Global phase portraits of planar piecewise linear refracting systems of saddle–saddle type

This paper deal with the global dynamics of planar piecewise linear refracting systems of saddle–saddle type with a straight line of separation. We investigate the singularities, limit cycles, homoclinic orbits, heteroclinic orbits and make the classification of global phase portraits in the Poincaré disk for the refracting systems. We prove that these systems have 18 topologically different global phase portraits.     

Control of chaos in a piecewise smooth nonlinear system

This paper shows the stabilization of the unstable periodic orbit of any given piecewise smooth system with linear and/or nonlinear characteristics. By utilizing the periodicity of the switching action, we construct the Poincaré mapping including all information of the original system. This mapping offers a first step toward extending a novel technique for controlling chaos based on the appropriate state feedback in piecewise smooth nonlinear systems. We also apply this approach to Rayleigh type oscillator described by the piecewise smooth nonlinear systems.     

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.     

Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line

This paper provides the classification of the phase portraits in the Poincaré disc of all piecewise linear continuous differential systems with two zones separated by a straight line having a unique finite singular point which is a node or a focus. The sufficient and necessary conditions for existence and uniqueness of limit cycles are also given.     

The planar discontinuous piecewise linear refracting systems have at most one limit cycle

In this paper we investigate the limit cycles of planar piecewise linear differential systems with two zones separated by a straight line. It is well known that when these systems are continuous they can exhibit at most one limit cycle, while when they are discontinuous the question about maximum number of limit cycles that they can exhibit is still open. For these last systems there are examples exhibiting three limit cycles.The aim of this paper is to study the number of limit cycles for a special kind of planar discontinuous piecewise linear differential systems with two zones separated by a straight line which are known as refracting systems. First we obtain the existence and uniqueness of limit cycles for refracting systems of focus-node type. Second we prove that refracting systems of focus–focus type have at most one limit cycle, thus we give a positive answer to a conjecture on the uniqueness of limit cycle stated by Freire, Ponce and Torres in Freire et al. (2013). These two results complete the proof that any refracting system has at most one limit cycle.     

The Poincaré-Bertrand Formula for the Bochner-Martinelli Integral

The Poincaré-Bertrand formula and the composition formula for the Bochner-Martinelli integral on piecewise smooth manifolds are obtained. As an application, the regularization problem for linear singular integral equation with Bochner-Martinelli kernel and variable coefficients is discussed.     

Factor complexity of S-adic words generated by the Arnoux–Rauzy–Poincaré algorithm

The Arnoux–Rauzy–Poincaré multidimensional continued fraction algorithm is obtained by combining the Arnoux–Rauzy and Poincaré algorithms. It is a generalized Euclidean algorithm. Its three-dimensional linear version consists in subtracting the sum of the two smallest entries from the largest if possible (Arnoux–Rauzy step), and otherwise, in subtracting the smallest entry from the median and the median from the largest (the Poincaré step), and by performing when possible Arnoux–Rauzy steps in priority. After renormalization it provides a piecewise fractional map of the standard 2-simplex. We study here the factor complexity of its associated symbolic dynamical system, defined as an S-adic system. It is made of infinite words generated by the composition of sequences of finitely many substitutions, together with some restrictions concerning the allowed sequences of substitutions expressed in terms of a regular language. Here, the substitutions are provided by the matrices of the linear version of the algorithm. We give an upper bound for the linear growth of the factor complexity. We then deduce the convergence of the associated algorithm by unique ergodicity.     

Existence of periodic orbits and chaos in a class of three-dimensional piecewise linear systems with two virtual stable node-foci

In this paper, we consider a new class of piecewise linear (PWL) systems with two virtual stable node-foci (the meaning of “virtual” is from Bernardo et al. (2008)) which exhibits periodic orbits and chaos. This fact that PWL systems have no unstable equilibria but has chaos will unavoidably make the exploration of this chaos more complicated. Particular values for bifurcation diagram are provided. Based on mathematical analysis and Poincaré map, periodic orbits of this kind of system without unstable equilibrium points are derived, the corresponding existence theorems are given, and the obtained results are applied to specific examples.     

Limit cycles and global dynamics of planar piecewise linear refracting systems of focus–focus type

The limit cycle problem of planar piecewise linear refracting systems has been solved except the focus–focus (FF) type. Here we concern this remaining FF type, and prove that such systems have at most one limit cycle and have 14 topologically different global phase portraits.     

Chaotic behavior analysis based on sliding bifurcations

In this paper, a mathematical analysis of a possible way to chaos for bounded piecewise smooth systems of dimension 3 submitted to one of its specific bifurcations, namely the sliding ones, is proposed. This study is based on period doubling method applied to the relied Poincaré maps.     

Pseudochaotic dynamics near global periodicity

In this paper, we study a piecewise linear version of kicked oscillator model: saw-tooth map. A special case of global periodicity, in which every phase point belongs to a periodic orbit, is presented. With few analytic results known for the corresponding map on torus, we numerically investigate transport properties and statistical behavior of Poincaré recurrence time in two cases of deviation from global periodicity. A non-KAM behavior of the system, as well as subdiffusion and superdiffusion, are observed through numerical simulations. Statistics of Poincaré recurrences shows Kac lemma is valid in the system and there is a relation between the transport exponent and the Poincaré recurrence exponent. We also perform careful numerical computation of capacity, information and correlation dimensions of the so-called exceptional set in both cases. Our results show that the fractal dimension of the exceptional set is strictly less than 2 and that the fractal structures are unifractal rather than multifractal.     

Isomonodromic confluence of singular points

We consider the problem of confluence of singular points under isomonodromic deformations of linear systems. We prove that a system with irregular singular points is a result of isomonodromic confluence of singular points with minimal Poincaré ranks, i.e., of singular points whose Poincaré rank does not decrease under gauge transformations.     

Intersection homology with general perversities

We study intersection homology with general perversities that assign integers to stratum components with none of the classical constraints of Goresky and MacPherson. We extend Goresky and MacPherson’s axiomatic treatment of Deligne sheaves, and use these to obtain Poincaré and Lefschetz duality results for these general perversities. We also produce versions of both the sheaf-theoretic and the piecewise linear chain-theoretic intersection pairings that carry no restrictions on the input perversities.     

A shadowing lemma for quasi-hyperbolic strings of flows

In this paper we prove a shadowing lemma for pseudo orbits made by quasi-hyperbolic strings. We allow singularities in question and hence, in particular, the quasi-hyperbolic strings are formulated by the rescaled linear Poincaré flow instead of the usual linear Poincaré flow. We also introduce the sectional Poincaré map and rescaled sectional Poincaré map for Lipschitz vector fields on Banach spaces in the article.     

Polynomial first integrals via the Poincaré series

In (Appl. Math. 28 (2001) 17) a method to compute the Poincaré–Liapunov constants for an arbitrary analytic differential system which has a linear center at the origin in function of the coefficients of the system was given. This method also computes the coefficients of the Poincaré series in function of the same coefficients. In this work we use this method to determine polynomial differential systems which have a polynomial first integral.     

General laws of the analytic linearization for random diffeomorphisms

We show that random dynamical systems obey the similar laws of the linear analytic linearization as non-autonomous difference systems, which naturally cover the validity and invalidity of Poincaré and Seigel type theorems for random diffeomorphisms, respectively.     

The several transformation formula in several complex variables and its applications

In this paper, by the method of global analysis, the authors give a new global integral transformation formula and obtain the Plemelj formula with Hadamard principal value of higher-order partial derivatives for the integral of Bochner-Martinelli type on a closed piecewise smooth orientable manifold C ⁿ . Moreover, the authors obtain the composition formula, Poincaré-Bertrand extended formula of the corresponding singular integral. As the application of some results, the authors also study a higher-order Cauchy boundary problem and a regularization problem of higher-order linear complex differential singular integral equation with variable coefficients.     

The construction of Poincaré-Chetayevand Smale bifurcation diagrams for conservative non-holonomic systems with symmetry

The problem of the existence of first integrals which are linear functions of the generalized velocities (momenta and quasi-velocities) is discussed for conservative non-holonomic Chaplygin systems with symmetry, as well as methods for investigating the existence, stability, and bifurcation of the steady motions of such systems. These methods are based on the classical methods of Routh-Salvadori, Poincaré-Chetayev, and Smale, but unlike the latter they do not require a knowledge of the explicit form of the linear integrals. The general conclusions are illustrated by the example of the problem of an ellipsoid of revolution moving on an absolutely rough horizontal surface. It is shown how in this case numerical techniques can be used to construct the Poincaré-Chetayev diagram — a surface in the space of generalized coordinates and constants of linear first integrals corresponding to motions in which the velocities of the non-cyclic coordinates vanish, while those of the cyclic coordinates are constant, and the Smale diagram — a surface in the space of constants of linear first integrals and the energy integral corresponding to these motions.     

Heteroclinic bifurcation in a class of planar piecewise smooth systems with multiple zones

We discuss heteroclinic bifurcation in a class of periodically excited planar piecewise smooth systems with discontinuities on finitely many smooth curves intersecting at the origin. Assume that the unperturbed system has a hyperbolic saddle in each subregion, and those saddles are connected by a heteroclinic cycle that crosses every switching curve transversally exactly once. We present a method of Melnikov type to derive sufficient conditions under which the perturbed stable and unstable manifolds intersect transversally. Such transversal intersections imply that the corresponding Poincaré map has a transverse heteroclinic cycle. As applications, we present examples with 2 and 4 switching curves respectively. Our numerical simulations suggest that such transversal intersections result in the appearance of chaotic motions in those example systems.     

Poincaré线性连续统直觉概念的公式化(英文)

本文研究了Poincaré著名注记中“内束”观念的数学表述法问题,通过构建Poincaré连续统模型,得到了这一问题的一种解答。文中还论述了有关数理哲学及方法论问题;文末特别指出了须继续研究的数学问题。