The normal forms of Hopf and generalized Hopf bifurcations have been extensively studied, and obtained using the method of normal form theory and many other different approaches. It is well known that if the normal forms of Hopf and generalized Hopf bifurcations are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal form. In this paper, three theorem...

Presently, there is a method based on Power Geometry that allows one to find asymptotic forms and expansions of solutions different kinds non-linear equations near their singularities. The contains three algorithms: (1) Reducing the equation its normal form, (2) separating truncated equations, (3) power transformations coordinates. Here, we describe for simplest case, single algebraic equation,...

The simplest models in finance come from continuous time models. In these models, the time variable, t, may take any value. The asset price, S(t) is defined for any value of t. In the simplest models, S(t) is a continuous and random function of t. Without randomness, we would describe the dynamics of S(t) using an ordinary differential equation. With randomness, we use instead a stochastic diff...

Recently, the nonlinearity continuation method has been used to numerically solve boundary value problems for steady-state Richards equation. The can be considered as a predictor-corrector procedure with simplest form which applied date having trivial, zeroth-order predictor. In this article, effect of more sophisticated predictor technique is examined. Numerical experiments are performed finit...

in this paper, the solution of the evolutionaryfourth-order in space, sivashinsky equation is obtained by meansof homotopy perturbation method (textbf{hpm}). the results revealthat the method is very effective, convenient  and quite accurateto systems of nonlinear partial differential equations.

in this paper, the convergence of zakharov-kuznetsov (zk) equation by homo- topy analysis method (ham) is investigated. a theorem is proved to guarantee the conver- gence of ham and to nd the series solution of this equation via a reliable algorithm.