20 research outputs found

    On Nonlinear Dynamics of the Pendulum with Periodically Varying Length

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    Dynamic behavior of a weightless rod with a point mass sliding along the rod axis according to periodic law is studied. This is the pendulum with periodically varying length which is also treated as a simple model of child's swing. Asymptotic expressions for boundaries of instability domains near resonance frequencies are derived. Domains for oscillation, rotation, and oscillation-rotation motions in parameter space are found analytically and compared with numerical study. Two types of transitions to chaos of the pendulum depending on problem parameters are investigated numerically.Comment: 8 pages, 8 figure

    Geometric phase around exceptional points

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    A wave function picks up, in addition to the dynamic phase, the geometric (Berry) phase when traversing adiabatically a closed cycle in parameter space. We develop a general multidimensional theory of the geometric phase for (double) cycles around exceptional degeneracies in non-Hermitian Hamiltonians. We show that the geometric phase is exactly π\pi for symmetric complex Hamiltonians of arbitrary dimension and for nonsymmetric non-Hermitian Hamiltonians of dimension 2. For nonsymmetric non-Hermitian Hamiltonians of higher dimension, the geometric phase tends to π\pi for small cycles and changes as the cycle size and shape are varied. We find explicitly the leading asymptotic term of this dependence, and describe it in terms of interaction of different energy levels.Comment: 4 pages, 1 figure, with revisions in the introduction and conclusio

    Complex magnetic monopoles, geometric phases and quantum evolution in vicinity of diabolic and exceptional points

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    We consider the geometric phase and quantum tunneling in vicinity of diabolic and exceptional points. We show that the geometric phase associated with the degeneracy points is defined by the flux of complex magnetic monopole. In weak-coupling limit the leading contribution to the real part of geometric phase is given by the flux of the Dirac monopole plus quadrupole term, and the expansion for its imaginary part starts with the dipolelike field. For a two-level system governed by the generic non-Hermitian Hamiltonian, we derive a formula to compute the non-adiabatic complex geometric phase by integral over the complex Bloch sphere. We apply our results to to study a two-level dissipative system driven by periodic electromagnetic field and show that in the vicinity of the exceptional point the complex geometric phase behaves as step-like function. Studying tunneling process near and at exceptional point, we find two different regimes: coherent and incoherent. The coherent regime is characterized by the Rabi oscillations and one-sheeted hyperbolic monopole emerges in this region of the parameters. In turn with the incoherent regime the two-sheeted hyperbolic monopole is associated. The exceptional point is the critical point of the system where the topological transition occurs and both of the regimes yield the quadratic dependence on time. We show that the dissipation brings into existence of pulses in the complex geometric phase and the pulses are disappeared when dissipation dies out. Such a strong coupling effect of the environment is beyond of the conventional adiabatic treatment of the Berry phase.Comment: 29 pages, 21 figure

    Parametric Resonance in Mechanics: Classical Problems and New Results

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    Analysis of the gyroscopic stabilization of a system of rigid bodies

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    Collapse of the Keldysh Chains and Stability of Continuous Non-Conservative Systems

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    In the present paper, eigenvalue problems for non-self-adjoint linear differential operators smoothly dependent on a vector of real parameters are considered. Bifurcation of eigenvalues along smooth curves in the parameter space is studied. The case of a multiple eigenvalue with the Keldysh chain of arbitrary length is investigated. Explicit expressions describing bifurcation of eigenvalues are found. The obtained formulas use eigenfunctions and associated functions of the adjoint eigenvalue problems as well as the derivatives of the differential operator taken at the initial point of the parameter space. These results are important for the stability problems and sensitivity analysis of nonconservative systems. As a mechanical application, the extended Beck problem of stability of an elastic column subjected to a partially tangential follower force is considered and discussed in detail

    Dissipation induced instabilities in continuous non-conservative systems

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    In this contribution we analyze the stabilizing and destabilizing effect of small damping for rather general class of continuous non-conservative systems, described by the non-self-adjoint boundary eigenvalue problems. Explicit asymptotic expressions obtained for the stability domain allow us to predict when a given combination of the damping parameters leads to increase or to decrease in the critical non-conservative load. The results obtained explain why different types of internal and external damping so surprisingly influence on the stability of non-conservative systems. As a mechanical example the stability of a viscoelastic rod with small internal and external damping, loaded by tangential follower force, is studied in detail. Weakly damped continuous non-conservative systems We consider the boundary eigenvalue problem arising in stability problems for viscoelastic systems The coefficients of the linear differential operators N , D, and M of order m, and of the matrices U N , U D , and Let the unperturbed circulatory system have discrete spectrum for 0 ≀ q < q 0 , consisting of simple purely imaginary eigenvalues λ (stability). For increasing load parameter q two simple purely imaginary eigenvalues move along the imaginary axis until they collide at q=q 0 forming the double eigenvalue iω 0 with the Keldysh chain [1], consisting of an eigenfunction u 0 (x) and associated function u 1 (x). After the collision the eigenvalues diverge in the directions perpendicular to the imaginary axis of the complex plane (flutter instability). Such a scenario is known as the strong interaction of eigenvalues and is a typical mechanism of the loss of stability for circulatory systems where the angular brackets denote the inner product of vectors in R n−1 . The components of the real vector f and the real quantity f are expressed by means of the eigenfunctions and associated functions of the adjoint boundary eigenvalue problems and the derivatives of the differential operators and the matrices of the boundary conditions where the asterisk denotes Hermitian conjugation. Similarly, one can find the real vector h and the real matrices H and G. As it follows from equation (3), the function q cr (k) is singular at the point k=0, and the critical load as a function of n−1 variable has no limit as k=(k 1 , . . . , k n−1 ) tends to zero. However, there exists the limit lim →0 q cr ( k) for an
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