To solve the problem of finding the maximum value of the bifurcation function, we consider a common example of a bifurcation function arising in saddle-node bifurcations:

Key Observations:

A typical bifurcation function (e.g., from the ODE $\frac{dx}{dt} = -x^3 + 3x + \lambda$) leads to the equilibrium condition $\lambda = x^3 - 3x$. This function $\lambda(x) = x^3 - 3x$ is the bifurcation function here.

Calculating the Maximum:

To find the maximum of $\lambda(x) = x^3 - 3x$:

1. Take the derivative: $\lambda'(x) = 3x^2 - 3$.
2. Set derivative to zero: $3x^2 - 3 = 0 \implies x^2 = 1 \implies x = \pm 1$.
3. Evaluate $\lambda(x)$ at these points:
   • $\lambda(-1) = (-1)^3 - 3(-1) = -1 + 3 = 2$.
   • $\lambda(1) = 1^3 - 3(1) = 1 - 3 = -2$.

The maximum value of the bifurcation function is 2.

Answer: $\boxed{2}$

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