Global analysis of the phase portrait for the Kuramoto-Sivashinsky equation

The global behavior of the Kuramoto-Sivashinsky equation is studied. The existence of an absorbing ball in every Sobolev norm is proved. The transition of energy from low modes to high ones is observed. An upper estimate for the Hausdorff dimension of the attractor is given. The main tool is to use the methods of the theory of ordinary differential equations in the investigation of partial differential equations.     

NONLINEAR RESPONSES OF A FLUID-CONVEYING PIPE EMBEDDED IN NONLINEAR ELASTIC FOUNDATIONS

The nonlinear responses of planar motions of a fluid-conveying pipe embedded in nonlinear elastic foundations are investigated via the differential quadrature method discretization （DQMD） of the governing partial differential equation. For the analytical model, the effect of the nonlinear elastic foundation is modeled by a nonlinear restraining force. By using an iterative algorithm, a set of ordinary differential dynamical equations derived from the equation of motion of the system are solved numerically and then the bifurcations are analyzed. The numerical results, in which the existence of chaos is demonstrated, are presented in the form of phase portraits of the oscillations. The intermittency transition to chaos has been found to arise.     

Periodic solutions of linear degenerate systems of ordinary differential equations of the second order

We propose sufficient conditions for the existence of a periodic solution of a system of linear ordinary differential equations of the second order with a degenerate symmetric matrix in the coefficient of the second-order derivative in the case of an arbitrary periodic inhomogeneity.     

On the mechanical modeling of anisotropic biological soft tissue and iterative parallel solution strategies

Biological soft tissues appearing in arterial walls are characterized by a nearly incompressible, anisotropic, hyperelastic material behavior in the physiological range of deformations. For the representation of such materials we apply a polyconvex strain energy function in order to ensure the existence of minimizers and in order to satisfy the Legendre–Hadamard condition automatically. The 3D discretization results in a large system of equations; therefore, a parallel algorithm is applied to solve the equilibrium problem. Domain decomposition methods like the Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) method are designed to solve large linear systems of equations, that arise from the discretization of partial differential equations, on parallel computers. Their numerical and parallel scalability, as well as their robustness, also in the incompressible limit, has been shown theoretically and in numerical simulations. We are using a dual-primal FETI method to solve nonlinear, anisotropic elasticity problems for 3D models of arterial walls and present some preliminary numerical results.     

Existence and Homogenisation of Travelling Waves Bifurcating from Resonances of Reaction–Diffusion Equations in Periodic Media

The existence of travelling wave type solutions is studied for a scalar reaction diffusion equation in \(\mathbb {R}^2\) with a nonlinearity which depends periodically on the spatial variable. We treat the coefficient of the linear term as a parameter and we formulate the problem as an infinite spatial dynamical system. Using a centre manifold reduction we obtain a finite dimensional dynamical system on the centre manifold with fully degenerate linear part. By phase space analysis and Conley index methods we find conditions on the parameter and nonlinearity for the existence of travelling wave type solutions with particular wave speeds. The analysis provides an approach to the homogenisation problem as the period of the periodic dependence in the nonlinearity tends to zero.     

Degenerate Fredholm boundary-value problems

We obtain a criterion for the existence of solutions of degenerate inhomogeneous Fredholm boundary-value problems for a system of ordinary differential equations under the assumption that the degenerate system of differential equations can be reduced to the central canonical form. The results are illustrated by examples. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 3, pp. 303–312, July–September, 2007.     

Preserving dissipation in approximate inertial forms for the Kuramoto-Sivashinsky equation

It has been observed, in earlier computations of bifurcation diagrams for dissipative partial differential equations, that the use of certain explicit approximate inertial forms can give rise to numerical artifacts such as spurious turning points and inaccurate solution branches. These shortcomings were attributed to a lack of dissipation in the forms used. We show analytically and verify numerically that with an appropriate adjustment we can eliminate these numerical artifacts. The motivation for this adjustment is to enforce dissipation, while maintaining the same order of approximation. We demonstrate with computations that the most natural remedy, namely, preparation of the equation, can be highly sensitive to assumptions on the size of the absorbing ball. In addition, we show that certain implicit forms are dissipative without any adjustment. As an illustrative example we use here the Kuramoto-Sivashinsky equation.     

Nonlinear flexural waves and chaos behavior in finite-deflection Timoshenko beam

Based on the Timoshenko beam theory, the finite-deflection and the axial inertia are taken into account, and the nonlinear partial differential equations for flexural waves in a beam are derived. Using the traveling wave method and integration skills, the nonlinear partial differential equations can be converted into an ordinary differential equation. The qualitative analysis indicates that the corresponding dynamic system has a heteroclinic orbit under a certain condition. An exact periodic solution of the nonlinear wave equation is obtained using the Jacobi elliptic function expansion. When the modulus of the Jacobi elliptic function tends to one in the degenerate case, a shock wave solution is given. The small perturbations are further introduced, arising from the damping and the external load to an original Hamilton system, and the threshold condition of the existence of the transverse heteroclinic point is obtained using Melnikov＇s method. It is shown that the perturbed system has a chaotic property under the Smale horseshoe transform.     

Analysis of Hopf and Takens–Bogdanov Bifurcations in a Modified van der Pol–Duffing Oscillator

We analyze a modified van der Pol–Duffing electronic circuit, modeled by a tridimensional autonomous system of differential equations with Z₂-symmetry. Linear codimension-one and two bifurcations of equilibria give rise to several dynamical behaviours, including periodic, homoclinic and heteroclinic orbits. The local analysis provides, in first approximation, the different bifurcation sets. These local results are used as a guide to apply the adequate numerical methods to obtain a global understanding of the bifurcation sets. The study of the normal form of the Hopf bifurcation shows the presence of cusps of saddle-node bifurcations of periodic orbits. The existence of a codimension-four Hopf bifurcation is also pointed out. In the case of the Takens–Bogdanov bifurcation, several degenerate situations of codimension-three are analyzed in both homoclinic and heteroclinic cases. The existence of a Hopf–Shil'nikov singularity is also shown.     

非线性动力系统的规范形和余维3退化分叉

本文里我们利用矩阵表示法计算了具有Z_(2~-)对称性时非线性动力系统的高阶规范形,求出了余维2退化和余维3退化情况下相应规范形的普适开折。最后利用所得到的规范形和普适开折讨论了非线性动力系统的余维3退化分叉。     

Random Attractors for Degenerate Stochastic Partial Differential Equations

We prove the existence of random attractors for a large class of degenerate stochastic partial differential equations (SPDE) perturbed by joint additive Wiener noise and real, linear multiplicative Brownian noise, assuming only the standard assumptions of the variational approach to SPDE with compact embeddings in the associated Gelfand triple. This allows spatially much rougher noise than in known results. The approach is based on a construction of strictly stationary solutions to related strongly monotone SPDE. Applications include stochastic generalized porous media equations, stochastic generalized degenerate $p$ -Laplace equations and stochastic reaction diffusion equations. For perturbed, degenerate $p$ -Laplace equations we prove that the deterministic, $\infty $ -dimensional attractor collapses to a single random point if enough noise is added.     

Pullback Exponential Attractors for Non-autonomous Lattice Systems

We first present some sufficient conditions for the existence and the construction of a pullback exponential attractor for the continuous process (non-autonomous dynamical system) on Banach spaces and weighted spaces of infinite sequences. Then we apply our results to study the existence of pullback exponential attractors for first order non-autonomous differential equations and partly dissipative differential equations on infinite lattices with time-dependent coupled coefficients and time-dependent external terms in weighted spaces.     

Pullback Attractors in Dissipative Nonautonomous Differential Equations Under Discretization

The effects of discretization on the nonautonomous pullback attractors of skew-product flows generated by a class of dissipative differential equations, are investigated, It is assumed that the vector, field of the differential equations varies in time due to the input of an autonomous dynamical system acting on a compact metric space. In particular, it is shown that the corresponding discrete time skew-product system generated by a one-step numerical scheme with variable timesteps also has a pullback attractor, the component subsets of which converge upper semicontinuously to their counterparts of the pullback attractor of the original continuous time system.     

Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations

Continuum mechanics with dislocations, with the Cattaneo-type heat conduction, with mass transfer, and with electromagnetic fields is put into the Hamiltonian form and into the form of the Godunov-type system of the first-order, symmetric hyperbolic partial differential equations (SHTC equations). The compatibility with thermodynamics of the time reversible part of the governing equations is mathematically expressed in the former formulation as degeneracy of the Hamiltonian structure and in the latter formulation as the existence of a companion conservation law. In both formulations the time irreversible part represents gradient dynamics. The Godunov-type formulation brings the mathematical rigor (the local well posedness of the Cauchy initial value problem) and the possibility to discretize while keeping the physical content of the governing equations (the Godunov finite volume discretization).     

Well-Posedness in Smooth Function Spaces for the Moving-Boundary Three-Dimensional Compressible Euler Equations in Physical Vacuum

We prove well-posedness for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) =  C _γ ρ ^γ for γ > 1. The physical vacuum singularity requires the sound speed c to go to zero as the square-root of the distance to the moving boundary, and thus creates a degenerate and characteristic hyperbolic free-boundary system wherein the density vanishes on the free-boundary, the uniform Kreiss–Lopatinskii condition is violated, and manifest derivative loss ensues. Nevertheless, we are able to establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary, and our estimates have no derivative loss with respect to initial data. Our proof is founded on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term, chosen to preserve as much of the geometric structure of the Euler equations as possible. We first construct solutions to this degenerate parabolic regularization using a higher-order version of Hardy’s inequality; we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter. Solutions to the compressible Euler equations are found in the limit as the artificial viscosity tends to zero. Our regular solutions can be viewed as degenerate viscosity solutions. Our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws.     

Numerical study of the strong evaporation of a binary mixture

The problem of the one-dimensional evaporation of a binary mixture is investigated by numerically solving a system of two coupled Boltzmann equations. The numerical method is based on the direct discretization of the Boltzmann equation and the Monte Carlo evaluation of the collision integrals. It is assumed that the fluid flows between an evaporating plate and a totally absorbing plate. The spatial profiles of macroscopic quantities as well as evaporation rates have been calculated for values of the Knudsen number between 1/2 and 1/20.     

A Smooth Center Manifold Theorem which Applies to Some Ill-Posed Partial Differential Equations with Unbounded Nonlinearities

We prove the existence of a smooth center manifold for several partial differential equations, including ill posed equations with unbounded nonlinearities. We also prove smooth dependence on parameters with respect to some perturbations, including unbounded ones. More concretely, we prove an abstract theorem and present applications to several concrete equations: ill posed Boussinesq, equation and system and nonlinear Laplace equations in cylindrical domains. We also consider the effect of some geometric structures.     

Transonic Shocks in Multidimensional Divergent Nozzles

We establish existence, uniqueness and stability of transonic shocks for a steady compressible non-isentropic potential flow system in a multidimensional divergent nozzle with an arbitrary smooth cross-section, for a prescribed exit pressure. The proof is based on solving a free boundary problem for a system of partial differential equations consisting of an elliptic equation and a transport equation. In the process, we obtain unique solvability for a class of transport equations with velocity fields of weak regularity (non-Lipschitz), an infinite dimensional weak implicit mapping theorem which does not require continuous Fréchet differentiability, and regularity theory for a class of elliptic partial differential equations with discontinuous oblique boundary conditions.     

A structured population model and a related functional differential equation: Global attractors and uniform persistence

A structured population model of a single population having two distinct life stages is considered. The model equations, consisting of a hyperbolic partial differential equation coupled to an ordinary differential equation, can be reduced to a single, scalar functional differential equation. This allows us to use the well-developed dynamical systems theory for functional differential equations in order to study the dynamical system generated by the more complicated coupled system. A precise relation is established between the dynamical systems generated by each system of equations and a correspondence between their respective global attractors is made. The two systems are topologically equivalent on their respective attractors. These relationships are used to determine sharp sufficient conditions for the uniform persistence of the population.     

On the construction of asymptotic solutions of two-point boundary-value problems for degenerate singularly perturbed systems of differential equations

We consider the problem of the existence of a solution of a two-point boundary-value problem for degenerate singularly perturbed linear systems of differential equations. We obtain asymptotic formulas for this solution.