Determining Lyapunov spectrum and Lyapunov dimension based on the Poincaré map in a vibro-impact system

We develop a method to compute the Lyapunov spectrum and Lyapunov dimension, which is effective for both symmetric and unsymmetric vibro-impact systems. The Poincaré section is chosen at the moment after impacting, and the six-dimensional Poincaré map is established. The time between two consecutive impacts is determined by the initial conditions and the impact condition, hence the Poincaré map is an implicit map. The Poincaré map is used to calculate all the Lyapunov exponents and the Lyapunov dimension. By numerical simulations, the attractors are represented in the projected Poincaré section, and the Lyapunov spectrum is obtained. The multi-degree-of-freedom vibro-impact system may exhibit complex quasi-periodic attractors, which can be characterized by the Lyapunov dimension.     

Analysis of a two-dimensional nonsmooth Poincaré-like map

A two-dimensional nonsmooth Poincaré-like map is investigated in the present work. The map generalizes in some sense the so-called Nordmark map, which is related to one-dimensional impacting oscillators near grazing points, and constitutes an intuitive basis for dynamics related to degenerate grazing in such oscillators.     

A hyperbolic Lindstedt–Poincaré method for homoclinic motion of a kind of strongly nonlinear autonomous oscillators

A hyperbolic Lindstedt-Poincare method is presented to determine the homoclinic solutions of a kind of nonlinear oscillators, in which critical value of the homoclinic bifurcation parameter can be determined. The generalized Lienard oscillator is studied in detail, and the present method＇s predictions are compared with those of Runge-Kutta method to illustrate its accuracy.     

Stabilization of the passive walking dynamics of the compass-gait biped robot by developing the analytical expression of the controlled Poincaré map

Nonlinear Dynamics - The compass-gait biped robot is a two-DoF legged mechanical system that has been known by its passive dynamic walking. This kind of passive biped robot is modeled by an...     

A fractional gradient representation of the Poincaré equations

A gradient representation and a fractional gradient representation of the Poincaré equations are studied. Firstly, the condition presented here for the Poincaré equation can be considered as a gradient system. Then, a condition under which the Poincaré equation can be considered as a fractional gradient system is obtained. Finally, two examples are given to illustrate applications of the result.     

Poincaré and the shortcomings of the Hamilton-Jacobi method for classical or quantized mechanics

A great theorem was proven by H. Poincaré in celestial mechanics. It states that, in the most general problems of mechanics, the total energy of the system is the only well behaved first integral of the system, while other so-called integrals cannot be represented by uniform and convergent series. This very important result can be explained and visualized by comparison with standard methods of discussion, as, for example, the Hamilton-Jacobi procedure. The discussion shows that there are serious limitations to the use of this procedure, which collapses in the most general problems (Poincaré theorem) and can be used only for “almost separated” variables. The Poincaré theorem appears to provide the distinction between determinism in mechanics and statistical mechanics according to Boltzmann. The research presented here done under Contract Nonr 266(56) and was first described in a Quarterly Report dated July 31, 1959.     

A Modified Poincaré Method for the Persistence of Periodic Orbits and Applications

This paper is devoted to the persistence of periodic orbits under perturbations in dynamical systems generated by evolutionary equations, which are not smoothing in finite time, but only asymptotically smoothing. When the periodic orbit of the unperturbed system is non-degenerate, we show the existence and uniqueness of a periodic orbit (with a minimal period near the minimal period of the unperturbed problem) by using “modified” Poincaré methods. Examples of applications, including the perturbed hyperbolic Navier–Stokes equations, systems of damped wave equations and the system of second grade fluids, are given.     

Poincaré oscillations and geostrophic adjustment in a rotating paraboloid

Free liquid oscillations (Poincaré oscillations) in a rotating paraboloid are investigated theoretically and experimentally. Within the framework of shallow-water theory, with account for the centrifugal force, expressions for the free oscillation frequencies are obtained and corrections to the frequencies related with the finiteness of the liquid depth are found. It is shown that in the rotating liquid, apart from the wave modes of free oscillations, a stationary vortex mode is also generated, that is, a process of geostrophic adjustment takes place. Solutions of the shallow-water equations which describe the wave dynamics of the adjustment process are presented. In the experiments performed the wave and vortex modes were excited by removing a previously immersed hemisphere from the central part of the paraboloid. Good agreement between theory and experiment was obtained.     

Spectral Analysis of the Neumann–Poincaré Operator and Characterization of the Stress Concentration in Anti-Plane Elasticity

When holes or hard elastic inclusions are closely located, stress which is the gradient of the solution to the anti-plane elasticity equation can be arbitrarily large as the distance between two inclusions tends to zero. It is important to precisely characterize the blow-up of the gradient of such an equation. In this paper we show that the blow-up of the gradient can be characterized by a singular function defined by the single layer potential of an eigenfunction corresponding to the eigenvalue 1/2 of a Neumann–Poincaré type operator defined on the boundaries of the inclusions. By comparing the singular function with the one corresponding to two disks osculating to the inclusions, we quantitatively characterize the blow-up of the gradient in terms of explicit functions. In electrostatics, our results apply to the electric field, which is the gradient of the solution to the conductivity equation, in the case where perfectly conducting or insulating inclusions are closely located.     

On the Spectrum of the Poincaré Variational Problem for Two Close-to-Touching Inclusions in 2D

We study the spectrum of the Poincaré variational problem for two close to touching inclusions in R ². We derive the asymptotics of its eigenvalues as the distance between the inclusions tends to zero.     

Nonzero mean PDF solution of nonlinear oscillators under external Gaussian white noise

This paper is concerned with the nonzero mean stationary probability density function (PDF) solution for nonlinear oscillators under external Gaussian white noise. The PDF solution is governed by the well-known Fokker–Planck–Kolmogorov (FPK) equation and this equation is numerically solved by the exponential-polynomial closure (EPC) method. Different types of oscillators are further investigated in the case of nonzero mean response. Either weak or strong nonlinearity is considered to show the effectiveness of the EPC method. When the polynomial order equals 2, the results of the EPC method are identical with those given by equivalent linearization (EQL) method. These results obtained with the EQL method differ significantly from exact solution or simulated results. When the polynomial order is 4 or 6, the PDFs obtained with the EPC method present a good agreement with the exact solution or simulated results, especially in the tail regions. The numerical analysis also shows that the nonzero mean PDF of the response is nonsymmetrically distributed about its mean unlike the case of the zero mean PDF reported in the references.     

On the Euler–Poincaré Equation with Non-Zero Dispersion

We consider the Euler–Poincaré equation on ${\mathbb{R}^d, \, d \geqq 2}$ R d , d ≧ 2 . For a large class of smooth initial data we prove that the corresponding solution blows up in finite time. This settles an open problem raised by Chae and Liu (Commun Math Phys 314:671–687, 2012). Our analysis exhibits some new concentration mechanisms and hidden monotonicity formulas associated with the Euler–Poincaré flow. In particular we show an abundance of blowups emanating from smooth initial data with certain sign properties. No size restrictions are imposed on the data. We also showcase a class of initial data for which the corresponding solution exists globally in time.     

Local modal participation analysis of nonlinear systems using Poincaré linearization

Nonlinear Dynamics - This study proposes a modified recurrence quantification analysis, called global recurrence quantification analysis (GRQA). It is well known that the recurrence threshold is an...     

Modeling the mechanics of HMX detonation using a Taylor–Galerkin scheme

Design of energetic materials is an exciting area in mechanics and materials science. Energetic composite materials are used as propellants, explosives, and fuel cell components. Energy release in these materials are accompanied by extreme events: shock waves travel at typical speeds of several thousand meters per second and the peak pressures can reach hundreds of gigapascals. In this paper, we develop a reactive dynamics code for modeling detonation wave features in one such material. The key contribution in this paper is an integrated algorithm to incorporate equations of state, Arrhenius kinetics, and mixing rules for particle detonation in a Taylor–Galerkin finite element simulation. We show that the scheme captures the distinct features of detonation waves, and the detonation velocity compares well with experiments reported in literature.     

Poincaré-Chetaev bifurcation diagrams in the problem of motion of an inhomogeneous dynamically and geometrically symmetric ellipsoid on a smooth plane

The existence, stability, and bifurcation of steady motions of an inhomogeneous dynamically and geometrically symmetric ellipsoid is considered. The mass center of the ellipsoid is shifted and located on its symmetry axis. The ellipsoid moves on a perfectly smooth horizontal plane.