Saddle Point Reduction Research Articles

The Existence of Periodic Solutions for Second-Order Delay Differential Systems.

In this paper, we consider a kind of second-order delay differential system. By taking some transforms, the property of delay is reflected in the boundary condition. The wonder is that the corrseponding first-order system is exactly the so-called P-boundary value problem of Hamiltonian system which has been studied deeply by many mathematicians, including the authors of this paper. Firstly, we define the relative Morse index for the delay system and give the relationship with the P-index of Hamiltonian system. Secondly, by this index, topology degree and saddle point reduction, the existence of periodic solutions is established for this kind of delay differential system.

Homoclinic solutions of discrete nonlinear Schrödinger equations with unbounded potentials

In this paper, the existence and multiplicity of homoclinic solutions for discrete nonlinear Schrödinger equations −Δuk+vkuk−ωuk=fk(uk)are considered. With asymptotical nonlinearity and unbounded potential V={vk:k∈Z} satisfying lim∣k∣→∞vk=∞, we achieve the existence of nontrivial homoclinic solution via the saddle point reduction method and Conley index theory. Furthermore, we can prove there are at least two nontrivial homoclinic solutions with extra assumptions about the nonlinearity fk.

Multiple brake orbits of even Hamiltonian systems on torus

In this paper, we establish a relationship between the Morse index at rest points in the saddle point reduction and the brake-orbit-type Maslov index at corresponding brake orbits. As an application, we give a criterion to find brake orbits which are contractible and start at $$\left\{0 \right\} \times \mathbb{T}{^n} \subset {\mathbb{T}^{2n}}$$ for even Hamiltonian on $${\mathbb{T}^{2n}}$$ by the methods of the Maslov-index theory and a critical point theorem formulated by Bartsch and Wang (1997). Explicitly, if all trivial solutions of a Hamiltonian are nondegenerate in the brake orbit boundary case, there are at least max{iL0 (z0)} pairs of nontrivial 1-periodic brake orbits if iL0 (z0) > 0 or at least max{−iL0 (z0) − n} pairs of nontrivial 1-periodic brake orbits if iL0 (z0) < −n. In the end, we give an example to find brake orbits for certain Hamiltonian via this criterion.

The existence of periodic solution for infinite dimensional Hamiltonian systems

In this paper, we will consider a kind of infinite dimensional Hamiltonian system (HS), by the method of saddle point reduction, topology degree and the index defined in Wang and Liu (2015), we will get the existence of periodic solution for (HS).

Existence and multiplicity of solutions for second-order Hamiltonian systems satisfying generalized periodic boundary value conditions at resonance

We investigate the existence and multiplicity of solutions for second-order Hamiltonian systems satisfying generalized periodic boundary value conditions at resonance by means of the index theory, the critical point theory without compactness assumptions, the least action principle, the saddle point reduction theorem, and the minimax method. Applying the results to second-order HS satisfying periodic boundary value conditions, we obtain some new results.

Existence of multiple periodic solutions to a semilinear wave equation with x-dependent coefficients

AbstractThis paper is concerned with the periodic (in time) solutions to an one-dimensional semilinear wave equation with x-dependent coefficients. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with saddle point reduction technique, we obtain the existence of at least three periodic solutions whenever the period is a rational multiple of the length of the spatial interval. Our method is based on a delicate analysis for the asymptotic character of the spectrum of the wave operator with x-dependent coefficients, and the spectral properties play an essential role in the proof.

Existence of Multiple Periodic Solutions for a Semilinear Wave Equation in an n-Dimensional Ball

Abstract This paper is devoted to the study of periodic solutions for a radially symmetric semilinear wave equation in an n-dimensional ball. By combining the variational methods and saddle point reduction technique, we obtain the existence of at least three periodic solutions for arbitrary space dimension n. The structure of the spectrum of the linearized problem plays an essential role in the proof, and the construction of a suitable working space is devised to overcome the restriction of space dimension.

Existence and multiplicity of rotating periodic solutions for resonant Hamiltonian systems

In the present paper, we consider a class of resonant Hamiltonian systems x′=JHx(t,x) in R2N. We will use saddle point reduction, Morse theory combining the technique of penalized functionals to obtain the existence of nontrivial rotating periodic solutions, i.e., x(t+T)=Qx(t) for any t∈R with T>0 and Q an symplectic orthogonal matrix. In the case: Qk≠I2N for any positive integer k, such a rotating periodic solution is just a quasi-periodic solution. Moreover, if H is even in x, we will give the multiplicity of nontrivial rotating periodic solutions by using two abstract critical theorems and previous techniques.

Existence of multiple periodic solutions to asymptotically linear wave equations in a ball

This paper is concerned with the Dirichlet problem of the asymptotically linear wave equation $$\begin{aligned} u_{tt}-\Delta u = g(t,x,u) \end{aligned}$$ in a n-dimensional ball with radius R, where \(n>1\) and g(t, x, u) is radially symmetric in x and T-periodic in time. An interesting feature is that the solvable of the problem depends on the space dimension n and the arithmetical properties of R and T. Based on the spectral properties of the radially symmetric wave operator, we use the saddle point reduction and variational methods to construct at least three radially symmetric solutions with time period T, when T is a rational multiple of R and g(t, x, u) satisfies some monotonicity and asymptotically linear conditions.

Relative index theories and applications

We develop some relative index theories for abstract operator equations. As applications, we prove a new Galerkin approximation formula and a new saddle point reduction formula for the $P$-index. We apply these new formulas to the minimal periodic problem for $P$-symmetric periodic solutions of nonlinear Hamiltonian systems.

Solutions of Dirac equations on compact spin manifolds via saddle point reduction

In this note, we apply the saddle point reduction to a class of nonlinear Dirac equation with a potential on a compact spin manifold. Two results are obtained, including a multiple theorem.

Fractional-Parabolic Deformations with Sinh-Acceleration

We develop a general simple methodology for very fast and accurate evaluation of integrals of functions that admit analytic continuation into wide regions of the complex plane or into appropriate Riemann surfaces. We use a family of fractional-parabolic deformations of the contours of integration to appropriate Riemann surfaces, make the corresponding conformal changes of variables and use the simplified trapezoid rule. An additional change variables leads to numerical schemes of complexity of the order of $A\pi^{-2}E^2$, where $E=\ln(1/\eps)$, $\eps$ is the error tolerance, and $A$ is, typically, in the range $[0.5,3]$, or even $A\pi^{-2}E \ln E$ (the Laplace and Fourier inversion, the Laplace transform and other integrals with exponentially decreasing or oscillating factors). Simple efficient error bounds make adaptive procedures unnecessary. As applications, we consider special functions of several classes, and outline applications to the Wiener-Hopf factorization, calculation of various probability distributions and pricing vanillas and exotics in the L'evy models, Heston model, and more general stochastic volatility models and affine models. The method can be applied to various boundary problems for partial differential and pseudo-differential equations, and in essentially all situations when the saddle point method or reduction to a cut are applied. Simple approximate recommendations for the choice of parameters of the scheme lead to programs that are accurate and competitive with Mathematica and Python, even without parallelization. In a number of situations, for the error tolerance $10^{-15}$, the resulting programs are dozens, hundreds and even thousands times faster than functions implemented in MATLAB, 8-15 times faster than SymPy. For $ _1F_1$ and $ _2F_1$, in some regions in the parameter space, the new programs are faster than functions implemented in Mathematica. Pricing of vanillas in the Heston model requires the summation of 6-15 terms for options of maturities T ≥ 0.004, in a wide region of strikes; the number of terms is a decreasing function of $T$.

Homology of saddle point reduction and applications to resonant elliptic systems

In the setting of saddle point reduction, we prove that the critical groups of the original functional and the reduced functional are isomorphic. As application, we obtain two nontrivial solutions for elliptic gradient systems which may be resonant both at the origin and at infinity. The difficulty that the variational functional does not satisfy the Palais–Smale condition is overcame by taking advantage of saddle point reduction. Our abstract results on critical groups are crucial.

Multiplicity result for asymptotically linear noncooperative elliptic systems

In this paper, we obtain a multiplicity result for a class of asymptotically linear noncooperative elliptic systems by using saddle point reduction technique and two abstract critical point theorems. Copyright © 2012 John Wiley &amp; Sons, Ltd.

Existence and multiplicity of nonzero periodic solution with saddle point character for some nonautonomous second order systems

The purpose of this paper is to study the existence and multiplicity of nonzero periodic solutions with saddle point character for the following nonautonomous second order systems: { u ¨ ( t ) = ∇ F ( t , u ( t ) ) , a.e. t ∈ [ 0 , T ] , u ( 0 ) − u ( T ) = u ˙ ( 0 ) − u ˙ ( T ) = 0 . Some new existence and multiplicity theorems are obtained by using saddle point reduction method and a three-critical-point theorem proposed by Brezis and Nirenberg.

Saddle point reduction method for some non-autonomous second order systems

The purpose of this paper is to study the existence of solutions with saddle point character for non-autonomous second order systems u ̈ (t)=∇F(t,u(t)), a.e. t∈[0,T], u(0)−u(T)= u ̇ (0)− u ̇ (T)=0. Some new solvability conditions are obtained.

Periodic solutions of asymptotically linear Hamiltonian systems

We prove existence and multiplicity results for periodic solutions of time dependent and time independent Hamiltonian equations, which are assumed to be asymptotically linear. The periodic solutions are found as critical points of a variational problem in a real Hilbert space. By means of a saddle point reduction this problem is reduced to the problem of finding critical points of a function defined on a finite dimensional subspace. The critical points are then found using generalized Morse theory and minimax arguments.