On bifurcation from infinity: a compactification approach

TL;DR

The paper addresses bifurcation from infinity for a scalar parabolic PDE on $(0,1)$ with nonlinear Robin boundary conditions, using a compactification via a Poincaré projection to analyze unbounded equilibrium branches that arise at Steklov eigenvalues. It constructs an induced semiflow at infinity in which Steklov eigenfunctions become equilibria, and shows that unbounded branches converge, after rescaling, to these eigenfunctions; it also proves the existence of infinite-time blow-up solutions aligning with leading eigenmodes. A five-regime classification of the global attractor as the parameter crosses critical values is developed, including equilibria at infinity and heteroclinic connections both finite and infinite in time. The results provide a robust framework for understanding large-amplitude dynamics and bifurcations from infinity, with implications for numerical schemes and potential extensions to higher dimensions and more general boundary nonlinearities.

Abstract

We consider a scalar parabolic partial differential equation on the interval with nonlinear boundary conditions that are asymptotically sublinear. As the parameter crosses critical values (e.g. the Steklov eigenvalues), it is known that there are large equilibria that arise through a bifurcation from infinity (i.e., such equilibria converge, after rescaling, to the Steklov eigenfunctions). We provide a compactification approach to the study of such unbounded bifurcation curves of equilibria, their stability, and heteroclinic orbits. In particular, we construct an induced semiflow at infinity such that the Steklov eigenfunctions are equilibria. Moreover, we prove the existence of infinite-time blow-up solutions that converge, after rescaling, to certain eigenfunctions that are equilibria of the induced semiflow at infinity.

Figures (7)

• Figure 2.1: The two Steklov eigenfunctions $\Phi_1,\Phi_2$, which possess the symmetries $\Phi_1(x)=\Phi_1(1-x)$ and $\Phi_2(x)=-\Phi_2(1-x)$. In particular, the minimum of $\Phi_1(x)$ and the zero of $\Phi_2(x)$ occur at $x=1/2$.
• Figure 2.2: The bifurcation at infinity for a few examples. Left: There are two (subcritical) bifurcation curves from infinity, each possessing a positive and a negative branch, $\pm c_1(\lambda)$ and $\pm c_2(\lambda)$, which respectively meet the origin at $\sigma^*_1:=\sigma_1-1=-2/(1+\mathrm{e})\approx -0.537\ldots$ and $\sigma^*_2:=\sigma_2-1=2/(\mathrm{e}-1)\approx 1.163\ldots$. Middle Left: The (supercritical) bifurcation curves from infinity meet $0$ at $\sigma^*_1:=\sigma_1+1,\sigma^*_2:=\sigma_2+1$. Middle Right: There are infinitely many turning points accumulating at infinity, see for example ARRB10CP17. Right: Infinitely many turning points nearby the origin, see CP12.
• Figure 2.3: Plot of the two graphs in \ref{['identity2']} for $\mu<-1$: the (black) left-hand side is given by $\tan(\sqrt{|1+\mu|})$ with asymptotes whenever $\mu=-((2k+1)\pi/2)^2-1,k\in\mathbb{N}_0$, and the (gray) right-hand side is given by $2(\lambda+1)\sqrt{|1+\mu|}/((\lambda+1)^2-|1+\mu|)$ with a (dashed) asymptote at $\mu=-(\lambda+1)^2-1$. The intersection (black dots) of both graphs yields eigenvalues $\mu=\mu(\lambda)<-1$. Note that there are resonances, which occurs whenever both the black and gray curves have asymptote at the same value, which occurs at $\lambda^\pm_k:=\pm (2k+1)\pi/2-1,k\in\mathbb{N}_0$. We emphasize that only two eigenvalues, $\mu_1$ and $\mu_2$, cross the value $\mu=-1$ whenever $\lambda=\pm 1$.
• Figure 2.4: Plot of the two graphs in \ref{['identity1']} for $\mu>-1$: the (black) left-hand side, $\mathrm{e}^{2\sqrt{1+\mu}}$, and the (gray) right-hand side, $(\sqrt{1+\mu}+\lambda+1)^2/(\sqrt{1+\mu}-\lambda-1)^2$, with a (dashed) asymptote at $\mu=(\lambda+1)^2-1$. Each intersection (black dots) of both graphs amounts to an eigenvalue $\mu=\mu(\lambda)>-1$. We emphasize that there are at most two eigenvalues that cross $\mu=0$, which occur exactly at the values $\lambda=\sigma^*_1,\sigma^*_2$, and thereby yield the bifurcation of the trivial equilibrium.
• Figure 2.5: The graphs in \ref{['identity1']} for $\mu>-1$: $\mathrm{e}^{2\sqrt{1+\mu}}$ in black and $(\sqrt{1+\mu}+\lambda+g'(u_*(0)))^2/(\sqrt{1+\mu}-\lambda-g'(u_*(0)))^2$ in gray, where $u_*\in \{\pm u^1,\pm u^2\}$. Left: For $u_*=\pm u^1$ with $\lambda \in (\sigma^*_1,\sigma_1)$, there is one intersections of both graphs yielding an eigenvalue $\mu_1=\mu_1(\lambda)\in (-1,0)$, and hence $\pm u^1$ are stable hyperbolic equilibra, since all other eigenvalues satisfy $\mu_k<-1,k=2,3,\ldots$. Moreover, $\mu_1 \to 0$ if either $\lambda\to\sigma^*_1$ or $\sigma_1$. Right: For $u_*=\pm u^2$ with $\lambda \in (\sigma^*_2,\sigma_2)$, there are two intersections of both graphs yielding the eigenvalues $\mu_1=\mu_1(\lambda)\in (0,\infty)$ and $\mu_2=\mu_2(\lambda)\in [-1,0)$, and thus $\pm u^1$ are hyperbolic saddles, since all other eigenvalues satisfy $\mu_k<-1,k=3,4,\ldots$. Moreover, $\mu_2 \to 0$ if either $\lambda\to\sigma^*_2$ or $\sigma_2$.

Theorems & Definitions (4)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 2.4