Bifurcation and stability of stationary solutions of nonlocal scalar reaction-diffusion equations

The stability of stationary solutions of nonlocal reaction-diffusion equations on a bounded intervalJ of the real line with homogeneous Dirichlet boundary conditions is studied. It is shown that it is possible to have stable stationary solutions which change sign once onJ in the case of constant diffusion when the reaction term does not depend explicitly on the space variable. The problem of the possible types of stable solutions that may exist is considered. It is also shown that Matano's result on the lap-number is still true in the case of nonlocal problems.On leave from IST, Lisbon, Portugal.     

On the multiple solutions to a three-coefficient spectral model of wind-driving ocean current equation

In this paper, a research on the problem of multiple solutions of the three-coefficient low-spectrum model for the quasi-geostrophic ocean current equation with forcing and dissipation terms is carried out. The state of the ocean current under wind conditions such as those of typhoon is discussed carefully and the relations between the multiple solutions and the coefficientsR and ε are analyzed. It is seen that in an approximate triangular region with the Rossby-coefficientR less than 0.5, and the friction-coefficient ε less than 0.22, there exist three equilibrium solutions, among which two are stable and one is unstable. For the former, the coefficientA orB in the expansion is rather large, while for the latter.A orB is relatively small. They respectively imply how much the ocean energy is fed back from the wind stress and the solution with a largeA is much more stable than that with a largerB. Institute of Systems Science, Academia Sinica The project partly supported by the national project of 75-76-01-03 “Study on numerical prediction of the South China Sea current”.     

The index of lyapunov stable fixed points in two dimensions

In this paper, we prove that a stable isolated fixed point of an orientation preserving local homeomorphism onR ₂ has fixed point index 1. We also give a number of applications to differential equations. In particular, we deduce that a number of existence methods for producing periodic solutions of differential equations in the plane always produce unstable solutions.     

Delay Equations with Rapidly Oscillating Stable Periodic Solutions

We prove analytically that there exist delay equations admitting rapidly oscillating stable periodic solutions. Previous results were obtained with the aid of computers, only for particular feedback functions. Our proofs work for stiff equations with several classes of feedback functions. Moreover, we prove that for negative feedback there exists a class of feedback functions such that the larger the stiffness parameter is, the more stable rapidly oscillating periodic solutions there are. There are stable periodic solutions with arbitrarily many zeros per unit time interval if the stiffness parameter is chosen sufficiently large.     

Continuation of Approximate Transformation Groups via Multiple Time Scales Method

Recently, the theory of approximate symmetries was developedfor tackling differential equations with a small parameter. This theoryfurnishes us with a tool, e.g. for constructing approximate groupinvariant solutions. Usually, these solutions are determined by powerseries in the small parameter and hence they are well defined only in asmall region of independent variables. In this paper, we modify theapproximate symmetry analysis by combining it with the multiple timescales method. In this way, we can extend the domain of definition ofapproximate symmetries of differential equations with a small parameterand of their invariant solutions. The method is illustrated by the vander Pol equation. It is shown that, in this example, our approachprovides a group theoretical background of ad hoc methods widelyused in perturbation techniques.     

Traveling Wave Solutions for a Class of One-Dimensional Nonlinear Shallow Water Wave Models

In this paper we consider a class of one-dimensional nonlinear shallow water wave models that support weak solutions. We construct new traveling wave solutions for these models. Moreover, we show that these new traveling wave solutions are stable.     

Periodic Solutions of Second Order Differential Equations with Discontinuous Nonlinearities

We consider the periodic solutions of

Heteroclinic Connection of Periodic Solutions of Delay Differential Equations

For a certain class of delay equations with piecewise constant nonlinearities we prove the existence of a rapidly oscillating stable periodic solution and a rapidly oscillating unstable periodic solution. Introducing an appropriate Poincaré map, the dynamics of the system may essentially be reduced to a two dimensional map, the periodic solutions being represented by a stable and a hyperbolic fixed point. We show that the two dimensional map admits a one dimensional invariant manifold containing the two fixed points. It follows that the delay equations under consideration admit a one parameter family of rapidly oscillating heteroclinic solutions connecting the rapidly oscillating unstable periodic solution with the rapidly oscillating stable periodic solution.      

Analytic solution for flow of Sisko fluid through a porous medium

This paper presents the analytic solution for flow of a magnetohydrodynamic (MHD) Sisko fluid through a porous medium. The non-linear flow problem in a porous medium is formulated by introducing the modified Darcy’s law for Sisko fluid to discuss the flow in a porous medium. The analytic solutions are obtained using homotopy analysis method (HAM). The obtained analytic solutions are explicitly expressed by the recurrence relations and can give results for all the appropriate values of material parameters of the examined fluid. Moreover, the well-known solutions for a Newtonian fluid in non-porous and porous medium are the limiting cases of our solutions.     

Nontrivial Solutions for a Semilinear Dirichlet Problem with Nonlinearity Crossing Multiple Eigenvalues

In this paper we prove that a semilinear elliptic boundary value problem has at least three nontrivial solutions when the range of the derivative of the nonlinearity includes at least the first two eigenvalues of the Laplacian and all solutions are nondegenerate. A pair are of one sign (positive and negative, respectively). The one sign solutions are of Morse index less than or equal to 1 and the third solution has Morse index greater than or equal to 2. Extensive use is made of the mountain pass theorem, and Morse index arguments of the type Lazer–Solimini (see Lazer and Solimini, Nonlinear Anal. 12(8), 761–775, 1988). Our result extends and complements a theorem of Cossio and Veléz, Rev. Colombiana Mat. 37(1), 25–36, 2003.AMS Subject classification: 35J20; 35J25; 35J60.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.     

Analytical solutions for non-Newtonian fluid flows in pipe-like domains

This paper deals with some unsteady unidirectional transient flows of an Oldroyd-B fluid in unbounded domains which geometrically are axisymmetric pipe-like. An expansion theorem of Steklov is used to obtain exact solutions for flows satisfying no-slip boundary conditions. The well known solutions for a Navier-Stokes fluid, as well as those corresponding to a Maxwell fluid and a second grade one, appear as limiting cases of our solutions. The steady state solutions are also obtained for t→∞.     

Bifurcation analysis of non-linear oscillators interacting via soft impacts

In this paper we present a bifurcation analysis of two periodically forced Duffing oscillators coupled via soft impact. The controlling parameters are the distance between the oscillators and the difference in the phase of the harmonic excitation. In our previous paper http://arXiv:1602.04214 (P. Brzeski et al. Controlling multistability in coupled systems with soft impacts [11]) we show that in the multistable system we are able to change the number of stable attractors and reduce the number of co-existing solutions via transient impacts. Now we perform a detailed path-following analysis to show the sequence of bifurcations which cause the destabilization of solutions when we decrease the distance between the oscillating systems. Our analysis shows that all solutions lose stability via grazing-induced bifurcations (period doubling, fold or torus bifurcations). The obtained results provide a deeper understanding of the mechanism of reduction of the multistability and confirmed that by adjusting the coupling parameters we are able to control the system dynamics.     

Natural convection from a horizontal permeable surface in a porous medium - numerical and asymptotic solutions

The equations which govern the similarity solution for free convection boundary-layer flow above a permeable, horizontal surface in a fluid-saturated porous medium are considered. These are seen to depend on the dimensionless parameters and m measuring mass transport rate and the wall temperature variation, respectively. Numerical solutions are presented for a wide range of values of and m. Asymptotic solutions are obtained for ¦¦ large (for both fluid injection, > 0, and fluid withdrawal, < 0) and for m large. These are compared with the numerical solutions.     

Existence of the vortex solutions of some semilinear elliptic equations

The purpose is to extend the existence result of vortex solutions to semilinear elliptic equations for a large class of nonlinearities. M. I. Weinstein used variational techniques to show the existence of nodal solutions for the specific nonlinear term f(¦¦)=(1–¦¦²). An ordinary differential equation phase space setting is used to show the unique transverse intersection of unstable and stable manifolds which contain the solutions satisfying the necessary boundary conditions under certain assumptions on the nonlinearity.     

Nonlinear Stability for Multidimensional Fourth-Order Shock Fronts

We consider the question of stability for planar wave solutions that arise in multidimensional conservation laws with only fourth-order regularization. Such equations arise, for example, in the study of thin films, for which planar waves correspond to fluid coating a pre-wetted surface. An interesting feature of these equations is that both compressive, and undercompressive, planar waves arise as solutions (compressive or undercompressive with respect to asymptotic behavior relative to the un-regularized hyperbolic system), and numerical investigation by Bertozzi, Münch, and Shearer indicates that undercompressive waves can be nonlinearly stable. Proceeding with pointwise estimates on the Green's function for the linear fourth-order convection–regularization equation that arises upon linearization of the conservation law about the planar wave solution, we establish that under general spectral conditions, such as appear to hold for shock fronts arising in our motivating thin films equations, compressive waves are stable for all dimensions d≧2 and undercompressive waves are stable for dimensions d≧3. (In the special case d=1, compressive waves are stable under a very general spectral condition.) We also consider an alternative spectral criterion (valid, for example, in the case of constant-coefficient regularization), for which we can establish nonlinear stability for compressive waves in dimensions d≧3 and undercompressive waves in dimensions d≧5. The case of stability for undercompressive waves in the thin films equations for the critical dimensions d=1 and d=2 remains an interesting open problem.     

On the construction of manufactured solutions for one and two‐equation eddy‐viscosity models

This paper presents manufactured solutions (MSs) for some well‐known eddy‐viscosity turbulence models, viz. the Spalart & Allmaras one‐equation model and the TNT and BSL versions of the two‐equation k–ω model. The manufactured flow solutions apply to two‐dimensional, steady, wall‐bounded, incompressible, turbulent flows. The two velocity components and the pressure are identical for all MSs, but various alternatives are considered for specifying the eddy‐viscosity and other turbulence quantities in the turbulence models. The results obtained for the proposed MSs with a second‐order accurate numerical method show that the MSs for turbulence quantities must be constructed carefully to avoid instabilities in the numerical solutions. This behaviour is model dependent: the performance of the Spalart & Allmaras and k–ω models is significantly affected by the type of MS. In one of the MSs tested, even the two versions of the k–ω model exhibit significant differences in the convergence properties. Copyright © 2006 John Wiley & Sons, Ltd.     

Topological Invariants and Solutions with a High Complexity for Scalar Semilinear PDE

In this paper topological invariants of dynamics of scalar parabolic equations are defined. Conservation of these invariants causes persistence of complex spatially chaotic patterns in the dynamics. It implies, in particular, the existence of extremely many stable essentially different spatially complex solutions of scalar semilinear parabolic, elliptic and hyperbolic equations in the situation when coefficients of equations depend on the spatial variable strongly enough and the domain is large or unbounded, the number of different solutions exponentially depends on the volume of domain. Another implication of the conserved quantities is persistence of complexity of solutions in the dynamics.     

A stability analysis of periodic solutions to the steady forced Korteweg–de Vries–Burgers equation

The stability of periodic solutions to the steady forced Korteweg–de Vries–Burgers (fKdVB) equation is investigated here. This family of periodic solutions was identified by Hattam and Clarke (2015) using a multi-scale perturbation technique. Here, Floquet theory is applied to the governing equation. Consequently, two criteria are found that determine when the periodic solutions are stable. This analysis is then confirmed by a numerical study of the steady fKdVB equation.     

Oscillatory modes in a nonlinear second-order differential equation with delay

Solution properties of the nonlinear second-order delay-differential equation x(t)=–ax(t)+f[x(t–)] are studied wheref is a piecewise constant function which mimics negative feedback. We show that the solutions can be obtained by a simple geometrical construction which, in principle, can be implemented using a ruler and a compass. Analytical results guarantee the existence and stability properties of limit cycle solutions. Computer-aided constructions reveal a remarkable richness of different types of dynamical behaviors including a variety of unconventional bifurcation schemes.     

On the efficiency and quality of numerical solutions in CFD problems using the interface strip preconditioner for domain decomposition methods

In this paper, some pathologies found for simple tests solved by means of preconditioned full iterative schemes are presented. According to these results (Sections 4 and 5), the accuracy deterioration observed should be considered as a warning for the final application given to these solutions. Even though it is well known that full iterative solvers are not the best selection for comparison, they were chosen because they are widely used by the computational fluid dynamic (CFD) community for a diversity of complex fluid dynamics applications. FEM simulated solutions are compared with analytical solutions or measured data for problems that have been considered as ‘benchmarks’ in the CFD literature. For this purpose, the study of the solution obtained via parallelized iterative methods that have been extensively used (e.g. conjugate gradients (CG), GMRes global iteration and its variants, ‘overlapping’ and ‘non‐overlapping’ additive Schwarz domain decomposition schemes) in CFD computations and those obtained with the new interface strip preconditioner (J. Comput. Meth. Sci. Engng 2003; Int. J. Numer. Meth. Engng 2005; 62 (13):1873–1894) for the Schur complement method is carried out. The idea is to present the new solver as an alternative to obtain more accurate and faster solutions in the context of monolithic and non‐monolithic schemes applied to a internal/external viscous compressible/incompressible flows around bodies of complex shapes. Therefore, the target of this work is to show how the reliability of CFD codes is affected by the solver selection and why domain decomposition methods should be viewed not only as a more efficient strategy, but also to guarantee the solution quality. Copyright © 2006 John Wiley & Sons, Ltd.