Relaxation oscillations are periodic orbits of multiple time scale dynamical systems that contain both slow and fast segments. The slow-fast decomposition of these orbits is defined in the singular limit. Geometric methods in singular perturbation theory classify degeneracies of these decompositions that occur in generic one parameter families of relaxation oscillations. This paper investigates...

We study a family of singular perturbation problems of the kind inf { 1 ε ∫ Ω f(u, ε∇u, ερ) dx : ∫

where Ω ⊂ R (N ≥ 2) is a smooth bounded domain, 1 < p < (N + 2)/(N − 2) for N ≥ 3, 1 < p < ∞ for N = 2 and ε > 0 is a positive small parameter. Our interest in (1.1) arises from two aspects. First, (1.1) is a typical singular perturbation problem. Singular perturbation problems have received much attention lately due to their significances in applications such as chemotaxis (see [18] and [19]),...

In this paper we show that stability for holomorphic vector bundles are equivalent to the existence of solutions to certain system of Monge Amp ere equations parametrized by a parameter k. We solve this fully nonlinear elliptic system by singular perturbation technique and show that the vanishing of obstructions for the perturbation is given precisely by the stability condition. This can be int...

We consider several model problems from a class of elliptic perturbation equations in two dimensions. The domains, the diierential operators, the boundary conditions, and so on, are rather simple, and are chosen in a way that the solutions can be obtained in the form of integrals or Fourier series. By using several techniques from asymptotic analysis (saddle point methods, for instance) we try ...

We consider least-squares problems where the coefficient matrices A, b are unknown but bounded. We minimize the worst-case residual error using (convex) second-order cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on t...

The spline element method with constraints is a discretization method where the unknowns are expanded as polynomials on each element and Lagrange multipliers are used to enforce the interelement conditions, the boundary conditions and the constraints in numerical solution of partial differential equations. Spaces of piecewise polynomials with global smoothness conditions are known as multivaria...

It is well known that the standard finite element method based on the space Vh of continuous piecewise linear functions is not reliable in solving singular perturbation problems. It is also known that the solution of a two-point boundary-value singular perturbation problem admits a decomposition into a regular part and a finite linear combination of explicit singular functions. Taking into acco...

The cusp singularity—a point at which two curves of fold points meet—is a prototypical example in Takens’ classification of singularities in constrained equations, which also includes folds, folded saddles, folded nodes, among others. In this article, we study cusp singularities in singularly perturbed systems for sufficiently small values of the perturbation parameter, in the regime in which t...

the current paper focuses on some analytical techniques to solve the non-linear duffing oscillator with large nonlinearity. four different methods have been applied for solution of the equation of motion; the variational iteration method, he’s parameter expanding method, parameterized perturbation method, and the homotopy perturbation method. the results reveal that approximation obtained by th...