Periodic-impact motions and bifurcations of vibro-impact systems near 1:4 strong resonance point

Two typical vibro-impact systems are considered. The periodic-impact motions and Poincaré maps of the vibro-impact systems are derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional one, and the normal form map associated with 1:4 strong resonance is obtained. Two-parameter bifurcations of fixed points in the vibro-impact systems, associated with 1:4 strong resonance, are analyzed. The results from simulation illustrate some interesting features of dynamics of the vibro-impact systems. Some complicated bifurcations, e.g., tangent, fold and Neimark–Sacker bifurcations of period-4 orbits are found to exist near the 1:4 strong resonance points of the vibro-impact systems.     

On computing Poincaré map by Hénon method

The trajectory of the autonomous chaotic system deviates from the original path leading to a deformation in its attractor while calculating Poincaré map using the method presented by Hénon [Hénon M. Physica D 1982;5:412]. Also, the Poincaré map obtained is found to be the Poincaré map of deformed attractor instead of the original attractor. In order to overcome these drawbacks, this method is slightly modified by introducing an important change in the existing algorithm. Then it is shown that the modified Hénon method calculates the Poincaré map of the original attractor and it does not affect the system dynamics (attractor). The modified method is illustrated by means of the Lorenz and Chua systems.     

Relativistic particles and commutator algebras with twisted Poincaré transformation

In this paper we consider the massive and massless action for relativistic particle in D-dimensional flat space–time. We show that the Poincaré space–time algebra in the commutator version, and the Killing field provides the generators of the Poincaré algebra. We apply the non-commutative version to action, which is not Poincaré invariant. This leads us to consider the twisted Poincaré transformation, finally by using this transformation, we see that the action is invariant. By using the non-commutative space in massless action, in contrast to the commutative case the scale and conformal in-variance is broken by massive term [For an interesting discussion of the cosmological constant problem see A. Zee, Dark energy and the nature of the graviton, <arXiv:hep-th/0309032>].     

Poincaré-Hopf inequalities for periodic orbits

The interplay between the dynamics of a nonsingular Morse-Smale flow on a smooth, closed, n-dimensional manifold, M, and the topology of M, was exhibited in Franks (Comment Math Helv 53(2):279?C294, 1978), Smale (Bull Am Math Soc 66:43?C49, 1960), by means of a collection of inequalities, which we refer to as Morse-Smale inequalities. These inequalities relate the number of closed orbits of each index to the Betti numbers of M. These well-known inequalities provide the necessary conditions for a given dynamical data in the form of a specified number of closed orbits of a given index to be realized as a nonsingular Morse-Smale flow on M. In this article we provide two inequalities, hereby referred to as Poincaré-Hopf inequalities for periodic orbits, which imposes constraints on the dynamics of periodic orbits without reference to the Betti numbers of the manifold M. The main theorem establishes the necessity and sufficiency of the Poincaré-Hopf inequalities in order for the Morse-Smale inequalities to hold.     

Poincaré’s Equations for Cosserat Media: Application to Shells

In 1901, Henri Poincaré discovered a new set of equations for mechanics. These equations are a generalization of Lagrange’s equations for a system whose configuration space is a Lie group which is not necessarily commutative. Since then, this result has been extensively refined through the Lagrangian reduction theory. In the present contribution, we apply an extended version of these equations to continuous Cosserat media, i.e. media in which the usual point particles are replaced by small rigid bodies, called microstructures. In particular, we will see how the shell balance equations used in nonlinear structural dynamics can be easily deduced from this extension of the Poincaré’s result. In future, these results will be used as foundations for the study of squid locomotion, which is an emerging topic relevant to soft robotics.     

Statistics of Poincaré recurrences in local and global approaches

The basic statistical characteristics of the Poincaré recurrence sequence are obtained numerically for the logistic map in the chaotic regime. The mean values, variance and recurrence distribution density are calculated and their dependence on the return region size is analyzed. It is verified that the Afraimovich–Pesin dimension may be evaluated by the Kolmogorov–Sinai entropy. The peculiarities of the influence of noise on the recurrence statistics are studied in local and global approaches. It is shown that the obtained numerical data are in complete agreement with the theoretical results. It is demonstrated that the Poincaré recurrence theory can be applied to diagnose effects of stochastic resonance and chaos synchronization and to calculate the fractal dimension of a chaotic attractor.     

Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm

We construct geodesics in the Wasserstein space of probability measures along which all the measures have an upper bound on their density that is determined by the densities of the endpoints of the geodesic. Using these geodesics we show that a local Poincaré inequality and the measure contraction property follow from the Ricci curvature bounds defined by Sturm. We also show for a large class of convex functionals that a local Poincaré inequality is implied by the weak displacement convexity of the functional.     

The Geometry of Systems of Third Order Differential Equations Induced by Second Order Regular Lagrangians

A system of third order differential equations, whose coefficients do not depend explicitly on time, can be viewed as a third order vector field, which is called a semispray, and lives on the second order tangent bundle. We prove that a regular second order Lagrangian induces such a semispray, which is uniquely determined by two associated Poincaré-Cartan one-forms. To study the geometry of this semispray, we construct a horizontal distribution, which is a Lagrangian subbundle for an associated Poincaré-Cartan two-form. Using this semispray and the associated nonlinear connection we define dynamical covariant derivatives of first and second order. With respect to this, the second order dynamical derivative of the Lagrangian metric tensor vanishes.     

Multi-index Filtrations and Generalized Poincaré Series

A multi-index filtration on the ring of germs of functions can be described by its Poincaré series. We consider a finer invariant (or rather two invariants) of a multi-index filtration than the Poincaré series generalizing the last one. The construction is based on the fact that the Poincaré series can be written as a certain integral with respect to the Euler characteristic over the projectivization of the ring of functions. The generalization of the Poincaré series is defined as a similar integral with respect to the generalized Euler characteristic with values in the Grothendieck ring of varieties. For the filtration defined by orders of functions on the components of a plane curve singularity C and for the so called divisorial filtration for a modification of (\Bbb C²,0)({\Bbb C}^2,0) by a sequence of blowing-ups there are given formulae for this generalized Poincaré series in terms of an embedded resolution of the germ C or in terms of the modification respectively. The generalized Euler characteristic of the extended semigroup corresponding to the divisorial filtration is computed giving a curious “motivic version” of an A’Campo type formula.     

THE POINCARÉ INEQUALITY FOR A VECTOR FIELD WITH ZERO TANGENTIAL OR NORMAL COMPONENT ON THE BOUNDARY

ABSTRACT

Poincaré's inequality is well known: given a bounded domain G, ∥u∥_p ? c∥?u∥_p provided u(x) vanishes on the boundary ?G. The case where u(x) is a vector field u(x) that does not vanish on the boundary ?G is considered. It is shown that when either the tangential component or the normal component vanishes on the boundary ?G, then the Poincaré inequality is satisfied.     

Existence of limit cycles in general planar piecewise linear systems of saddle–saddle dynamics

The existence and number of limit cycles in a class of general planar piecewise linear systems constituted by two linear subsystems with saddle–saddle dynamics are investigated. Using the Liénard-like canonical form with seven parameters, the parametric regions of the existence of limit cycles are given by constructing proper Poincaré maps. In particular, the existence of at least two limit cycles is proved and some parameter regions where two nested limit cycles exist are given.     

Optimization of Poincaré sections for discriminating between stochastic and deterministic behavior of dynamical systems

This paper studies the problem of finding optimal parameters for a Poincaré section used for determining the type of behavior of a time series: a deterministic or stochastic one. To reach that goal optimization algorithms are coupled with the Poincaré & Higuchi (P&H) method, which calculates the Higuchi dimension using points obtained by performing a Poincaré section of a certain attractor. The P&H method generates distinctive patterns that can be used for determining if a given attractor is produced by a deterministic or a stochastic system, but this method is sensitive to the parameters of the Poincaré section. Patterns generated by the P&H method can be characterized using numerical measures which in turn can be used for finding such parameters for the Poincaré section for which the patterns produced by the P&H method are the most prominent. This paper studies several approaches to parameterization of the Poincaré section. Proposed approaches are tested on twelve time series, six produced by deterministic chaotic systems and six generated randomly. The obtained results show, that finding good parameters of the Poincaré section is important for determining the type of behavior of a time series. Among the tested methods the evolutionary algorithm was able to find the best Poincaré sections for use with the P&H method.     

Atiyah–Bott index on stratified manifolds

In this paper, the Poincaré isomorphism in K-theory on manifolds with edges is constructed. It is shown that the Poincaré isomorphism can be naturally constructed in terms of noncommutative geometry. More precisely, we obtain a correspondence between a manifold with edges and a noncommutative algebra and establish an isomorphism between the K-group of this algebra and the K-homology group of the manifold with edges, which is considered as a compact topological space.     

Self-improvement of Poincaré Type Inequalities Associated with Approximations of the Identity and Semigroups

The purpose of this paper is to present a general method that allows us to study self-improving properties of generalized Poincaré inequalities. When measuring the oscillation in a given cube, we replace the average by an approximation of the identity or a semigroup scaled to that cube and whose kernel decays fast enough. We apply the method to obtain self-improvement in the scale of Lebesgue spaces of Poincaré type inequalities. In particular, we propose some expanded Poincaré estimates that take into account the lack of localization of the approximation of the identity or the semigroup. As a consequence of this method we are able to obtain global pseudo-Poincaré inequalities.     

Multi-index Filtrations and Generalized Poincaré Series

A multi-index filtration on the ring of germs of functions can be described by its Poincaré series. We consider a finer invariant (or rather two invariants) of a multi-index filtration than the Poincaré series generalizing the last one. The construction is based on the fact that the Poincaré series can be written as a certain integral with respect to the Euler characteristic over the projectivization of the ring of functions. The generalization of the Poincaré series is defined as a similar integral with respect to the generalized Euler characteristic with values in the Grothendieck ring of varieties. For the filtration defined by orders of functions on the components of a plane curve singularity C and for the so called divisorial filtration for a modification of by a sequence of blowing-ups there are given formulae for this generalized Poincaré series in terms of an embedded resolution of the germ C or in terms of the modification respectively. The generalized Euler characteristic of the extended semigroup corresponding to the divisorial filtration is computed giving a curious “motivic version” of an A’Campo type formula. First two authors were partially supported by the grant MEC, PN I + D + i MTM2004-00958. Partially supported by the grants RFBR-04-01-00762, NSh-4719.2006.1 The author is thankful to the University of Valladolid for hospitality.     

Global periodic structure of integrable hyperbolic map

Integrable hyperbolic mappings are constructed within a scheme presented by Suris. The Cosh map is a singular map, of which fixed point is unstable. The global behavior of periodic orbits of the Sinh map is investigated referring to the Poincaré–Birkhoff resonance condition. Close to the fixed point, the periodicity is indeed determined from the Poincaré–Birkhoff resonance condition. Increasing the distance from the fixed point, the orbit is affected by the nonlinear effect and the average periodicity varies globally. The Fourier transformation of the individual orbits determines overall spectrum of global variation of the periodicity.     

Sporadic randomness,Maxwell's Demon and the Poincaré recurrence times

In the case of fully chaotic systems, the distribution of the Poincaré recurrence times is an exponential whose decay rate is the Kolmogorov–Sinai (KS) entropy. We address the discussion of the same problem, the connection between dynamics and thermodynamics, in the case of sporadic randomness, using the Manneville map as a prototype of this class of processes. We explore the possibility of relating the distribution of Poincaré recurrence times to “thermodynamics”, in the sense of the KS entropy, also in the case of an inverse power-law. This is the dynamic property that Zaslavsky [Physics Today 52 (8) (1999) 39] finds to be responsible for a striking deviation from ordinary statistical mechanics under the form of Maxwell's Demon effect. We show that this way of establishing a connection between thermodynamics and dynamics is valid only in the case of strong chaos, where both the sensitivity to initial conditions and the distribution of the Poincaré recurrence times are exponential. In the case of sporadic randomness, resulting at long times in the Lévy diffusion processes, the sensitivity to initial conditions is initially a power-law, but it becomes exponential again in the long-time scale, whereas the distribution of Poincaré recurrence times keeps, or gets, its inverse power-law nature forever, including the long-time scale where the sensitivity to initial condition becomes exponential. We show that a non-extensive version of thermodynamics would imply the Maxwell's Demon effect to be determined by memory, and thus to be temporary, in conflict with the dynamic approach to Lévy statistics. The adoption of heuristic arguments indicates that this effect is possible, as a form of genuine equilibrium, after completion of the process of memory erasure.     

Pro-p completions of Poincaré duality groups

We consider some sufficient conditions for the pro-p completion of an orientable Poincaré duality group of dimension n ≥ 3 to be a virtually pro-p Poincaré duality group of dimension at most n ? 2.     

Complex dynamics of a Holling type II prey–predator system with state feedback control

The complex dynamics of a Holling type II prey–predator system with impulsive state feedback control is studied in both theoretical and numerical ways. The sufficient conditions for the existence and stability of semi-trivial and positive periodic solutions are obtained by using the Poincaré map and the analogue of the Poincaré criterion. The qualitative analysis shows that the positive periodic solution bifurcates from the semi-trivial solution through a fold bifurcation. The bifurcation diagrams, Lyapunov exponents, and phase portraits are illustrated by an example, in which the chaotic solutions appear via a cascade of period-doubling bifurcations. The superiority of the state feedback control strategy is also discussed.