327 research outputs found

    Dynamics of Simple Balancing Models with State Dependent Switching Control

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    Time-delayed control in a balancing problem may be a nonsmooth function for a variety of reasons. In this paper we study a simple model of the control of an inverted pendulum by either a connected movable cart or an applied torque for which the control is turned off when the pendulum is located within certain regions of phase space. Without applying a small angle approximation for deviations about the vertical position, we see structurally stable periodic orbits which may be attracting or repelling. Due to the nonsmooth nature of the control, these periodic orbits are born in various discontinuity-induced bifurcations. Also we show that a coincidence of switching events can produce complicated periodic and aperiodic solutions.Comment: 36 pages, 12 figure

    New existence and multiplicity of homoclinic solutions for second order non-autonomous systems

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    In this paper, we study the second order non-autonomous system \begin{eqnarray*} \ddot{u}(t)+A\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0, \ \ \forall t\in\mathbb{R}, \end{eqnarray*} where AA is an antisymmetric N×NN\times N constant matrix, L∈C(R,RN×N)L\in C(\mathbb{R},\mathbb{R}^{N\times N}) may not be uniformly positive definite for all t∈Rt\in\mathbb{R}, and W(t,u)W(t,u) is allowed to be sign-changing and local superquadratic. Under some simple assumptions on AA, LL and WW, we establish some existence criteria to guarantee that the above system has at least one homoclinic solution or infinitely many homoclinic solutions by using mountain pass theorem or fountain theorem, respectively. Recent results in the literature are generalized and significantly improved

    Integrability, localisation and bifurcation of an elastic conducting rod in a uniform magnetic field

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    The classical problem of the buckling of an elastic rod in a magnetic ¯eld is investigated using modern techniques from dynamical systems theory. The Kirchhoff equations, which describe the static equilibrium equations of a geometrically exact rod under end tension and moment are extended by incorporating the evolution of a fixed external vector (in the direction of the magnetic field) that interacts with the rod via a Lorentz force. The static equilibrium equations (in body cordinates) are found to be noncanonical Hamiltonian equations. The Poisson bracket is generalised and the equilibrium equations found to sit, as the third member, in a family of rod equations in generalised magnetic fields. When the rod is linearly elastic, isotropic, inextensible and unshearable the equations are completely integrable and can be generated by a Lax pair. The isotropic system is reduced using the Casimirs, via the Euler angles, to a four-dimensional canonical system with a first integral provided the magnetic field is not aligned with the force within the rod at any point as the system losses rank. An energy surface is specified, defning three-dimensional flows. Poincare sections then show closed curves. Through Mel'nikov analysis it is shown that for an extensible rod the presence of a magnetic field leads to the transverse intersection of the stable and unstable manifolds and the loss of complete integrability. Consequently, the system admits spatially chaotic solutions and a multiplicity of multimodal homoclinic solutions exist. Poincare sections associated with the loss of integrability are displayed. Homoclinic solutions are computed and post-buckling paths found using continutaion methods. The rods buckle in a Hamiltonian-Hopf bifurcation about a periodic solution. A codimension-two point, which describes a double Hamiltonian-Hopf bifurcation, determines whether straight rods buckle into localised configurations at either two critical values of the magnetic field, a single critical value or do not buckle at all. The codimension-two point is found to be an organising centre for primary and multimodal solutions

    Invariant manifolds of homoclinic orbits: super-homoclinics and multi-pulse homoclinic loops

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    Consider a Hamiltonian flow on R4 with a hyperbolic equilibrium O and a transverse homoclinic orbit Γ. In this thesis, we study the dynamics near Γ in its energy level when it leaves and enters O along strong unstable and strong stable directions, respectively. In particular, we provide necessary and sufficient conditions for the existence of the local stable and unstable invariant manifolds of Γ. We then consider the case in which both of these manifolds exist. We globalize them and assume they intersect transversely. We prove that near any orbit of this intersection, called super-homoclinic, there exist infinitely many multi-pulse homoclinic loops.Open Acces

    Shilnikov Lemma for a nondegenerate critical manifold of a Hamiltonian system

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    We prove an analog of Shilnikov Lemma for a normally hyperbolic symplectic critical manifold M⊂H−1(0)M\subset H^{-1}(0) of a Hamiltonian system. Using this result, trajectories with small energy H=μ>0H=\mu>0 shadowing chains of homoclinic orbits to MM are represented as extremals of a discrete variational problem, and their existence is proved. This paper is motivated by applications to the Poincar\'e second species solutions of the 3 body problem with 2 masses small of order μ\mu. As μ→0\mu\to 0, double collisions of small bodies correspond to a symplectic critical manifold of the regularized Hamiltonian system
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