We consider existence and stability of an almost periodic solution of the quasilinear system of differential equations with piecewise constant argument of generalized type. The associated linear homogeneous system satisfies exponential dichotomy. The deviations of the argument are not restricted by any sign assumption when existence is considered. The problem of positive (almost periodic) solutions of the logistic equation is discussed as an example.
A new technique of investigation of equations with piecewise argument, based on integral representation, is developed.
We consider a new model for shunting inhibitory cellular neural networks, retarded functional differential equations with piecewise constant argument. The existence and exponential stability of almost periodic solutions are investigated. An illustrative example is provided.
In this paper we introduce a general type of differential equations with piecewise constant argument (EPCAG), and consider the problem of backward continuation of solutions. We establish the existence of global integral manifolds of quasilinear EPCAG, consisting of infinitely back continued solutions, while the solutions starting outside the invariant sets may not be back continued. The smoothness of the manifolds is investigated. The existence of bounded and periodic solutions is considered. A new technique of investigation of equations with piecewise argument, based on an integral representa…
This work deals with the existence of an almost periodic solution for certain kind of differential equations with generalized piecewise constant argument, almost periodic coefficients which are seen as a perturbation of a linear equation of that kind satisfying an exponential dichotomy on a difference equation. The stability of that solution in a semi-axis studied.
On existence, uniqueness and numerical approximation of impulsive differential equations with adaptive state-dependent delays using equations with piecewise-constant arguments
In this paper, the general perturbation problem of piecewise smooth integrable differential systems with two switching planes is considered. Firstly, when the unperturbed system has a family of periodic orbits, the first order Melnikov vector function is derived which can be used to study the number of periodic orbits bifurcated from the period annuli. Then, by using the obtained Melnikov vector function, we get an upper bound of the number of periodic orbits of a concrete $n$-dimensional piecewise smooth differential system.
In this paper, we introduce some concepts of square-mean S-asymptotically $ω$-periodic stochastic processes. Using the stochastic analysis method and the Banach contraction mapping principle, we establish the existence and uniqueness results of the mild solution and the square-mean S-asymptotically $ω$-periodic solution for a semilinear nonautonomous stochastic fractional evolution equation driven by Lévy noise.
The Conley theory has a tool to guarantee the existence of periodic trajectories in isolating neighborhoods of semi-dynamical systems. We prove that the positive trajectories generated by a piecewise-smooth vector field $Z=(X, Y)$ defined in a closed manifold of three dimensions without the scape region produces a semi-dynamical system. Thus, we have built a semiflow that allows us to apply the classical Conley theory. Furthermore, we use it to guarantee the existence of periodic orbits in this class of piecewise-smooth vector fields.
We prove that for continuous maps on the interval, the existence of an n-cycle, implies the existence of n-1 points which interwind the original ones and are permuted by the map. We then use this combinatorial result to show that piecewise affine maps (with no zero slope) cannot be infinitely renormalizable.
For flows, the singular cycles connecting saddle periodic orbit and saddle equilibrium can poten- tially result in the so-called singular horseshoe, which means the existence of a non-uniformly hyperbolic chaotic invariant set. However, it is very hard to find a specific dynamical system that exhibits such singular cycles in general. In this paper, the existence of the singular cycles involved in saddle periodic orbits is studied by two types of piecewise affine systems: one is the piecewise affine system having an admissible saddle point with only real eigenvalues and an admissible saddle per…
By varying a parameter of a one-dimensional piecewise smooth map, stable periodic orbits are observed. In this paper, complete analytic characterization of these stable periodic orbits is obtained. An interesting relationship between the cardinality of orbits and their period is established. It is proved that for any $n$, there exist $ϕ(n)$ distinct admissible patterns of cardinality $n$. An algorithm to obtain these distinct admissible patterns is outlined. Additionally, a novel algorithm to find the range of parameter for which the orbit exists is proposed.
In this paper, we study the number of limit cycles that can bifurcating from a periodic annulus in discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines. We prove that if the central subsystem, i.e. the system defined between the two parallel lines, has a real center and the others subsystems have centers or saddles, then we have at least three limit cycles that appear after perturbations of periodic annulus. For this, we study the number of zeros of a Melnikov function for piecewise Hamiltonian system and present a norm…
We study periodic, piecewise linear maps on the plane starting with the Mort Brown's map. We show that if the number of pieces is two, there is only a short list of possible periods (this fact can be seen as the crystallographic restriction for this class of maps). Otherwise, without the restriction on the number of pieces, a map can have any period. We show how to construct such maps using binary trees and so called admissible sequences.
In this paper, we first show that any nonlinear monotonic increasing contracting maps with one discontinuous point on a unit interval which has an unique periodic point with period $n$ conjugates to a piecewise linear contracting map which has periodic point with same period. Second, we consider one parameter family of monotonic increasing contracting maps, and show that the family has the periodic structure called Arnold tongue for the parameter which is associated with the Farey series. This implies that there exist a parameter set with a positive Lebesgue measure such that the map has a per…
In this paper we introduce a new type of differential equations with piecewise constant argument (EPCAG), more general than EPCA. The Reduction Principle is proved for EPCAG. The structure of the set of solutions is specified. We establish also the existence of global integral manifolds of quasilinear EPCAG in the so called critical case and investigate the stability of the zero solution.
We consider the dynamics of `nonlinear tent maps': piecewise smooth unimodal maps with nowhere vanishing derivative. We show that if a nonlinear tent map $f$ is not infinitely renormalizable, then all its periodic orbits of sufficiently high period are hyperbolic repelling. If additionally all periodic orbits of $f$ are hyperbolic, then $f$ has at most finitely many periodic attractors and there is a hyperbolic expansion outside the basins of these periodic attractors. In particular, if a nonlinear tent map $f$ is not infinitely renormalizable and all its periodic orbits are hyperbolic repelli…
In this paper, we study the number of limit cycles that can bifurcate from a periodic annulus in discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines, such that the linear differential systems that define the piecewise one have a center and two saddles. That is, the linear differential system in the region between the two parallel lines (i.e. the central subsystem) has a center and the others subsystems have saddles. We prove that if the central subsystem has a real or a boundary center, then we have at least six limit …
Motivated by problems coming from different areas of the applied science we study the periodic solutions of the following differential system $$x'(t)=F_0(t,x)+\varepsilon F_1(t,x)+\varepsilon^2 R(t,x,\varepsilon),$$ when $F_0$, $F_1$, and $R$ are discontinuous piecewise functions, and $\varepsilon$ is a small parameter. It is assumed that the manifold $\mathbb{Z}$ of all periodic solutions of the unperturbed system $x'=F_0(t,x)$ has dimension $n$ or smaller then $n$. The averaging theory is one of the best tools to attack this problem. This theory is completely developed when $F_0$, $F_1$ and …
We establish an equivalence between infinitely many asymptotically stable periodic solutions and subsumed homoclinic connections for $N$-dimensional piecewise-linear continuous maps. These features arise as a codimension-three phenomenon. The periodic solutions are single-round: they each involve one excursion away from a central saddle-type periodic solution. The homoclinic connection is subsumed in the sense that one branch of the unstable manifold of the saddle solution is contained entirely within its stable manifold. The results are proved by using exact expressions for the periodic solut…
We consider the iterates of a generic injective piecewise contraction of the interval defined by a finite family of contractions. Let $ϕ_i:[0,1]\to (0,1)$, $1\le i\le n$, be $C^2$-diffeomorphisms with $\sup_{x\in (0,1)} \vert Dϕ_i(x)\vert<1$ whose images $ϕ_1([0,1]), \ldots, ϕ_n([0,1])$ are pairwise disjoint. Let $0<x_1<\cdots<x_{n-1}<1$ and let $I_1,\ldots, I_n$ be a partition of the interval $[0,1)$ into subintervals $I_i$ having interior $(x_{i-1},x_i)$, where $x_0=0$ and $x_n=1$. Let $f_{x_1,\ldots,x_{n-1}}$ be the map given by $x\mapsto ϕ_i(x)$ if $x\in I_i$, for $1\le i\le n$. Among othe…
We consider piecewise $C^2$ non-flat maps of the interval and show that, for Lebesgue almost every point, its omega-limit set is either a periodic orbit, a cycle of intervals or the closure of the orbits of a subset of the critical points. In particular, every piecewise $C^2$ non-flat map of the interval displays only a finite number of non-periodic attractors.
Piecewise smooth maps are known to exhibit a wide range of dynamical features including numerous types of periodic orbits. Predicting regions in parameter space where such periodic orbits might occur and determining their stability is crucial to characterize the dynamics of the system. However, obtaining the conditions of existence and stability of these periodic orbits generally use brute force methods which require successive application of the iterative map on a starting point. In this article, we propose a faster and more elegant way of obtaining those conditions without iterating the comp…
Based on the results about the invariant cones appeared in the literature this paper analyses the existence of periodic orbits in three-dimensional continuous piecewise linear homogeneous systems with two zones, and a necessary and sufficient condition for the existence of periodic orbits of such systems is given.
We consider a class of discontinuous piecewise linear differential systems in $\mathbb{R}^3$ with two pieces separated by a plane. In this class we show that there exist differential systems having: a unique limit cycle, a unique one-parameter family of periodic orbits, scrolls, invariant cylinders foliated by orbits which can be periodic or no.
Let $ϕ_1,\ldots,ϕ_n:[0,1]\to (0,1)$ be Lipschitz contractions. Let $I=[0,1)$, $x_0=0$ and $x_n=1$. We prove that for Lebesgue almost every $(x_1,...,x_{n-1})$ satisfying $0<x_1<\cdots <x_{n-1}<1$, the piecewise contraction $f:I\to I$ defined by $x\in [x_{i-1},x_i)\mapsto ϕ_i(x)$ is asymptotically periodic. More precisely, $f$ has at least one and at most $n$ periodic orbits and the $ω$-limit set $ω_f(x)$ is a periodic orbit of $f$ for every $x\in I$.
The averaging theory has been extensively employed for studying periodic solutions of smooth and nonsmooth differential systems. Here, we extend the averaging theory for studying periodic solutions a class of regularly perturbed non-autonomous $n$-dimensional discontinuous piecewise smooth differential system. As a fundamental hypothesis, it is assumed that the unperturbed system has a manifold $\mathcal{Z}\subset\mathbb{R}^n$ of periodic solutions satisfying $\dim(\mathcal{Z})<n.$ Then, we apply this result to study limit cycles bifurcating from periodic solutions of linear differential syste…
Existence and uniqueness of Remotely Almost Periodic solutions of differential equations with piecewise constant argument
Facultad de Ingeniería, Universidad de la República Oriental del Uruguay（乌拉圭东岸共和国大学工程学院）；Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción（康塞普西翁大学理学院数学工程系）
For piecewise-linear maps, the phenomenon that a branch of a one-dimensional unstable manifold of a periodic solution is completely contained in its stable manifold is codimension-two. Unlike codimension-one homoclinic corners, such `subsumed' homoclinic connections can be associated with stable periodic solutions. The purpose of this paper is to determine the dynamics near a generic subsumed homoclinic connection in two dimensions. Assuming the eigenvalues associated with the periodic solution satisfy $0 < |λ| < 1 < σ< \frac{1}{|λ|}$, in a two-parameter unfolding there exists an infinite sequ…
In this paper, we investigate a class of nonlinear impulsive stochastic differential evolution equations with infinite delay in Banach space. Based on the Krasnoselskii's fixed point theorem, sufficient conditions of the existence of the square mean piecewise almost periodic solutions to this type of equations are derived. Moreover, the exponential stability of the square mean piecewise almost periodic solution is investigated.
Let $-1<λ<1$ and $f:[0,1)\to\mathbb{R}$ be a piecewise $λ$-affine map, that is, there exist points $0=c_0<c_1<\cdots <c_{n-1}<c_n=1$ and real numbers $b_1,\ldots,b_n$ such that $f(x)=λx+b_i$ for every $x\in [c_{i-1},c_i)$. We prove that, for Lebesgue almost every $δ\in\mathbb{R}$, the map $f_δ=f+δ\,({\rm mod}\,1)$ is asymptotically periodic. More precisely, $f_δ$ has at most $2n$ periodic orbits and the $ω$-limit set of every $x\in [0,1)$ is a periodic orbit.
For any transitive piecewise monotonic map for which the set of periodic measures is dense in the set of ergodic invariant measures (such as monotonic mod one transformations and piecewise monotonic maps with two monotonic pieces), we show that the set of points for which the Birkhoff average of a continuous function does not exist (called the irregular set) is either empty or has full topological entropy. This generalizes Thompson's theorem for irregular sets of $β$-transformations, and reduces a complete description of irregular sets of transitive piecewise monotonic maps to Hofbauer-Raith p…
Planar piecewise linear systems with two linearity zones separated by a straight line and with a periodic orbit at infinity are considered. By using some changes of variables and parameters, a reduced canonical form with five parameters is obtained. Instead of the usual Bendixson transformation to work near infinity, a more direct approach is introduced by taking suitable coordinates for the crossing points of the possible periodic orbits with the separation straight line. The required computations to characterize the stability and bifurcations of the periodic orbit at infinity are much easier…
We determine periodic and aperiodic points of certain piecewise affine maps in the Euclidean plane. Using these maps, we prove for $λ\in\{\frac{\pm1\pm\sqrt5}2,\pm\sqrt2,\pm\sqrt3\}$ that all integer sequences $(a_k)_{k\in\mathbb Z}$ satisfying $0\le a_{k-1}+λa_k+a_{k+1}<1$ are periodic.
In this paper we consider a one dimensional liner piecewise-smooth discontinuous map. It is well known that stable periodic orbits exist in this type of map for a specific parameter region. It is also known that the corresponding bifurcation phenomena termed as "period adding bifurcation" exhibits a special structure for such case. In last couple of years, various authors have analyzed this structure using border collision bifurcation (BCB) curves and given characterization for various parameter regions. In this paper we have analyzed a specific parameter space using a different method. We sho…
Resonance tongues are mode-locking regions of parameter space in which stable periodic solutions occur; they commonly occur, for example, near Neimark-Sacker bifurcations. For piecewise-smooth, continuous maps these tongues typically have a distinctive lens-chain (or sausage) shape in two-parameter bifurcation diagrams. We give a symbolic description of a class of "rotational" periodic solutions that display lens-chain structures for a general $N$-dimensional map. We then unfold the codimension-two, shrinking point bifurcation, where the tongues have zero width. A number of codimension-one bif…
The existence and bifurcation of homoclinic orbits in planar piecewise linear homogeneous systems with two regions separated by a discontinuity boundary are investigated in this paper. In addition, existence of periodic orbits and stability of the origin are also discussed without the assumption of continuity for the planar piecewise linear homogeneous systems.
We consider a network of identical piecewise smooth systems that synchronizes on the manifold given by a periodic orbit of a single agent. We explicitly characterize the fundamental matrix solution of the network along the synchronous solution and extend the Master Stability Function tool to the present case of non-smooth dynamics.
Let $I_1=[a_0,a_1),\ldots,I_{k}= [a_{k-1},a_k)$ be a partition of the interval $I=[0,1)$ into $k$ subintervals. Let $f:I\to I$ be a map such that each restriction $f|_{I_i}$ is an increasing Lipschitz contraction. We prove that any $f$ admits at most $k$ periodic orbits, where the upper bound is sharp. We are also interested in the dynamics of piecewise linear $λ$-affine maps, where $0<λ<1$. Let $b_1,\ldots,b_k$ be real numbers and let $F_λ: I\to \mathbb{R}$ be a function such that each restriction $F_λ|_{I_i}(x)=λx +b_i$. Under a generic assumption on the parameters $a_1,\ldots,a_{k-1},b_1,\l…
We study the existence of periodic solutions in a class of planar Filippov systems obtained from non-autonomous periodic perturbations of reversible piecewise smooth differential systems. It is assumed that the unperturbed system presents a simple two-fold cycle, which is characterized by a closed trajectory connecting a visible two-fold singularity to itself. It is shown that under certain generic conditions the perturbed system has sliding and crossing periodic solutions. In order to get our results, Melnikov's ideas were applied together with tools from the geometric singular perturbation t…
In this work we consider a two-dimensional piecewise smooth system, defined in two domains separated by the switching manifold $x=0$. We assume that there exists a piecewise-defined continuous Hamiltonian that is a first integral of the system. We also suppose that the system possesses an invisible fold-fold at the origin and two heteroclinic orbits connecting two hyperbolic critical points on either side of $x=0$. Finally, we assume that the region closed by these heteroclinic connections is fully covered by periodic orbits surrounding the origin, whose periods monotonically increase as they …
On the periodic orbits of C0-typical impulsive semiflows
Departamento de Matematica da Univesidade Aberta（开放大学数学系）；Departamento de Matemática, Instituto de Matemática, Universidade Federal do Rio de Janeiro（里约热内卢联邦大学数学研究所数学系）；CMAT（数学中心）；Departamento de Matemática, Universidade do Minho（米尼奥大学数学系）；CIDMA（阿威罗数学与几何中心）；Departamento de Matemática da Universidade de Aveiro（阿威罗大学数学系）
In this paper some piecewise smooth perturbations of a three-dimensional differential system are considered. The existence of invariant manifolds filled by periodic orbits is obtained after suitable small perturbations of the original differential system. These manifolds emerge from a continuum of cylinders of $\mathbb{R}^3$ which does exist for the piecewise smooth differential systems after a rotation of some planar algebraic polynomial curves. The main tool used in order to obtain the results is the averaging theory for piecewise smooth differential systems.
In this paper the existence and unicity of a stable periodic orbit is proven, for a class of piecewise affine differential equations in dimension 3 or more, provided their interaction structure is a negative feedback loop. It is also shown that the same systems converge toward a unique stable equilibrium point in dimension 2. This extends a theorem of Snoussi, which showed the existence of these orbits only. The considered class of equations is usually studied as a model of gene regulatory networks. It is not assumed that all decay rates are identical, which is biologically irrelevant, but has…
We give a condition on a piecewise constant roof function and an irrational rotation by $α$ on the circle to give rise to a special flow having the mild mixing property. Such flows will also satisfy Ratner's property. As a consequence we obtain a class of mildly mixing singular flows on the two-torus that arise from quasi-periodic Hamiltonians flows by velocity changes.
We consider piecewise expanding maps of the interval with finitely many branches of monotonicity and show that they are generically combinatorially stable, i.e., the number of ergodic attractors and their corresponding mixing periods do not change under small perturbations of the map. Our methods provide a topological description of the attractor and, in particular, give an elementary proof of the density of periodic orbits.
Consider a multimodal interval map $f$ of $C^3$ with non-flat critical points. We establish several characterizations of the map $f$ is quasi-symmetrically conjugated to a piecewise affine map in the case $f$ is topologically exact and all of its periodic points are hyperbolic repelling. In particular, we give a negative answer to a question posted by Henk Bruin in \cite{Bru96}.
The mode-locking regions of a dynamical system are the subsets of the parameter space of the system within which there exists an attracting periodic solution. For piecewise-linear continuous maps, these regions have a curious chain structure with points of zero width called shrinking points. In this paper we perform a local analysis about an arbitrary shrinking point. This is achieved by studying the symbolic itineraries of periodic solutions in nearby mode-locking regions and performing an asymptotic analysis on one-dimensional slow manifolds in order to build a comprehensive theoretical fram…
