Weakly nonlinear analysis

The weakly nonlinear analysis is a procedure to approximate a branch of a bifurcation diagram.

To fix ideas, suppose that we have a potential energy $W(u,\lambda)$, where $\lambda$ is the loading parameter, and $u$ is the unknown. We assume that $u=0$ is the fundamental branch, so that $D_u W(0,\lambda)=0$ for every $\lambda$.

Any bifurcated branch can be represented as a parametrized curve $\xi\mapsto (u_b(\xi),\lambda_b(\xi))$. This branch satisfies the virtual work equation:

(here we think of the Frechet derivative of $W$ as a linear form acting on virtual variations) and is assumed to cross the fundamental branch $u=0$ at $(0,\lambda_c)$. Weakly nonlinear analysis assumes that the solution branch admits the expansion:

and extracts information about the coefficients by plugging this expansion into $\eqref{eq:1}$.

As a consequence of the above ansatz, the function

can be written in power-series expansion:

where the coefficients $F_i$ depend on the derivatives of $W$ at $(0,\lambda_c)$ and on the coefficients in the expansion of $u_b$ and $\lambda_b$. The equations

and

From the practical point of view, the calculation of the derivatives can be carried out easily through the formula: