On Hopf bifurcation in non-smooth planar systems

As we know, for non-smooth planar systems there are foci of three different types, called focus-focus (FF), focus-parabolic (FP) and parabolic-parabolic (PP) type respectively. The Poincaré map with its analytical property and the problem of Hopf bifurcation have been studied in Coll et al. (2001) [3] and Filippov (1988) [6] for general systems and in Zou et al. (2006) [13] for piecewise linear systems. In this paper we also study the problem of Hopf bifurcation for non-smooth planar systems, obtaining new results. More precisely, we prove that one or two limit cycles can be produced from an elementary focus of the least order (order 1 for foci of FF or FP type and order 2 for foci of PP type) (Theorem 2.3), different from the case of smooth systems. For piecewise linear systems we prove that 2 limit cycles can appear near a focus of either FF, FP or PP type (Theorem 3.3).     

On the period of the limit cycles appearing in one-parameter bifurcations

The generic isolated bifurcations for one-parameter families of smooth planar vector fields {X_μ} which give rise to periodic orbits are: the Andronov-Hopf bifurcation, the bifurcation from a semi-stable periodic orbit, the saddle-node loop bifurcation and the saddle loop bifurcation. In this paper we obtain the dominant term of the asymptotic behaviour of the period of the limit cycles appearing in each of these bifurcations in terms of μ when we are near the bifurcation. The method used to study the first two bifurcations is also used to solve the same problem in another two situations: a generalization of the Andronov-Hopf bifurcation to vector fields starting with a special monodromic jet; and the Hopf bifurcation at infinity for families of polynomial vector fields.     

Perturbations of symmetric elliptic Hamiltonians of degree four

In this paper four-parameter unfoldings Xλ of symmetric elliptic Hamiltonians of degree four are studied. We prove that in a compact region of the period annulus of X₀ the displacement function of Xλ is sign equivalent to its principal part, which is given by a family induced by a Chebychev system; and we describe the bifurcation diagram of Xλ in a full neighborhood of the origin in the parameter space, where at most two limit cycles can exist for the corresponding systems.     

Multiplicity of limit cycles and analytic m-solutions for planar differential systems

This work deals with limit cycles of real planar analytic vector fields. It is well known that given any limit cycle Γ of an analytic vector field it always exists a real analytic function f₀(x,y), defined in a neighborhood of Γ, and such that Γ is contained in its zero level set. In this work we introduce the notion of f₀(x,y) being an m-solution, which is a merely analytic concept. Our main result is that a limit cycle Γ is of multiplicity m if and only if f₀(x,y) is an m-solution of the vector field. We apply it to study in some examples the stability and the bifurcation of periodic orbits from some non-hyperbolic limit cycles.     

Dynamic Hopf bifurcations generated by nonlinear terms

We consider the so-called delayed loss of stability phenomenon for singularly perturbed systems of differential equations in case that the associated autonomous system with a scalar parameter undergoes the Hopf bifurcation at the zero equilibrium point. It is assumed that the linearization of the associated system is independent of the parameter and the next terms in the expansion of the right-hand parts at zero are positive homogeneous of order α>1. Simple formulas are presented to estimate the asymptotic delay for the delayed loss of stability phenomenon. More precisely, we suggest sufficient conditions which ensure that zeros of a simple function ψ defined by the positive homogeneous nonlinear terms are the Hopf bifurcation points of the associated system, the sign of ψ at other points determines stability of the zero equilibrium, and the asymptotic delay equals the distance between the bifurcation point and a zero of some primitive of ψ.     

Impulsive semilinear differential inclusions: Topological structure of the solution set and solutions on non-compact domains

This paper deals with an impulsive Cauchy problem governed by the semilinear evolution differential inclusion x^′(t)∈A(t)x(t)+F(t,x(t))$x^{\prime }\left(t\right)\in A\left(t\right)x\left(t\right)+F(t,x\left(t\right))$, where _{{A(t)}t∈[0,b]}${\left\{A\left(t\right)\right\}}_{t\in [0,b]}$ is a family of linear operators (not necessarily bounded) in a Banach space E$E$ generating an evolution operator and F$F$ is a Carathéodory type multifunction. First a theorem on the compactness of the set of all mild solutions for the problem is given. Then this result is applied to obtain the existence of mild solutions for the impulsive Cauchy problem defined on non-compact domains.     

Configurations of limit cycles in Liénard equations

We show that every finite configuration of disjoint simple closed curves in the plane is topologically realizable as the set of limit cycles of a polynomial Liénard equation. The related vector field X is Morse–Smale. Moreover it has the minimum number of singularities required for realizing the configuration in a Liénard equation. We provide an explicit upper bound on the degree of X, which is lower than the results obtained before, obtained in the context of general polynomial vector fields.     

A proof of Perkoʼs conjectures for the Bogdanov–Takens system

The Bogdanov–Takens system has at most one limit cycle and, in the parameter space, it exists between a Hopf and a saddle-loop bifurcation curves. The aim of this paper is to prove the Perko?s conjectures about some analytic properties of the saddle-loop bifurcation curve. Moreover, we provide sharp piecewise algebraic upper and lower bounds for this curve.     

Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop

In this paper, we study limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with the center at the origin and a homoclinic loop around the origin. By using the first Melnikov function of piecewise near-Hamiltonian systems, we give lower bounds of the maximal number of limit cycles in Hopf and homoclinic bifurcations, and derive an upper bound of the number of limit cycles that bifurcate from the periodic annulus between the center and the homoclinic loop up to the first order in ε$\epsilon$. In the case when the degree of perturbing terms is low, we obtain a precise result on the number of zeros of the first Melnikov function.     

Bifurcations and distribution of limit cycles which appear from two nests of periodic orbits

In this paper, we study the distribution and simultaneous bifurcation of limit cycles bifurcated from the two periodic annuli of the holomorphic differential equation , after a small polynomial perturbation. We first show that, under small perturbations of the form , where is a polynomial of degree 2m−1 in which the power of z is odd and the power of is even, the only possible distribution of limit cycles is (u,u) for all values of u=0,1,2,…,m−3. Hence, the sharp upper bound for the number of limit cycles bifurcated from each two period annuli of is m−3, for m≥4. Then we consider a perturbation of the form , where is a polynomial of degree m in which the power of z is odd and obtain the upper bound m−5, for m≥6. Moreover, we show that the distribution (u,v) of limit cycles is possible for 0≤u≤m−5, 0≤v≤m−5 with u+v≤m−2 and m≥9.     

Maximal degree variational principles and Liouville dynamics

Let M be smooth n-dimensional manifold, fibered over a k-dimensional submanifold B as π:M→B, and ?∈Λ^k(M); one can consider the functional on sections φ of the bundle π defined by , with D a domain in B. We show that for k=n−2 the variational principle based on this functional identifies a unique (up to multiplication by a smooth function) nontrivial vector field in M, i.e., a system of ODEs. Conversely, any vector field X on M satisfying for some ?∈Λⁿ⁻²(M) admits such a variational characterization. We consider the general case, and also the particular case M=P×R where one of the variables (the time) has a distinguished role; in this case our results imply that any Liouville (volume-preserving) vector field on the phase space P admits a variational principle of the kind considered here.     

Asymmetric type II periodic motions for nonlinear impact oscillators

In this paper a general class of nonlinear impact oscillators is considered for Type II periodic motions. This system can be used to model an inverted pendulum impacting on rigid walls under external periodic excitation. The unperturbed system possesses a pair of homoclinic cycles and three separate families of periodic orbits inside and outside the homoclinic cycles via the identification given by the impact law. By approximating the Poincaré map to O(ε)$O\left(\epsilon \right)$ directly, a general method of Melnikov type for detecting the existence of asymmetric Type II subharmonic orbits outside the homoclinic cycles is presented.     

On the critical points of the flight return time function of perturbed closed orbits

We deal here with planar analytic systems $\dot{x}=X(x,\epsilon )$ which are small perturbations of a period annulus. For each transversal section Σ to the unperturbed orbits we denote by $T^{\mathrm{\Sigma }}(q,\epsilon )$ the time needed by a perturbed orbit that starts from $q\in \mathrm{\Sigma }$ to return to Σ. We call this the flight return time function. We say that the closed orbit Γ of $\dot{x}=X(x,0)$ is a continuable critical orbit in a family of the form $\dot{x}=X(x,\epsilon )$ if, for any $q\in \mathrm{\Gamma }$ and any Σ that passes through q, there exists $q^{\epsilon }\in \mathrm{\Sigma }$ a critical point of $T^{\mathrm{\Sigma }}(?,\epsilon )$ such that $q^{\epsilon }\to q$ as $\epsilon \to 0$. In this work we study this new problem of continuability.In particular we prove that a simple critical periodic orbit of $\dot{x}=X(x,0)$ is a continuable critical orbit in any family of the form $\dot{x}=X(x,\epsilon )$. We also give sufficient conditions for the existence of a continuable critical orbit of an isochronous center $\dot{x}=X(x,0)$.     

Travelling waves of auto-catalytic chemical reaction of general order—An elliptic approach

In this paper we study the existence and non-existence of travelling wave to parabolic system of the form at=axx−af(b), bt=Dbxx+af(b), with f a degenerate nonlinearity. In the context of an auto-catalytic chemical reaction, a is the density of a chemical species called reactant A, b that of another chemical species B called auto-catalyst, and D=DB/DA>0 is the ratio of diffusion coefficients, DB of B and DA of A, respectively. Such a system also arises from isothermal combustion. The nonlinearity is called degenerate, since f(0)=f^′(0)=0. One case of interest in this article is the propagating wave fronts in an isothermal auto-catalytic chemical reaction of order with 1<n<2, and D≠1 due to different molecular weights and/or sizes of A and B. The resulting nonlinearity is f(b)=bn. Explicit bounds v_∗ and v^∗ that depend on D are derived such that there is a unique travelling wave of every speed v?v^∗ and there does not exist any travelling wave of speed v<v_∗. New to the literature, it is shown that v_∗∝v^∗∝D when D<1. Furthermore, when D>1, it is shown rigorously that there exists a vmin such that there is a travelling wave of speed v if and only if v?vmin. Estimates on vmin improve significantly that of early works. Another case in which two different orders of isothermal auto-catalytic chemical reactions are involved is also studied with interesting new results proved.     

Periodic solutions for a higher order p-Laplacian neutral functional differential equation with a deviating argument

In this paper, a higher order p-Laplacian neutral functional differential equation with a deviating argument:

[φ_p([x(t)−c(t)x(t−σ)]⁽ⁿ⁾)]^(m)+f(x(t))x^′(t)+g(t,x(t−τ(t)))=e(t)     

Oscillation susceptibility analysis along the path of longitudinal flight equilibriums

In this paper the oscillation susceptibility of an aircraft in a longitudinal flight with constant forward velocity is analyzed in different flight models. Conditions which ensure such a flight, and equations governing the flight are presented. The stability of the equilibriums appearing is analyzed and the existence of Hopf bifurcations and saddle-node bifurcations is researched. For two aircrafts in a simplified model it is shown that saddle-node bifurcations are present and there are no Hopf bifurcations. It is shown that for the elevator deflection there are two turning points , having the property that if , then the angle of attack α and the pitch rate q oscillate with the same period, while the pitch angle θ increases (decreases) tending to . The behavior of the aircraft is simulated in the simplified model when the elevator deflection δ_e varies in the range and when δ_e leaves this range. For one of the aircrafts the analysis is performed also in the not simplified model, showing the differences between the results obtained in different models.     

The cyclicity of the elliptic segment loops of the reversible quadratic Hamiltonian systems under quadratic perturbations

Denote by Q^H and Q^R the Hamiltonian class and reversible class of quadratic integrable systems. There are several topological types for systems belong to Q^H∩Q^R. One of them is the case that the corresponding system has two heteroclinic loops, sharing one saddle-connection, which is a line segment, and the other part of the loops is an ellipse. In this paper we prove that the maximal number of limit cycles, which bifurcate from the loops with respect to quadratic perturbations in a conic neighborhood of the direction transversal to reversible systems (called in reversible direction), is two. We also give the corresponding bifurcation diagram.     

Smoothness of the carrying simplex for discrete-time competitive dynamical systems: A characterization of neat embedding

The paper is concerned with the question of smoothness of the carrying simplex S for a discrete-time dissipative competitive dynamical system. We give a necessary and sufficient criterion for S being a C¹ submanifold-with-corners neatly embedded in the nonnegative orthant, formulated in terms of inequalities between Lyapunov exponents for ergodic measures supported on the boundary of the orthant. This completes one thread of investigation occasioned by a question posed by M.W. Hirsch in 1988. Besides, amenable conditions are presented to guarantee the Cr (r?1) smoothness of S in the time-periodic competitive Kolmogorov systems of ODEs. Examples are also presented, one in which S is of class C¹ but not neatly embedded, the other in which S is not of class C¹.