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.

An adaptive homotopy method for computing bifurcations of nonlinear parametric systems In this paper, we present an adaptive stepsize homotopy tracking method for computing bifurcation points of nonlinear systems. There are four components in this new method: 1) an adaptive tracking technique is developed near bifurcation points; 2) an inflation technique is backed up when the adaptive tracking fails; 3) Puiseux series interpolation is used to compute bifurcation points; and 4)the tangent cone structure of the bifurcation point is approximated numerically to compute solutions on different branches. Various numerical examples of nonlinear systems are given to illustrate the efficiency of this new approach. This new adaptive homotopy tracking method is also applied to a system of nonlinear PDEs and shows robustness and efficiency for largescale nonlinear discretized systems.