Related papers: A shooting approach to the Lorenz equations
By variational methods, we provide a simple proof of existence of a heteroclinic orbit to the Hamiltonian system $u''=\nabla W(u)$ that connects the two global minima of a double-well potential $W$. Moreover, we consider several…
This article describes a method for constructing approximations to periodic solutions of dynamic Lorenz system with classical values of the system parameters. The author obtained a system of nonlinear algebraic equations in general form…
We use a geometric construction to exhibit examples of autonomous Lagrangian systems admitting exactly two homoclinics emanating from a nondegenerate maximum of the potential energy and reaching a regular level of the potential having the…
This paper investigates the dynamics of a particle orbiting around a rotating homogeneous cube, and shows fruitful results that have implications for examining the dynamics of orbits around non-spherical celestial bodies. This study can be…
We investigated the effects of gravitational lensing for a system in which a lens is a point mass and a homogeneous disc with a central hole. In such system there is a variety of cases resulting in formation of one, two and three Einstein…
We present a method for proving the existence of symmetric periodic, heteroclinic or homoclinic orbits in dynamical systems with the reversing symmetry. As an application we show that the Planar Restricted Circular Three Body Problem…
The classical Lorenz lowest order system of three nonlinear ordinary differential equations, capable of producing chaotic solutions, has been generalized by various authors in two main directions: (i) for number of equations larger than…
We give an elementary proof of a Landesman-Lazer type result for systems by means of a shooting argument and explore its connection with the fundamental theorem of algebra.
We analyze existence, multiplicity and oscillatory behavior of positive radial solutions to a class of quasilinear equations governed by the Lorentz-Minkowski mean curvature operator. The equation is set in a ball or an annulus of $\mathbb…
We study the relationship between homoclinic orbits associated to repellors, usually called snap-back repellors, and expanding sets of smooth endomorphisms. Critical homoclinic orbits constitutes an interesting bifurcation that is locally…
We study the existence of patterns (nontrivial, stationary solutions) for one-dimensional Swift-Hohenberg Equation in a directional quenching scenario, that is, on $x\leq 0$ the energy potential associated to the equation is bistable,…
This paper extends the model reduction method by the operator projection to the three-dimensional special relativistic Boltzmann equation. The derivation of arbitrary order moment system is built on our careful study of infinite families of…
Index theory revealed its outstanding role in the study of periodic orbits of Hamiltonian systems and the dynamical consequences of this theory are enormous. Although the index theory in the periodic case is well-established, very few…
Homoclinic snaking is a widespread phenomenon observed in many pattern-forming systems. Demonstrating its occurrence in non-perturbative regimes has proven difficult, although a forcing theory has been developed based on the identification…
One way to understand more about spacetime singularities is to construct solutions of the Einstein equations containing singularities with prescribed properties. The heuristic ideas of the BKL picture suggest that oscillatory singularities…
The paper provides a detailed proof that complicated motion exists in Shilnikov's scenario of a smooth vectorfield $V$ on $mathbb{R}^3$ with $V(0)=0$ so that the equation $x'=V(x)$ has a homoclinic solution $h$ with…
We consider the Earth-Moon planar circular restricted three body problem and present a proof of the existence orbits, which approach arbitrarily close to one of the primary masses, and at the same time after each approach they move away…
In this paper, we prove existence of symmetric homoclinic orbits for the suspension bridge equation $u""+\beta u" + e^u-1=0$ for all parameter values $\beta \in [0.5,1.9]$. For each $\beta$, a parameterization of the stable manifold is…
Semiclassical sum rules, such as the Gutzwiller trace formula, depend on the properties of periodic, closed, or homoclinic (heteroclinic) orbits. The interferences embedded in such orbit sums are governed by classical action functions and…
The purpose of this paper is to prove a new, more general version of the Morse index theorem for heteroclinic, homoclinic, and half-clinic solutions in general Lagrangian systems. In the final section, we compute the Morse index for…