Monotonicity in the damped pendulum type equations

In this paper we investigate the monotonicity in the pendulum type equations with position dependent damping. We show that the system is strongly monotone under the overdamped condition. In the underdamped case, the Poincaré map P^T is strongly monotone in a forward invariant region provided the average of the external force is large enough. Combining the strong monotonicity with the dissipation property we show that the Poincaré map has in the cylindrical phase space an invariant circle, on which P^T is actually an orientation preserving circle homeomorphism. A series of consequences has then been obtained, including the existence and uniqueness of the average velocity. Furthermore, we discuss the smoothness of this invariant curve and the ergodicity of P^T on this curve. Supported by National Natural Science Foundation of China (10771155, 10571131) and Natural Science Foundation of Jiangsu Province (BK 2006046).     

Cotangent cohomology of rational surface singularities

In this paper we show that the number of generators of the cotangent cohomology groups T _Y ⁿ , n≥2, is the same for all rational surface singularities Y of fixed multiplicity. For a large class of rational surface singularities, including quotient singularities, this number is also the dimension. For them we obtain an explicit formula for the Poincaré series P _Y (t)=∑dim T ⁿ _Y ·t ⁿ . In the special case of the cone over the rational normal curve we give the multigraded Poincaré series. Oblatum: 18-XI-1998 & 25-III-1999 / Published online: 6 July 1999     

Remarks on p-cyclically monotone operators

ABSTRACT

In this paper, we deal with three aspects of p-cyclically monotone operators. First, we introduce a notion of monotone polar adapted for p-cyclically monotone operators and study these kinds of operators with a unique maximal extension (called pre-maximal), and with a convex graph. We then deal with linear operators and provide characterizations of p-cyclical monotonicity and maximal p-cyclical monotonicity. Finally, we show that the Brézis-Browder theorem preserves p-cyclical monotonicity in reflexive Banach spaces.     

Three results on the regularity of the curves that are invariant by an exact symplectic twist map

A theorem due to G. D. Birkhoff states that every essential curve which is invariant under a symplectic twist map of the annulus is the graph of a Lipschitz map. We prove: if the graph of a Lipschitz map h:T→R is invariant under a symplectic twist map, then h is a little bit more regular than simply Lipschitz (Theorem 1); we deduce that there exists a Lipschitz map h:T→R whose graph is invariant under no symplectic twist map (Corollary 2). Assuming that the dynamic of a twist map restricted to a Lipschitz graph is bi-Lipschitz conjugate to a rotation, we obtain that the graph is even C ¹ (Theorem 3). Then we consider the case of the C ⁰ integrable symplectic twist maps and we prove that for such a map, there exists a dense G _δ subset of the set of its invariant curves such that every curve of this G _δ subset is C ¹ (Theorem 4).     

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.     

Mapping Surgery to Analysis I: Analytic Signatures

We develop the theory of analytically controlled Poincaré complexes over C ^*-algebras. We associate a signature in C ^*-algebra K-theory to such a complex, and we show that it is invariant under bordism and homotopy. The authors were supported in part by NSF Grant DMS-0100464.     

Properties of a quasi-uniformly monotone operator and its application to the electromagnetic p-curl systems

In this paper we propose a new concept of quasi-uniform monotonicity weaker than the uniform monotonicity which has been developed in the study of nonlinear operator equation Au = b. We prove that if A is a quasi-uniformly monotone and hemi-continuous operator, then A^?1 is strictly monotone, bounded and continuous, and thus the Galerkin approximations converge. Also we show an application of a quasi-uniformly monotone and hemi-continuous operator to the proof of the well-posedness and convergence of Galerkin approximations to the solution of steady-state electromagnetic p-curl systems.

     

Notes on the difference of two monotone operators

In this note, we present some results on maximality of the difference of two monotone operators. It is shown that, for two multifunctions S and T from X to X ^*, maximal monotonicity of S and monotonicity of both T and S?T imply maximal monotonicity of S?T.     

Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized Hénon maps

We study bifurcations of periodic orbits in two parameter general unfoldings of a certain type homoclinic tangency (called a generalized homoclinic tangency) to a saddle fixed point. We apply the rescaling technique to first return (Poincaré) maps and show that the rescaled maps can be brought to so-called generalized Hénon maps which have non-degenerate bifurcations of fixed points including those with multipliers e ^±iϕ . On the basis of this, we prove the existence of infinite cascades of periodic sinks and periodic stable invariant circles.      

A computer-assisted proof of chaos in Josephson junctions

This paper presents a rigorous verification of chaos in the RCLSJ model for studying dynamics of the Josephson junction. By carefully picking a suitable cross-section with respect to the attractor, it is shown that for the corresponding Poincaré map P obtained in terms of second return time, there exists a closed invariant set Λ in this cross-section such that P∣Λ is semi-conjugate to a 2-shift map, thus showing existence of chaos in the RCLSJ model.     

Unfolding the torus: Oscillator geometry from time delays

Summary We present a simple method of plotting the trajectories of systems of weakly coupled oscillators. Our algorithm uses the time delays between the “firings” of the oscillators. For any system ofn weakly coupled oscillators there is an attracting invariantn-dimensional torus, and the attractor is a subset of this invariant torus. The invariant torus intersects a suitable codimension-1 surface of section at an (n−1)-dimensional torus. The dynamics ofn coupled oscillators can thus be reduced,in principle, to the study of Poincaré maps of the (n−1)-dimensional torus. This paper gives apractical algorithm for measuring then−1 angles on the torus. Since visualization of 3 (or higher) dimensional data is difficult we concentrate onn=3 oscillators. For three oscillators, a standard projection of the Poincaré map onto the plane yields a projection of the 2-torus which is 4-to-1 over most of the torus, making it difficult to observe the structure of the attractor. Our algorithm allows a direct measurement of the 2 angles on the torus, so we can plot a 1-to-1 map from the invariant torus to the “unfolded torus” where opposite edges of a square are identified. In the cases where the attractor is a torus knot, the knot type of the attractor is obvious in our projection.     

Global Phase Portraits of Quadratic Systems with a Complex Ellipse as Invariant Algebraic Curve

In this paper, we study a new class of quadratic systems and classify all its phase portraits.More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse x~2+ y~2+ 1 = 0 as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincar′e disc.     

Discrete isoperimetric and Poincaré-type inequalities

We study some discrete isoperimetric and Poincaré-type inequalities for product probability measures μ ⁿ on the discrete cube {0, 1} ⁿ and on the lattice Z ⁿ . In particular we prove sharp lower estimates for the product measures of boundaries of arbitrary sets in the discrete cube. More generally, we characterize those probability distributions μ on Z which satisfy these inequalities on Z ⁿ . The class of these distributions can be described by a certain class of monotone transforms of the two-sided exponential measure. A similar characterization of distributions on R which satisfy Poincaré inequalities on the class of convex functions is proved in terms of variances of suprema of linear processes. Received: 30 April 1997 / Revised version: 5 June 1998     

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>].     

Chaotic Motion Generated by Delayed Negative Feedback Part II: Construction of Nonlinearities

We construct a smooth function g* : IR ? IR with such that the equation has a slowly oscillating periodic solution y, and a slowly oscillating solution z* whose phase curve is homoclinic with respect to the orbit o of y in the space C = C⁰([1,0],IR). For an associated Poincaré map we obtain a transversal homoclinic loop. The proof of transversality employs a criterion which uses oscillation properties of solutions of variational equations. The main result is that the trajectories (ψn)^∞_-∞ of the Poincaré map in a neighbourhood of the homoclinic loop form a hyperbolic set on which the motion is chaotic.     

Preserving maximal monotonicity with applications in sum and composition rules

In this paper, we study maximal monotonicity preserving mappings on the Banach space X × X ^*. Indeed, for a maximal monotone set ${M \subset X\times X^*}$ and for a multifunction ${T: X \times X^* \multimap Y \times Y^*}$ , under some sufficient conditions on M and T we show that T(M) is maximal monotone. As two consequences of this result we get sum and composition rules for maximal monotone operators.     

Approximating Solutions of Maximal Monotone Operators in Hilbert Spaces

Let H be a real Hilbert space and let T: H→2^H be a maximal monotone operator. In this paper, we first introduce two algorithms of approximating solutions of maximal monotone operators. One of them is to generate a strongly convergent sequence with limit vT⁻¹0. The other is to discuss the weak convergence of the proximal point algorithm. Next, using these results, we consider the problem of finding a minimizer of a convex function. Our methods are motivated by Halpern's iteration and Mann's iteration.     

Ornstein�CUhlenbeck equations with time-dependent coefficients and L��vy noise in finite and infinite dimensions

We solve a time-dependent linear SPDE with additive Lévy noise in the mild and weak sense. Existence of a generalized invariant measure for the associated transition semigroup is established and the generator is studied on the corresponding L ²-space. The square field operator is characterized, allowing to derive a Poincaré and a Harnack inequality.     

Generalized Invexity and Generalized Invariant Monotonicity

In this paper, several kinds of invariant monotone maps and generalized invariant monotone maps are introduced. Some examples are given which show that invariant monotonicity and generalized invariant monotonicity are proper generalizations of monotonicity and generalized monotonicity. Relationships between generalized invariant monotonicity and generalized invexity are established. Our results are generalizations of those presented by Karamardian and Schaible.