure of coalescence, breakage and volume scattering kernels appearing in the evolution equation related to a new model for the dynamics of liquid–liquid dispersions and show some numerical simulations. The mathematical model has been presented in [3, 4], where a proof of the existence and uniqueness for a classical solution to the integro–differential equation describing the physical phenomenon is provided as well as a complete analysis of the general characteristics of the integral kernels. Numerical simulations agree with experimental data and with the expected asymptotical behavior of the solution.     

The Asymptotic Behavior of Solution for the Singularly Perturbed Initial Boundary Value Problems of the Reaction Diffusion Equations in a Part of Domain

Ｔｈｅａｕｔｈｏｒｓｓｔｕｄｉｅｄａｃｌａｓｓｏｆｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓｉｎ [1 ] -[7] .Ｎｏｗｗｅｒａｉｓｅａｃｌａｓｓｏｆｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓｏｎａｐａｒｔｏｆｄｏｍａｉｎ .Ｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｉｎｉｔｉａｌｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｆｏｒｔｈｅｒｅａｃｔｉｏｎｄｉｆｆｕｓｉｏｎｅｑｕａｔｉｏｎｓ　　ｕｔ-λε(ｘ) ( μ(ｕ)ｕｘ) ｘ Ｋｘ(ｕ) ｆ(ｘ ,ｔ,ｕ) =0 ,(ｔ,ｘ) ∈ ( 0 ,Ｔ)× ( ( 0 ,α) ∪ (…     

Advanced Proper Orthogonal Decomposition Tools: Using Reduced Order Models to Identify Normal Modes of Vibration and Slow Invariant Manifolds in the Dynamics of Planar Nonlinear Rods

Reduced order models for the dynamics of geometrically exact planar rods are derived by projecting the nonlinear equations of motion onto a subspace spanned by a set of proper orthogonal modes. These optimal modes are identified by a proper orthogonal decomposition processing of high-resolution finite element dynamics. A three-degree-of-freedom reduced system is derived to study distinct categories of motions dominated by a single POD mode. The modal analysis of the reduced system characterizes in a unique fashion for these motions, since its linear natural frequencies are near to the natural frequencies of the full-order system. For free motions characterized by a single POD mode, the eigen-vector matrix of the derived reduced system coincides with the principal POD-directions. This property reflects the existence of a normal mode of vibration, which appears to be close to a slow invariant manifold. Its shape is captured by that of the dominant POD mode. The modal analysis of the POD-based reduced order system provides a potentially valuable tool to characterize the spatio-temporal complexity of the dynamics in order to elucidate connections between proper orthogonal modes and nonlinear normal modes of vibration.     

一类慢变参数振子系统的同宿分叉及其安全盆侵蚀

分析一个具有慢变参数的非线性系统,利用Ｍｅｌｎｋｏｖ方法,分析了系统在参数发生变化时的同宿分叉,同时利用分叉结果,数值讨论了当系统参数发生变化时安全盆的侵蚀及分叉,混沌的联系。     

A Kinetic Approach to the Plane Poiseuille Flow Over a Porous Matrix

The Poiseuille–Couette gas flow in a channel and the gas flow through an adjacent porous medium are considered when the governing equations are obtained via a molecular kinetic approach based on the Boltzmann equation. The mass continuity, momentum balance and energy conservation are written for the gas in the contiguous regions, whereas the behavior of the solid matrix obeys to the heat diffusion equation. Two different space scalings lead to different forms of the equations for the steady flow through the fully saturated matrix. The boundary conditions at the interface between the two domains are investigated via a matching procedure.     

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.     

Application of the Discrete Fourier Transform to Study the Dynamics of Wave Packets

A method for solving equations that describe the dynamics of wave packets of the Tollmien–Schlichting waves in the boundary layer is proposed. The method of splitting the initial problem into the linear and nonlinear parts at each time step is used. The linear part is resolved by using an equation for spectral components of the wave packet with a subsequent Fourier transform from the space of wavenumbers to the physical space. A system of ordinary differential equations is solved in the physical space. The Fourier transform is performed by means of the library procedure of the fast Fourier transform. As examples, the problems solved were the linear dynamics of the wave packet concentrated in the vicinity of the instability region (i.e., a set of wave vectors in the space of wavenumbers for which the imaginary part of the eigenfrequency of the Tollmien–Schlichting waves is positive) and the nonlinear dynamics of the wave packet overlapping the instability region.     

A Simple Approach to the 1:1 Resonance Bifurcation in Follower-Load Problems

We consider the problem of 1:1 resonance in autonomous, timereversible systems. We first present an abstract treatment for n-dimensionalsecond-order systems, and then apply our method to two simplemechanical examples involving follower loads. As the magnitude of the follower load is increased past a criticalvalue, the trivial solution loses stability as the real-valuedfrequencies of the linearized system first coalesce and then splitapart with complex-conjugate values. In Hamiltonian systems this isusually referred to as the Hamiltonian–Hopf bifurcation. Some novelfeatures of our analysis are the direct exploitation of reversibilityand a Liapunov–Schmidt reduction of the second-order (Newtonian)equations of motion, the latter of which requires no complexification.The analysis of the resulting two-parameter, one-dimensionalbifurcation equation yields the possibility that families ofnontrivial periodic solutions may exist for load values in excess of the critical value.     

Existence of Standing Pulse Solutions to an Inhomogeneous Reaction–Diffusion System

We prove the existence of locally unique, symmetric standing pulse solutions to homogeneous and inhomogeneous versions of a certain reaction–diffusion system. This system models the evolution of photoexcited carrier density and temperature inside the cavity of a semiconductor Fabry–Pérot interferometer. Such pulses represent the fundamental nontrivial mode of pattern formation in this device. Our results follow from a geometric singular perturbation approach, based largely on Fenichel's theorems and the Exchange Lemma.     

多体系统动力学逆问题的一种处理方法

首先用控制力法建立多体系统动力学逆问题的运动微分方程，然后给出奇异值分解在求系统动力学响应中的应用及作用在系统中的控制力的解法，最后举了一个算例。     

A Fractal Approach to Model Soil Structure and to Calculate Thermal Conductivity of Soils

Heat transport in soils depends on the spatial arrangement of solids, ice, air and water. In this study, we present a modified fractal approach to model the pore structure of soils and to describe its influence on the thermal conductivity. Three different fractal generators were sequentially applied to characterize a wide range of particle- and pore-size distributions. The given porosity and particle-size distribution of a clay, clay loam, silt loam and loamy sand were successfully modeled. The thermal conductivity of the fractal soil model was calculated using a network of resistors. We applied a renormalization approach to include the effects of smaller scale structures. The predictions were compared with the empirical Johansen' model (Johansen, 1975), that postulates a simple linear relationship between ice content and thermal conductivity. For high ice-saturated conditions, the calculated thermal conductivity agrees well with the empirical model. To describe partial ice saturation, we assumed that some pores were coated by ice films enclosing the air-filled center. In addition, we introduced a reduced heat exchange coefficient of the particles for unsaturated conditions. The ice-saturated and -unsaturated thermal conductivity calculated with this approach was very similar to that estimated by the empirical model. The variation of the thermal conductivities for different spatial arrangements of pores and particles in the prefractals were determined. Extreme values deviate more than 50% from the mean values.     

A non‐linear dynamical system approach to finite amplitude Taylor‐Vortex flow of shear‐thinning fluids

The effect of shear thinning on the stability of the Taylor–Couette flow is explored for a Carreau–Bird fluid in the narrow‐gap limit. The Galerkin projection method is used to derive a low‐order dynamical system from the conservation of mass and momentum equations. In comparison with the Newtonian system, the present equations include additional non‐linear coupling in the velocity components through the viscosity. It is found that the critical Taylor number, corresponding to the loss of stability of the circular Couette flow, becomes lower as the shear‐thinning effect increases. That is, shear thinning tends to precipitate the onset of Taylor vortex flow, which coincides with the onset of a supercritical bifurcation. Comparison with existing measurements of the effect of shear thinning on the critical Taylor and wave numbers show good agreement. The Taylor vortex cellular structure loses its stability in turn, as the Taylor number reaches a critical value. At this point, an inverse Hopf bifurcation emerges. In contrast to Newtonian flow, the bifurcation diagrams exhibit a turning point that sharpens with shear‐thinning effect. Copyright © 2004 John Wiley & Sons, Ltd.     

A Volterra Series Approach to the Approximation of Stochastic Nonlinear Dynamics

A response approximation method for stochastically excited, nonlinear, dynamic systems is presented. Herein, the output of the nonlinear system isapproximated by a finite-order Volterra series. The original nonlinear system is replaced by a bilinear system in order to determine the kernels of this series. The parameters of the bilinear system are determined by minimizing, in a statistical sense,the difference between the original system and the bilinear system. Application to a piecewise linear modelof a beam with a nonlinear one-sided supportillustrates the effectiveness of this approach in approximatingtruly nonlinear, stochastic response phenomena in both the statistical momentsand the power spectral density of the response of this system in case ofa white noise excitation.