Normal form of the Bautin bifurcation
We follow the paper[Kuznetsov] and consider a Cauchy problem
\[\dot x=\mathbf F(x,p).\]
We denote by $\mathbf L$ the jacobian of $\mathbf F$ at the bifurcation point $(x_0,p_0)$. We choose a basis such that:
\[\mathbf L q=i \omega_{0} q, \quad \mathbf L^{T} p=-i \omega_{0} p, \quad \langle p, q\rangle=1.\]
Under some conditions, $x(t)\approx x_0+2\Re w(t)q$ where $w$ satisfies the normal form:
\[\dot{w}=i \omega_{0} w+\frac{1}{2} G_{21} w|w|^{2}+\frac{1}{12} G_{32} w|w|^{4}+O\left(|w|^{6}\right).\tag{E}\]
The second Lyapunov coefficient is
\[l_2:=\frac{1}{12} \operatorname{Re} G_{32}.\]
Normal form computation
The normal form (E) can be automatically computed as follows
get_normal_form(br::ContResult, ind_bif::Int;
verbose = false,
ζs = nothing,
lens = getlens(br),
kwargs...)
br
is a branch computed after a call to continuation
with detection of bifurcation points enabled and ind_bif
is the index of the bifurcation point on the branch br
. The above call returns a point with information needed to compute the bifurcated branch. For more information about the optional parameters, we refer to get_normal_form
. The result returns an object of type Bautin
.
You should not need to call get_normal_form
except if you need the full information about the branch point.
Predictor
The predictor for a non trivial guess at distance $\delta p$ from the bifurcation point is provided by the method
BifurcationKit.predictor
— Methodpredictor(gh, , ϵ; verbose, ampfactor)
Compute the predictor for the curve of Folds of periodic orbits near the Bautin bifurcation point.
Reference
Kuznetsov, Yu A., H. G. E. Meijer, W. Govaerts, and B. Sautois. “Switching to Nonhyperbolic Cycles from Codim 2 Bifurcations of Equilibria in ODEs.” Physica D: Nonlinear Phenomena 237, no. 23 (December 2008): 3061–68. https://doi.org/10.1016/j.physd.2008.06.006.
References
- Kuznetsov
Kuznetsov, Yu. A. “Numerical Normalization Techniques for All Codim 2 Bifurcations of Equilibria in ODE’s.” SIAM Journal on Numerical Analysis 36, no. 4 (January 1, 1999): 1104–24. https://doi.org/10.1137/S0036142998335005.