Asymptotic Analysis and Numerical Methods for Oscillatory Infinite Generalized Bessel Transforms with an Irregular Oscillator In this work, we perform a complete asymptotic analysis and the construction of affordable quadrature rules for a class of oscillatory infinite Bessel transform with a general oscillator. Especially in the presence of critical points, e.g., endpoints, zeros and stationary points, we first derive a series of useful asymptotic expansions in inverse powers of the frequency parameter omega . The resulting asymptotic expansions clarify the large omega  behavior of the transform and provide powerful tools for designing quadrature rules and conducting error analysis. As a consequence, efficient and affordable new modified Filontype methods for computing the transform numerically are proposed. Particularly, we carry out the rigorous error analysis and obtain asymptotic error estimates in inverse powers of omega . Numerical examples can confirm our analysis. The accuracy can be improved greatly by either adding more derivatives interpolation at endpoints or adding more interior nodes. Moreover, only using a small number of nodes and multiplicities, we can obtain the required accuracy level. For fixed number of nodes and multiplicities, the higher accuracy can be achieved with the larger values of omega .

Analytical approximation of cuspidal loops using a nonlinear time transformation method Abstract   In this work we consider cuspidal loops, i.e., homoclinic orbits to cuspidal singular points. We develop an iterative procedure, founded on the nonlinear time transformation method, to estimate such codimensionthree global bifurcations up to any wanted order, not only in the space of parameters but also in the phase plane. As far as we know, this is the first time in the literature that this theoretical result is achieved for these global connections. The existence and uniqueness of the perturbed solution obtained are proved. To illustrate the effectiveness of the method we study cuspidal loops in two normal forms of degenerate Takensxe2x80x93Bogdanov bifurcations. Excellent agreement is found between our analytical predictions and the corresponding numerical continuations.