One-dimensional centre manifolds, part 4

In this post I consider the case of the transcritical bifurcation, starting with the case of a one-dimensional dynamical system depending on one parameter $\dot x=f(x,\alpha)$ . I consider the situation where for structural reasons $x=0$ is always a steady state. Hence $f(x,\alpha)=xg(x,\alpha)$ for a smooth function $g$ . The equation $g(x,\alpha)=0$ could potentially have a bifurcation at the origin but here I specifically consider the case where it does not. The assumptions are $g(0,0)=0$ and $g_x(0,0)\ne 0$ . Moreover it is assumed that $g_\alpha(0,0)\ne 0$ . By the implicit function theorem the zero set of $g$ is a function of $\alpha$ . By the last assumption made the derivative of this function at zero is non-zero. The graph crosses both axes transversally at the origin. It will be supposed, motivated by intended applications, that $g_x(0,0)<0$ . If $g_\alpha<0$ there exist positive steady states for $\alpha>0$ and if $g_\alpha>0$ there exist positive steady states with $\alpha<0$ . The conditions which have been required of $g$ can be translated to conditions on $f$ . Of course $f(0,0)=0$ and $f_\alpha (0,0)=0$ . Since $g(0,0)=0$ we also have $f_x(0,0)=0$ and the origin is a bifurcation point for $f$ . The other two conditions on $g$ translate to $f_{xx}(0,0)\ne 0$ and $f_{\alpha x}(0,0)\ne 0$ . It follows from the implicit function theorem, applied to $g$ that the bifurcation can be reduced by a coordinate transformation to the standard form $\dot x=x(\alpha\pm x)$ . Thus we see how many steady states there are for each value of $\alpha$ and what their stability properties are.

My primary motivation for discussing this here is to throw light on the concept of backward bifurcation. At a bifurcation point of this kind there is a one-dimensional centre manifold. I want to explain the relation of general centre manifold theory to Theorem 4 of in the paper of van den Driessche and Watmough. Background for this discussion can be found here. It is explicitly assumed in the discussion leading up to the theorem that the centre manifold at the bifurcation point is one-dimensional. All but one of the non-zero eigenvalues of the linearization have negative real parts close to the bifurcation point. The remaining one changes sign as the bifurcation point is passed. The main idea is to reduce the bifurcation to the two-dimensional extended centre manifold at the bifurcation point. In equations (23) and (24) the key diagnostic quantities $a$ and $b$ for the bifurcation are defined. They are expressed in invariant form both in terms of the extrinsic dynamical system and in a form intrinsic to the centre manifold. In the coordinate form used above $a=\frac12 f_{xx}(0,0)$ and $b=f_{\alpha x}(0,0)$ . Theorem 4 of the paper makes statements about the existence and stability of positive steady states which are essentially equivalent to those made above when it is taken into account that in the situation of the theorem the centre manifold is asymptotically stable with respect to perturbations in the transverse directions. It does not say more than that. The fact, often seen in pictures of backward bifurcation that the unstable branch undergoes a fold bifurcation is not included. In particular the fact that for some parameter values there exists more than one positive steady state is not included.

This entry was posted on April 30, 2024 at 2:56 pm and is filed under dynamical systems, mathematical biology. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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