The Center Manifold Reduction Method for Hopf Bifurcation Theorem

In this paper we apply the center manifold reduction method to prove a Hopf bifurcation theorem for infinite dimensional problem. The asymptolic expression of bifurcation formulae and stability condition are given. The Hopf bifurcation problem for a system of parabolic equations is considered.     

A priori estimates for solutions of the first initial boundary-value problem for systems of fully nonlinear partial differential equations

We prove a priori estimates for a solution of the first initial boundary-value problem for a system of fully nonlinear partial differential equations (PDE) in a bounded domain. In the proof, we reduce the initial boundary-value problem to a problem on a manifold without boundary and then reduce the resulting system on the manifold to a scalar equation on the total space of the corresponding bundle over the manifold. St. Petersburg Architecture Building University, St. Petersburg. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 3, pp. 338–363, March, 1997.     

Integral manifolds of systems of differential equations with a random right-hand side in a Banach space

The concept of an integral manifold of a system of differential equations with a random right-hand side is introduced. The problem of the existence of an integral manifold of a certain class of differential equations in a Banach space and several of its properties are investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 1, pp. 17–21, January, 1992.     

“Invariance-Inducing” Control of Nonlinear Discrete-Time Dynamical Systems

The present study aims at the derivation of model-based control laws that attain the invariance objective for nonlinear skew-product discrete-time dynamical systems. The problem under consideration naturally arises in a variety of control problems pertaining to physical/chemical systems, and in the present study, it is conveniently formulated and addressed in the context of functional equations theory. In particular, the mathematical formulation of the problem of interest is realized via a system of nonlinear functional equations (NFEs), and a rather general set of necessary and sufficient conditions for solvability is derived. The solution to the above system of NFEs can be proven to be a unique locally analytic one, and this enables the development of a series solution method that is easily programmable with the aid of a symbolic software package such as MAPLE. It is also shown that, on the basis of the solution to the above system of NFEs, a locally analytic manifold and a nonlinear control law can be explicitly derived that renders the manifold invariant for the class of skew-product systems considered. Furthermore, the restriction of the system dynamics on the aforementioned invariant manifold represents exactly the target controlled system dynamics. Finally, the proposed method is applied to the HF molecular system classically modeled as a rotationless Morse oscillator in the presence of an external laser-field, where the primary objective is molecular dissociation.     

Equations of Maxwell type

For an elliptic complex of first order differential operators on a smooth manifold X, we define a system of two equations which can be thought of as abstract Maxwell equations. The formal theory of this system proves to be very similar to that of classical Maxwell's equations. The paper focuses on boundary value problems for the abstract Maxwell equations, especially on the Cauchy problem.     

Identification of shock profile solutions for bidisperse suspensions

This contribution is a condensed version of an extended paper, where a contact manifold emerging in the interior of the phase space of a specific hyperbolic system of two nonlinear conservation laws is examined. The governing equations are modelling bidisperse suspensions, which consist of two types of small particles differing in size and viscosity that are dispersed in a viscous fluid. Based on the calculation of characteristic speeds, the elementary waves with the origin as left Riemann datum and a general right state in the phase space are classified. In particular, the dependence of the solution structure of this Riemann problem on the contact manifold is elaborated.     

A Primal-Dual Formulation for Certifiable Computations in Schubert Calculus

Formulating a Schubert problem as solutions to a system of equations in either Plücker space or local coordinates of a Schubert cell typically involves more equations than variables. We present a novel primal-dual formulation of any Schubert problem on a Grassmannian or flag manifold as a system of bilinear equations with the same number of equations as variables. This formulation enables numerical computations in the Schubert calculus to be certified using algorithms based on Smale’s \(\alpha \)-theory.     

Synthesis of virtual measurements in nonlinear observation problem

The method of the observation problems reducing to the algebraic ones is considered for the systems, which are linear with respect to unknown components of the phase vector. The approach proposed is based on the methods of the controlled stabilization of nonlinear system with respect to the part of variables. The equations of the initial observable system are supplemented by the equations of its controlled prototype. Then control law synthesied in such way that any given manifold becomes an invariant for extended system. For ensuring of this manifold attracting property the partial differential equations are obtained. Finally, the chosen in such way algebraic relations are considered as additional virtual measurements of unknown state. As example the problem of the angular velocity determination of a rigid body is considered. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a system of differential equations with operator coefficients and its applications to multidimensional systems of composite type

A mixed problem for a system of differential equations with operator coefficients is considered on an interval. Necessary and sufficient conditions for the existence of at least one solution of the given problem are investigated. It is established that the linear manifold of the solutions of the homogeneous problem is finite-dimensional. The obtained results are applied to multidimensional systems of differential equations of composite type, defined in cylindrical domains.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 16, pp. 25–69, 1992.     

A SINGULAR SINGULARLY-PERTURBED BOUNDARY VALUE PROBLEM

ＡＳＩＮＧＵＬＡＲＳＩＮＧＵＬＡＲＬＹ－ＰＥＲＴＵＲＢＥＤＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭ￥ＬｉｎＷｕｚｈｏｎｇ（林武忠）；ＷａｎｇＺｈｉｍｉｎｇ（汪志鸣）（ＥａｓｔＣｈｉｎａＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，上海华东师范大学，邮编：２０００...     

一类可再生资源系统的最优动态平衡收获

研究一类可再生资源系统的最优利用问题．首先,引进一个新的效用函数, 它依赖于收获努力度和资源量,由此导出最优控制问题．其次证明该控制问题最优解的存在性．然后,利用无穷区间上控制问题的最大值原理,得到一个非线性的四维最优系统．通过对上述系统正平衡解的详细分析,借助 Hopf 分支定理证明了极限环的存在性．之后考虑中心流形上的简化系统, 分析极限环的稳定性．最后,解释所得结果的生物经济学意义．     

The effect of calmness on the solution set of systems of nonlinear equations

We address the problem of solving a continuously differentiable nonlinear system of equations under the condition of calmness. This property, also called upper Lipschitz-continuity in the literature, can be described by a local error bound and is being widely used as a regularity condition in optimization. Indeed, it is known to be significantly weaker than classic regularity assumptions that imply that solutions are isolated. We prove that under this condition, the rank of the Jacobian of the function that defines the system of equations must be locally constant on the solution set. In addition, we prove that locally, the solution set must be a differentiable manifold. Our results are illustrated by examples and discussed in terms of their theoretical relevance and algorithmic implications.     

Asymptotically stable invariant manifold for coupled nonlinear parabolic-hyperbolic partial differential equations

This article considers a coupled system of nonlinear parabolic and hyperbolic partial differential equations which arises in the study of wave phenomena which are heat generating or temperature related. Under appropriate conditions, for example high thermal diffusivity, it is proved that there exists an invariant manifold for the full system of equations. The asymptotic stability of the invariant manifold is also considered. Moreover, it is shown that an equilibrium which is asymptotically stable for flows on the invariant manifold will be asymptotically stable for the full system.     

Formal analysis of the Cauchy problem for a system associated with the (2+1)-dimensional Krichever-Novikov equation

The singularity manifold equation of the Kadomtsev-Petviashvili equation, the so-called Krichever-Novikov equation, has an exact linearization to an overdetermined system of partial differential equations in three independent variables. We study in detail the Cauchy problem for this system as an example for the use of the formal theory of differential equations. A general existence and uniqueness theorem is established. Formal theory is then contrasted with Janet-Riquier theory in the formulation of Reid. Finally, the implications of the results for the Krichever-Novikov equation are outlined.     

The Euler-Lagrange method in Pontryagin’s formulation

The classical variational problem with nonholonomic constraints is solvable by the Euler-Lagrange method in Pontryagin’s formulation; however, in this case Lagrange multipliers are merely measurable functions. In this paper, we put forward a modified Euler-Lagrange method, in which the original problem involves a Lagrangian dependent only on the independent components of the velocity vector. Under this approach, the Lagrange multipliers make up an absolutely continuous vector function. Our method is applied to the problem of horizontal geodesics for a nonholonomic distribution on a manifold. These equations are established as having two types of connections: connection on the distribution and connection on the manifold; this was not accounted for by other researchers.     

On the Asymptotic Solution of Non-Hamiltonian Systems Exhibiting Sustained Resonance

With the exception of some special examples, much of the literature on the formal construction of asymptotic solutions of systems exhibiting sustained resonance concerns Hamiltonian problems, for which the reduced problem is of order two when a single resonance is present. In the Hamiltonian case, the resonance manifold is a curve that is explicitly defined by the governing equations and is independent of the actual sustained resonance solution. When the basic standard form system is non-Hamiltonian, with M slow and N fast variables, the corresponding reduced problem is of order M + 1; in general it involves all of the slow variables, P₁,…, P_M, plus the resonant phase Q. In this paper, the solution of a general non-Hamiltonian system in standard form is formally constructed for the case of a single sustained resonance. First, a well-known example is reviewed, for which the projection of the solutions on the resonance manifold can be derived a priori, independent of the evolution of Q. Then, the general case is solved, using a generalization of the multiple scale method of Kuzmak-Luke, where knowledge of the asymptotic solution for Q (as well as higher-order terms) is needed to define the projection of the solution on the resonance manifold. The results simplify significantly when initial conditions are chosen exactly on the resonance manifold; the modifications necessary for arbitrary initial conditions are also given. Two examples are discussed in detail to illustrate the procedure. The asymptotic results are confirmed for several test cases by comparison with numerical integrations of the exact equations.     

Traveling fronts of a real supercritical quintic Ginzburg-Landau equation coupled by a slow diffusion mode

In this paper, we investigate the existence of traveling front solutions for a class of quintic Ginzburg-Landau equations coupled with a slow diffusion mode. By employing the theory of geometric singular perturbations, we turn the problem into a geometric perturbation problem. We demonstrate the intersection property of the critical manifold and further validate the existence of heteroclinic orbits by computing the zeros of the Melnikov function on the critical manifold. The results demonstrate that under certain parameters, there is 1 or 2 heteroclinic solutions, confirming the existence of traveling front solutions for the considered quintic Ginzburg-Landau equation coupled with a slow diffusion mode.     

Existence of dichotomies and invariant splittings for linear differential systems I

This paper is concerned with linear time-varying ordinary differential equations. Sufficient conditions are given for the existence of an exponential dichotomy for a class of equations which includes those with Bohr almost-periodic coefficients. The problem is treated in the context of linear skew-product flows, where it becomes clear how to generalize to the case of fiber-preserving flows on vector bundles. Both continuous and discrete flows are treated and the results apply to the linearized variational equation for a time-varying vector field on a manifold as well as the linearization of a diffeomorphism acting on a manifold. Sufficient conditions are given for a diffeomorphism on a manifold to be an Anosov diffeomorphism. For linear skew-product flows arising from ordinary differential equations our theory is a partial generalization of Floquet theory to the almost-periodic case.     

Positive solutions of asymptotically linear equations via Pohozaev manifold

We present a new method for finding positive solutions of nonlinear elliptic equations, which are non-homogeneous and asymptotically linear at infinity, by using projections on a Pohozaev manifold rather than the Nehari manifold associated with the problem.     

A note on Fischer-Marsden's conjecture

In this paper, we borrowed some ideas from general relativity and find a Robinson-type identity for the overdetermined system of partial differential equations in the Fischer-Marsden conjecture. We proved that if there is a nontrivial solution for such an overdetermined system on a 3-dimensional, closed manifold with positive scalar curvature, then the manifold contains a totally geodesic 2-sphere.