STABILITY OF SUBHARMONIC SOLUTIONS OF FIRST-ORDER HAMILTONIAN SYSTEMS WITH ANISOTROPIC GROWTH

Using the dual Morse index theory, we study the stability of subharmonic solutions of first-order autonomous Hamiltonian systems with anisotropic growth, that is, we obtain a sequence of elliptic subharmonic solutions(that is, all its Floquet multipliers lying on the unit circle on the complex plane C).     

Existence and stability of nontrivial periodic solutions of periodically forced discrete dynamical Systems

The local existence and local asymptotic stability of nontrivial p-periodic solutions of p-periodically forced discrete systems are proven using Liapunov-Schmidt methods. The periodic solutions bifurcate transcritically from the trivial solution at the critical value n=n^cr of the bifurcation parameter with a typical exchange of stability. If the trivial solution loses (gains) stability as n is increased through n^cr , then the periodic solutions on the nontrivial bifurcating branch are locally asymptotically stable if and only if they correspond to n>n^cr (n n^cr ).     

Dynamic behaviors of the periodic predator–prey system with distributed time delays and impulsive effect

In this paper, a periodic predator–prey system with distributed time delays and impulsive effect is investigated. By using the Floquet theory of linear periodic impulsive equation, some conditions for the linear stability of trivial periodic solution and semi-trivial periodic solutions are obtained. It is proved that the system can be permanent if all the trivial and semi-trivial periodic solutions are linearly unstable. We improve some results in Guo and Chen (2009) [1].     

Stable Modulating Multipulse Solutions for Dissipative Systems with Resonant Spatially Periodic Forcing

Summary.    We show the existence and stability of modulating multipulse solutions for a class of bifurcation problems given by a dispersive Swift-Hohenberg type of equation with a spatially periodic forcing. Equations of this type arise as model problems for pattern formation over unbounded weakly oscillating domains and, more specifically, in laser optics. As associated modulation equation, one obtains a nonsymmetric Ginzburg-Landau equation which possesses exponentially stable stationary n—pulse solutions. The modulating multipulse solutions of the original equation then consist of a traveling pulselike envelope modulating a spatially oscillating wave train. They are constructed by means of spatial dynamics and center manifold theory. In order to show their stability, we use Floquet theory and combine the validity of the modulation equation with the exponential stability of the n—pulses in the modulation equation. The analysis is supplemented by a few numerical computations. In addition, we also show, in a different parameter regime, the existence of exponentially stable stationary periodic solutions for the class of systems under consideration. Received November 30, 1999; accepted December 4, 2000 Online publication March 23, 2001     

Complementary Regions of Knot and Link Diagrams

An increasing sequence of integers is said to be universal for knots and links if every knot and link has a reduced projection on the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. In this paper, it is proved that the following infinite sequences are each universal for knots and links: (3, 5, 7, . . .), (2, n, n + 1, n + 2, . . .) for each n ≥ 3, (3, n, n + 1, n + 2, . . .) for each n ≥ 4. Moreover, the finite sequences (2, 4, 5) and (3, 4, n) for each n ≥ 5 are universal for all knots and links. It is also shown that every knot has a projection with exactly two odd-sided faces, which can be taken to be triangles, and every link of n components has a projection with at most n odd-sided faces if n is even and n + 1 odd-sided faces if n is odd.     

Continuity in weak topology:higher order linear systems of ODE

We will introduce a type of Fredholm operators which are shown to have a certain con- tinuity in weak topologies.From this,we will prove that the fundamental matrix solutions of k-th, k≥2,order linear systems of ordinary differential equations are continuous in coefficient matrixes with weak topologies.Consequently,Floquet multipliers and Lyapunov exponents for periodic systems are continuous in weak topologies.Moreover,for the scalar Hill's equations,Sturm-Liouville eigenvalues, periodic and anti-periodic eigenvalues,and rotation numbers are all continuous in potentials with weak topologies.These results will lead to many interesting variational problems.     

Stable closed characteristics on partially symmetric convex hypersurfaces

Let n be a positive integer and P=diag(−I_n−κ,I_κ,−I_n−κ,I_κ) for some integer κ∈[0,n]. In this paper, we prove that for any convex compact smooth hypersurface Σ in R²ⁿ with n?2 there always exists at least one closed characteristic on Σ which possesses at least 2n−4κ Floquet multipliers on the unit circle of the complex plane, provided Σ is P-symmetric, i.e., x∈Σ implies Px∈Σ.     

Quasi-Stable Structures in Circular Gene Networks

A new mathematical model is proposed for a circular gene network representing a system of unidirectionally coupled ordinary differential equations. The existence and stability of special periodic motions (traveling waves) for this system is studied. It is shown that, with a suitable choice of parameters and an increasing number m of equations in the system, the number of coexisting traveling waves increases indefinitely, but all of them (except for a single stable periodic solution for odd m) are quasistable. The quasi-stability of a cycle means that some of its multipliers are asymptotically close to the unit circle, while the other multipliers (except for a simple unit one) are less than unity in absolute value.     

Set of ambiguous points for functions in ℝ n

It is known that an arbitrary function in the open unit disk can have at most countable set of ambiguous points. Point ζ on the unit circle is an ambiguous point of a function if there exist two Jordan arcs, lying in the unit ball, except the endpoint ζ, such that cluster sets of function along these arcs are disjoint. We investigate whether it is possible to modify the notion of ambiguous point to keep the analogous result true for functions defined in the n-dimensional Euclidean unit ball.     

Periodic solutions of Hamiltonian systems on hypersurfaces in a torus

The existence of periodic solutions to Hamiltonian systems on the symplectic manifold (T ²ⁿ, ω) is studied. We show that on a class of hypersurfaces in the torusT ²ⁿ there is a periodic solution, which generalizes the results due to Long and Zehnder. Supported by NNSF of China     

On Periodic solutions for a reduction of Benney chain

We study periodic solutions for a quasi-linear system, which naturally arises in search of integrable Hamiltonian systems of the form H = p ²/2 + u(q, t). Our main result classifies completely periodic solutions for such a 3 by 3 system. We prove that the only periodic solutions have the form of traveling waves so, in particular, the potential u is a function of a linear combination of t and q. This result implies that the there are no nontrivial cases of the existence of a fourth power integral of motion for H: if it exists, then it is equal necessarily to the square of a quadratic integral. Our main observation for the quasi-linear system is the genuine non-linearity of the maximal and minimal eigenvalues in the sense of Lax. We use this observation in the hyperbolic region, while the “elliptic” region is treated using the maximum principle.     

Analytic Continuation of Dirichlet Series with Almost Periodic Coefficients

We consider Dirichlet series z_g,a(s)=?_n=1^￥ g(na) e^{-l_n s}{\zeta_{g,\alpha}(s)=\sum_{n=1}^\infty g(n\alpha) e^{-\lambda_n s}} for fixed irrational α and periodic functions g. We demonstrate that for Diophantine α and smooth g, the line Re(s) = 0 is a natural boundary in the Taylor series case λ _n  = n, so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series ?_n=1^￥ g(na) zⁿ{\sum_{n=1}^{\infty} g(n\alpha) z^n}. We prove that a Dirichlet series z_g,a(s) = ?_n=1^￥ g(n a)/n^s{\zeta_{g,\alpha}(s) = \sum_{n=1}^{\infty} g(n \alpha)/n^s} has an abscissa of convergence σ ₀ = 0 if g is odd and real analytic and α is Diophantine. We show that if g is odd and has bounded variation and α is of bounded Diophantine type r, the abscissa of convergence σ ₀ satisfies σ ₀ ≤ 1 − 1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and α is Diophantine, then the Dirichlet series ζ _g,α (s) has an analytic continuation to the entire complex plane.     

Weak hyperbolicity on periodic orbits for polynomials

We prove that if the multipliers of the repelling periodic orbits of a complex polynomial grow at least like n^5+ε with the period, for some ε>0, then the Julia set of the polynomial is locally connected when it is connected. As a consequence for a polynomial the presence of a Cremer cycle implies the presence of a sequence of repelling periodic orbits with “small” multipliers. Somewhat surprisingly the proof is based on measure theorical considerations. To cite this article: J. Rivera-Letelier, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1113–1118.     

Sequences of minimal surfaces in S2n+1

For a minimal surface immersed into an odd-dimensional unit sphere S ²ⁿ⁺¹ with the first (n−2) higher-order ellipses of curvature being a circle, we construct a sequence of such surfaces and investigate if some two minimal surfaces in such a sequence can be congruent by an orientation-reversing isometry.     

Permutation Matrices, Wreath Products, and the Distribution of Eigenvalues

We consider a class of random matrix ensembles which can be constructed from the random permutation matrices by replacing the nonzero entries of the n×n permutation matrix matrix with M×M diagonal matrices whose entries are random Kth roots of unity or random points on the unit circle. Let X be the number of eigenvalues lying in a specified arc I of the unit circle, and consider the standardized random variable (X–E[X])/(Var(X))^1/2. We show that for a fixed set of arcs I ₁,...,I _N , the corresponding standardized random variables are jointly normal in the large n limit, and compare the covariance structures which arise with results for other random matrix ensembles.     

Asymptotic estimates and stability analysis of Kuramoto-Sivashinsky type models

We first show asymptotic L ² bounds for a class of equations, which includes the Burger-Sivashinsly model for odd solutions with periodic boundary conditions. We consider the conditional stability of stationary solutions of Kuramoto-Sivashinsky equation in the periodic setting. We establish rigorously the general structure of the spectrum of the linearized operator, in particular the linear instability of steady states. In addition, we show conditional asymptotic stability with asymptotic phase, under a natural spectral hypothesis for the corresponding linearized operator. For the zero solution, we have more precise results. Namely, in the non-resonant regime L ≠ n π, we prove conditional asymptotic stability, provided one considers only mean value zero data. If, however, L = n ₀ π (but still ò\nolimits_-L^L u₀(x) dx=0{\int\nolimits_{-L}^L u_0(x) dx=0}), then we have conditional orbital stability. More specifically, the solutions relax to a small (but generally non-zero) function as long as the initial data are small and lie on a center-stable manifold of codimension 2(n ₀ − 1).     

Morse theory for periodic solutions of hamiltonian systems and the maslov index

In this paper we prove Morse type inequalities for the contractible 1-periodic solutions of time dependent Hamiltonian differential equations on those compact symplectic manifolds M for which the symplectic form and the first Chern class of the tangent bundle vanish over π₂ (M). The proof is based on a version of infinite dimensional Morse theory which is due to Floer. The key point is an index theorem for the Fredholm operator which plays a central role in Floer homology. The index formula involves the Maslov index of nondegenerate contractible periodic solutions. This Maslov index plays the same role as the Morse index of a nondegenerate critical point does in finite dimensional Morse theory. We shall use this connection between Floer homology and Maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every 1-periodic solution has at least one Floquet multiplier which is not equal to 1.     

The lubrication approximation for thin viscous films: Regularity and long-time behavior of weak solutions

We consider the fourth-order degenerate diffusion equation, in one space dimension. This equation, derived from a lubrication approximation, models the surface-tension-dominated motion of thin viscous films and spreading droplets [15]. The equation with f(h) = |h| also models a thin neck of fluid in the Hele-Shaw cell [10], [11], [23]. In such problems h(x,t) is the local thickness of the the film or neck. This paper considers the properties of weak solutions that are more relevant to the droplet problem than to Hele-Shaw. For simplicity we consider periodic boundary conditions with the interpretation of modeling a periodic array of droplets. We consider two problems: The first has initial data h₀ ≥ 0 and f(h) = |h|ⁿ, 0 < n < 3. We show that there exists a weak nonnegative solution for all time. Also, we show that this solution becomes a strong positive solution after some finite time T*, and asymptotically approaches its means as t → ∞. The weak solution is in the classical sense of distributions for 3/8 < n < 3 and in a weaker sense introduced in [1] for the remaining 0 < n ≤ 3/8. Furthermore, the solutions have high enough regularity to just include the unique source-type solutions [2] with zero slope at the edge of the support. They do not include any of the less regular solutions with positive slope at the edge of the support. Second, we consider strictly positive initial data h₀ ≥ m > 0 and f(h) = |h|ⁿ, 0 < n < ∞. For this problem we show that even if a finite-time singularity of the form h → 0 does occur, there exists a weak nonnegative solution for all time t. This weak solution becomes strong and positive again after some critical time T*. As in the first problem, we show that the solution approaches its mean as t → ∞. The main technical idea is to introduce new classes of dissipative entropies to prove existence and higher regularity. We show that these entropies are related to norms of the difference between the solution and its mean to prove the relaxation result. © 1996 John Wiley & Sons, Inc.     

Factoriality condition of some nodal threefolds in {\mathbb{P}^4}

We prove that for n = 8, 9, 10, 11, a nodal hypersurface of degree n in is factorial if it has at most (n − 1)² − 1 nodes. The author is grateful to Ivan Cheltsov for valuable comments and suggestions.