Geometric Approach to the Bifurcation at Infinity: A Case Study

Abstract

In this paper, the reaction-diffusion equation in space 1D with negative power nonlinearity is presented as an example the stationary solutions of which are characterized by the bifurcation at infinity. The all dynamics, including to infinity, of the ordinary differential equation concerning with the stationary problem of the reaction-diffusion equation are obtained by the Poincaré-type compactification. The results of this paper give relationship between the bifurcation at infinity of the stationary solutions for the reaction-diffusion equation and the connecting orbits of the ordinary differential equations, especially, that connect the equilibria at infinity. Then, for the bifurcation at infinity, the question of what kind of bifurcation structures are present in this ODEs not only before the solutions go to infinity but also after they go to infinity is answered by a precise analysis. Moreover, this result also answers the question of what kind of solutions appear (or disappear) associated with the bifurcation. In addition, from the bifurcation theory view point, to clarify the bifurcation structures of the equilibria in the ordinary differential equation, the dynamics of it through the one-point compactification called the Bendixson compactification is also shown.

Appendix A Overview of the Poincaré-type compactification.

The Poincaré compactification is one of the compactifications of the original phase space (the embedding of $\mathbb {R}^{n}$ into the unit upper hemisphere of $\mathbb {R}^{n+1}$). In this appendix, we briefly introduce the Poincaré compactification. Here Section 2 of [5] are reproduced. Also, it should be noted that we refer [4] for more details. Let

$$\begin{aligned} X = P(\phi ,\psi ) \dfrac{\partial }{\partial \phi } + Q(\phi ,\psi ) \dfrac{\partial }{\partial \psi }\end{aligned}$$

be a polynomial vector field on $\mathbb {R}^{2}$, or in other words

$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{\phi } = P(\phi ,\psi ), \\ \dot{\psi } = Q(\phi ,\psi ), \end{array}\right. } \end{aligned}$$

where $\dot{~~}$ denotes d/dt, and P, Q are polynomials of arbitrary degree in the variables $\phi$ and $\psi$.

We consider $\mathbb {R}^{2}$ as the plane in $\mathbb {R}^{3}$ defined by $(y_{1},y_{2},y_{3})=(\phi ,\psi ,1)$. We consider the sphere $\mathbb {S}^{2} = \{ y \in \mathbb {R}^{3} \, |\, y_{1}^{2} + y_{2}^{2}+y_{3}^{2}=1\}$ which we call Poincaré sphere. We divide the sphere into

$$\begin{aligned} H_{+} = \{ y \in \mathbb {S}^{2}\,|\,y_{3}>0\}, \quad H_{-} = \{ y \in \mathbb {S}^{2}\,|\,y_{3}<0\} \end{aligned}$$

and

$$\begin{aligned}\mathbb {S}^{1} = \{y \in \mathbb {S}^{2}\, | \, y_{3}=0\}.\end{aligned}$$

Let us consider the embedding of vector field X from $\mathbb {R}^{2}$ to $\mathbb {S}^{2}$ given by

$$\begin{aligned} f^{+}:\mathbb {R}^{2} \rightarrow \mathbb {S}^{2}, \quad f^{-}:\mathbb {R}^{2} \rightarrow \mathbb {S}^{2},\end{aligned}$$

where

$$\begin{aligned}f^{\pm }(\phi ,\psi ):= \pm \left( \dfrac{\phi }{\Delta (\phi ,\psi )},\dfrac{\psi }{\Delta (\phi ,\psi )},\dfrac{1}{\Delta (\phi ,\psi )} \right) \end{aligned}$$

with $\Delta (\phi ,\psi ) = \sqrt{\phi ^{2}+\psi ^{2}+1}$.

Then we consider six local charts on $\mathbb {S}^{2}$ given by $U_{k} = \{y \in \mathbb {S}^{2} \, | \, y_{k}>0\}$, $V_{k} = \{y \in \mathbb {S}^{2} \, | \, y_{k}<0\}$ for $k=1,2,3$. Consider the local projection

$$\begin{aligned} g^{+}_{k}: U_{k} \rightarrow \mathbb {R}^{2}, \quad g^{-}_{k}: V_{k} \rightarrow \mathbb {R}^{2} \end{aligned}$$

defined as

$$\begin{aligned}g^{+}_{k}(y_{1},y_{2},y_{3}) = - g^{-}_{k}(y_{1},y_{2},y_{3}) = \left( \dfrac{y_{m}}{y_{k}},\dfrac{y_{n}}{y_{k}} \right) \end{aligned}$$

for $m<n$ and $m,n \not = k$. The projected vector fields are obtained as the vector fields on the planes

$$\begin{aligned} \overline{U}_{k} = \{y \in \mathbb {R}^{3} \, | \, y_{k} = 1\}, \quad \overline{V}_{k} = \{y \in \mathbb {R}^{3} \, | \, y_{k} = -1\} \end{aligned}$$

for each local chart $U_{k}$ and $V_{k}$. We denote by $(x,\lambda )$ the value of $g^{\pm }_{k}(y)$ for any k.

For instance, it follows that

$$\begin{aligned} (g^{+}_{2} \circ f^{+})(\phi ,\psi ) = \left( \dfrac{\phi }{\psi },\dfrac{1}{\psi }\right) = (x,\lambda ), \end{aligned}$$

therefore, we can obtain the dynamics on the local chart $\overline{U}_{2}$ by the change of variables $\phi = x/\lambda$ and $\psi = 1/\lambda$. The locations of the Poincaré sphere, $(\phi ,\psi )$-plane and $\overline{U}_{2}$ are expressed as Fig. 9. Throughout this paper, we follow the notations used here for the Poincaré compactification. It is sufficient to consider the dynamics on $H_{+}\cup \mathbb {S}^{1}$, which is called Poincaré disk.

Fig. 9

Locations of the Poincaré sphere and chart $\overline{U}_{2}$

Next, we consider the case that a vector field is quasi-homogeneous. In this case, it should be noted that we choose appropriate compactifications to consider the information about dynamics at infinity. That is, when the vector field is quasi-homogeneous, the information at infinity may not be reflected correctly in the Poincaré compactification. Then, we introduce the Poincaré-Lyapunov compactification (the directional compactification) that is based on asymptotically quasi-homogeneous vector fields. Then we define a class of vector fields that are quasi-homogeneous near infinity, which is determined by types and orders. In the following, we reproduce the definitions given in [8] as an aid to understanding the methods used in this paper. Following [8, 9], we cite the discussion for the general dimension case. See [8, 9] for details.

Definition 1

([8], Definition 2.1) Let $f:\mathbb {R}^{n}\rightarrow \mathbb {R}$ be a smooth function. Let $\alpha _{1}, \alpha _{2}, \ldots , \alpha _{n} \ge 0$ with $(\alpha _{1}, \alpha _{2}, \ldots \alpha _{n})\ne (0,0, \ldots , 0)$ be integers and $k \ge 1$. We say that f is a quasi-homogeneous function of type $(\alpha _{1},\alpha _{2}, \ldots , \alpha _{n})$ and order k if

$$\begin{aligned} f(R^{\alpha _{1}}x_{1}, R^{\alpha _{2}}x_{2}, \ldots , R^{\alpha _{n}}x_{n})=R^{k}f(x_{1}, x_{2}, \ldots , x_{n}),\quad \forall x\in \mathbb {R}^{n},\quad R\in \mathbb {R}. \end{aligned}$$

Next, let

$$\begin{aligned}X=\sum _{j=1}^{n}f_{j}(x)\dfrac{\partial }{\partial x_{j}} \iff X:\left( \begin{array}{l} f_{1}(x_{1},x_{2}, \ldots , x_{n}) \\ f_{2}(x_{1},x_{2}, \ldots , x_{n}) \\ \quad \quad \quad \vdots \\ f_{n}(x_{1}, x_{2}, \ldots , x_{n}) \end{array} \right) \end{aligned}$$

be a smooth vector field. We say that X, or $f=(f_{1}, f_{2}, \ldots , f_{n})$ is a quasi-homogeneous vector field of type $(\alpha _{1},\alpha _{2}, \ldots , \alpha _{n})$ and order $k+1$ if each component $f_{j}$ is a quasi-homogeneous function of type $(\alpha _{1}, \alpha _{2}, \ldots , \alpha _{n})$ and order $k+\alpha _{j}$.

For applications to general vector fields, we define the following notion.

Definition 2

([8], Definition 2.2) Let $\alpha =(\alpha _{1}, \alpha _{2}, \ldots , \alpha _{n})$ be a set of nonnegative integers. Let the index set $I_{\alpha }$ as

$$\begin{aligned}I_{\alpha }:=\{ i\in \{1,2,\ldots , n\} \mid \alpha _{i}>0 \}, \end{aligned}$$

which we shall call the set of homogeneity indices associated with $\alpha =(\alpha _{1}, \alpha _{2},\ldots , \alpha _{n})$. Let $U\subset \mathbb {R}^{n}$. We say the domain $U\subset \mathbb {R}^{n}$ is admissible with respect to the sequence $\alpha$ if

$$\begin{aligned} U:=\{ x=(x_{1}, x_{2}, \ldots , x_{n})\in \mathbb {R}^{n} \mid x_{i}\in \mathbb {R}, \,\,\text{ if }\,\, i\in I_{\alpha },\, (x_{j_{1}}, x_{j_{2}}, \ldots , x_{j_{n-l}})\in \tilde{U} \}, \end{aligned}$$

where $\{j_{1}, j_{2}, \ldots , j_{n-l}\}=\{1,2, \ldots , n\} \backslash I_{\alpha }$ and $\tilde{U}$ is an $(n-l)$-dimensional open set.

Assumptions in Definition 1 indicate $I_{\alpha }\ne \emptyset$. The notion of asymptotic quasi-homogeneity defined below provides a systematic validity of scalings at infinity in many practical applications.

Definition 3

([8], Definition 2.3) Let $f=(f_{1}, f_{2}, \ldots , f_{n}): U\rightarrow \mathbb {R}^{n}$ be a smooth function with an admissible domain $U\subset \mathbb {R}^{n}$ with respect to $\alpha$ such that f is uniformly bounded for each $x_{i}$ with $i\in I_{\alpha }$, where $I_{\alpha }$ is the set of homogeneity indices associated with $\alpha$. We say that

$$\begin{aligned} X=\sum _{j=1}^{n}f_{j}(x)\dfrac{\partial }{\partial x_{j}} \iff X:\left( \begin{array}{l} f_{1}(x_{1},x_{2}, \ldots , x_{n}) \\ f_{2}(x_{1},x_{2}, \ldots , x_{n}) \\ \quad \quad \quad \vdots \\ f_{n}(x_{1}, x_{2}, \ldots , x_{n}) \end{array} \right) \end{aligned}$$

or simply f is an asymptotically quasi-homogeneous vector field of type $(\alpha _{1}, \alpha _{2}, \ldots , \alpha _{n})$ and order $k+1$ at infinity if

$$\begin{aligned} \lim _{R\rightarrow +\infty }R^{-(k+\alpha _{j})} \bigl \{ f_{j}(R^{\alpha _{1}}x_{1}, R^{\alpha _{2}}x_{2}, \ldots , R^{\alpha _{n}}x_{n}) -R^{k+\alpha _{j}}(f_{\alpha ,k})_{j}(x_{1}, x_{2}, \ldots x_{n}) \bigm \}=0 \end{aligned}$$

holds for any $(x_{1}, x_{2}, \ldots , x_{n})\in U_{1}$, where $f_{\alpha ,k}=((f_{\alpha ,k})_{1}, (f_{\alpha ,k})_{2}, \ldots , (f_{\alpha ,k})_{n}): U\rightarrow \mathbb {R}^{n}$ is a quasi-homogeneous vector field of type $(\alpha _{1},\alpha _{2}, \ldots , \alpha _{n})$ and order $k+1$, and

$$\begin{aligned} U_{1}:=\{ x=(x_{1}, x_{2}, \ldots , x_{n})\in \mathbb {R}^{n} \mid (x_{i_{1}}, x_{i_{2}}, \ldots , x_{i_{l}})\in \mathbb {S}^{l-1},\, (x_{j_{1}}, x_{j_{2}}, \ldots , x_{j_{n-l}})\in \tilde{U} \}, \end{aligned}$$

where $\{i_{1},i_{2},\ldots , i_{l} \}=I_{\alpha }$.