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.     

Perturbations of May–Leonard system

In this paper we study the quadratic homogeneous perturbations of the 3-dimensional May–Leonard system with α+β=2$\alpha +\beta =2$. It is shown that there are perturbed systems having exactly one or two limit cycles bifurcated from the periodic orbits of May–Leonard system. This is proved by estimating the number of zeros of the first and the second order Melnikov functions.     

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.     

Existence of weak solutions of two-point boundary value problems for second-order dynamic equations on time scales

In this paper, by using critical point theory, we establish the existence of weak solutions of the two-point boundary value problem for a second-order dynamic equation on an arbitrary time scale T$\mathbb{T}$, so that the well known case of differential dynamic systems (T=R)$(\mathbb{T}=\mathbb{R})$ and the recently developed case of discrete dynamic systems (T=Z)$(\mathbb{T}=\mathbb{Z})$ are unified. To the best of our knowledge, this is the first time that boundary value problems of dynamic equations on time scales have been dealt with by using critical point theory.     

Asymptotic expansions for elliptic-like regularizations of semilinear evolution equations

Consider in a real Hilbert space H the Cauchy problem (_P0$P_0$): u^′(t)+Au(t)+Bu(t)=f(t)$u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, where −A   is the infinitesimal generator of a _C0$C_0$-semigroup of contractions, B is a nonlinear monotone operator, and f is a given H-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem (_P0$P_0$) the following regularization (_Pε$P_{\epsilon }$): −εu^″(t)+u^′(t)+Au(t)+Bu(t)=f(t)$-\epsilon u^{″}(t)+u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, u^′(T)=_uT$u^{\prime }(T)=u_T$, where ε>0$\epsilon >0$ is a small parameter. We investigate existence, uniqueness and higher regularity for problem (_Pε$P_{\epsilon }$). Then we establish asymptotic expansions of order zero, and of order one, for the solution of (_Pε$P_{\epsilon }$). Problem (_Pε$P_{\epsilon }$) turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of C([0,T];H)$C([0,T];H)$. However, the boundary layer of order one is not visible through the norm of L²(0,T;H)$L^2(0,T;H)$.     

Step size strategies for the numerical integration of systems of differential equations

In this study, the step size strategies are obtained such that the local error is smaller than the desired error level in the numerical integration of a type of nonlinear equation system in interval [_t0,T]$[t_0,T]$. The algorithms are given for calculating step sizes and numerical solutions according to these strategies. The algorithms are supported by the numerical examples.     

Extended rank reduction formulas containing Wedderburn and Abaffy–Broyden–Spedicato rank reducing processes

The Wedderburn rank reduction formula and the Abaffy–Broyden–Spedicato (ABS) algorithms are powerful methods for developing matrix factorizations and many fundamental numerical linear algebra processes such as Gram–Schmidt, conjugate direction and Lanczos methods. We present a rank reduction formula for transforming the rows and columns of A, extending the Wedderburn rank reduction formula and the ABS approach. By repeatedly applying the formula to reduce the rank, an extended rank reducing process is derived. The biconjugation process associated with the Wedderburn rank reduction process and the scaled extended ABS class of algorithms are shown to be in our proposed rank reducing process, while the process is more general to produce several other effective reduction algorithms to compute various structured factorizations. The process provides a general finite iterative approach for constructing factorizations of A   and A^T$A^T$ under a common framework of a general decomposition V^TAP=Ω$V^{T}AP=\Omega$. We also show that the biconjugation process associated with the Wedderburn rank reduction process can be derived from the scaled ABS class of algorithms applied to A   or A^T$A^T$. Finally, we provide a list of some well-known reduction procedures as special cases of our extended rank reducing process. The approach is general enough to produce various structured decompositions as well.     

Reflectionless Sturm–Liouville equations

We consider compactly supported perturbations of periodic Sturm–Liouville equations. In this context, one can use the Floquet solutions of the periodic background to define scattering coefficients. We prove that if the reflection coefficient is identically zero, then the operators corresponding to the periodic and perturbed equations, respectively, are unitarily equivalent. In some appendices, we also provide the proofs of several basic estimates, e.g., bounds and asymptotics for the relevant m$m$-functions.