2 Resolution of singularities

In this section, we explain how to construct the space of initial values for the system (1.2). The notion of initial value spaces described in Definition 2.2 is based on foliation theory, and we start by first motivating the reason for this construction. We then explain how to construct such a space by carrying out resolutions or blow-ups, based on the process described in Definition 2.3.

The system (1.2) is a system of two first-order ODEs for $(y(x), z(x))$ . Given initial values $(y_0, z_0)$ at $x_0$ , local existence and uniqueness theorems provide a solution that is defined on a local polydisk $U\times V$ in $\mathbb C\times \mathbb C^2$ , where $x_0\in U\subset \mathbb C\setminus \{0,1\}$ and $(y_0, z_0)\in V\subset (\mathbb C\setminus \{0\})\times \mathbb {C}$ . Our interest lies in global extensions of these local solutions. However, the occurrence of movable poles in the Painlevé transcendents acts as a barrier to the extension of $U\times V$ to the whole domain of (1.2). The first step to overcome this obstruction is to compactify the space $\mathbb C^2$ , in order to include the poles. We carry this out by embedding $\mathbb C^2$ into the first Hirzebruch surface $\mathbb {F}_1$ [Reference Beauville1, Reference Hirzebruch14]. $\mathbb {F}_1$ is a projective space covered by four affine coordinate charts (given in Section 3).

The next step in this process results from the occurrence of singularities in the Painlevé vector field (1.2) in V. By the term singularity, we mean points where $(y', z')$ becomes either unbounded or undefined because at least one component approaches the undefined limit $0/0$ . We are led, therefore, to construct a space in which the points where the singularities appear are regularised. The process of regularisation is called “blowing up” or resolving a singularity.

The appearance of these singularities is related to the irreducibility of the solutions of Painlevé equations, originally due to Painlevé [Reference Painlevé28], which we have restated below in modern terminology. A function is said to be reducible to another function if it is related to it through a series of allowable operations (described by Painlevé and itemised as (O), (P1)–(P5) in [Reference Umemura30, p.33]).

Modern proofs of the irreducibility of the Painlevé equations have been developed by many authors, including Malgrange [Reference Malgrange21], Umemura [Reference Umemura30, Reference Umemura31] and Watanabe [Reference Watanabe32]. Since the Hirzebruch surface is a compact rational variety, the above theorem implies that it cannot be the space of initial values for (1.2). We are now in a position to define the notion of initial value space.

The properties listed in Definition 2.2 imply that each leaf of the foliation is isomorphic to the base $\mathcal {B}$ . Since the transcendental solutions of the sixth Painlevé equation can be globally extended as meromorphic functions of $x\in \mathbb {C}\setminus \{0,1\}$ , we search for the fibration with the base equal to $\mathbb {C}\setminus \{0,1\}$ .

In order to construct the fibration, we apply the blow-up procedure defined below [Reference Duistermaat3, Reference Griffiths and Harris12, Reference Hartshorne13] to the singularities of the system (1.2) that occur where at least one component becomes undefined of the form $0/0$ . Okamoto [Reference Okamoto25] showed that such singular points are contained in the closure of infinitely many leaves. Moreover, these leaves are holomorphically extended at such a point.

Definition 2.3. The blow-up of the plane $\mathbb {C}^2$ at point $(0,0)$ is the closed subset X of $\mathbb {C}^2\times \mathbb {CP}^1$ defined by the equation $u_1t_2=u_2t_1$ , where $(u_1,u_2)\in \mathbb {C}^2$ and $[t_1:t_2]\in \mathbb {CP}^1$ (see Figure 1). There is a natural morphism $\varphi : X\to \mathbb {C}^2$ , which is the restriction of the projection from $\mathbb {C}^2\times \mathbb {CP}^1$ to the first factor. $\varphi ^{-1}(0,0)$ is the projective line $\{(0,0)\}\times \mathbb {CP}^1$ , called the exceptional line.

Figure 1

The blow-up of the plane at a point.

Notice that the blow-up construction separates the lines containing the point $(0,0)$ in Definition 2.3, as shown in Figure 1. In this way, the solutions of (1.2) containing the same point can be separated. Additional blow-ups may be required if the solutions have a common tangent line or a tangency of higher order at such a point. The explicit resolution of the vector field (1.2) is carried out in Appendix B. Okamoto described so-called singular points of the first class that are not contained in the closure of any leaf of the foliation given by the system of differential equations. At such points, the corresponding vector field is infinite.

3 The construction of Okamoto’s space

In this section, we construct Okamoto’s space of initial values in such a way as to ensure that the process yields a well-defined compact surface if we set $x_0=0$ . We start by defining a new time coordinate $t=\ln x$ , or $x=\exp (t)$ , suitable for taking the limit $x\to 0$ , and rewrite the dependent variables as

$$\begin{align*}u(t)=y(x),\ v(t)=z(x). \end{align*}$$

For conciseness, we continue to use the notation $x=e^t$ where needed.

Denoting t-derivatives by dots, we get $\dot {u}=x\frac {\partial H}{\partial v},\ \dot {v}=-x\frac {\partial H}{\partial u},$ or, equivalently

(3.1) $$ \begin{align} \dot{u}=\dfrac{\partial E}{\partial v},\ \dot{v}=-\dfrac{\partial E}{\partial u}, \end{align} $$

where

$$ \begin{align*} E=&\frac{u(u-1)(u-e^t)}{e^t-1}\Bigl\{ v^2-v\left(\frac{\theta_0}{u} +\frac{\theta_1}{u-1}+\frac{\theta_x-1}{u-e^t}\right) +{\frac{\theta \overline{\theta}}{u(u-1)}} \Bigr\}. \end{align*} $$

Suppose we are given $x_0=e^{t_0}\in \mathbb {C}\setminus \{0,1\}$ . We compactify the space of initial values $ (u(t_0),v(t_0))\in \mathbb {C}^2$ to the first Hirzebruch surface $\mathbb {F}_1$ [Reference Hirzebruch14], which is covered by four affine charts in $\mathbb C^2$ [Reference Beauville1]

$$ \begin{align*} &(u_0,v_0)=(u,v),&(u_1,v_1)&=\left(u,\frac{1}{v}\right),\\ &(u_2,v_2)=\left(\frac{1}{u},\frac{1}{uv}\right), &(u_3,v_3)&=\left(\frac{1}{u},uv\right). \end{align*} $$

Let L be the unique section of the natural projection $\mathbb {F}_1\to \mathbb {P}^1$ defined by $(u,v)\mapsto u$ . Then, L is given by $\{v_0=0\}\cup \{v_3=0\}$ and the self-intersection of its divisor class is $-1$ . We identify four particular fibres of this projection:

$$ \begin{align*}\mathcal{V}_j:=\{u_0=j\}\cup\{u_1=j\}\quad \forall j\in \{0,x,1\}\,, \quad \mathcal{D}_\infty :=\{u_2=0\}\cup\{u_3=0\}.\end{align*} $$

Note that as fibres of the projection, these lines all have self-intersection $0$ . Then $\mathbb {F}_1\setminus \mathbb {C}^2$ is given by $\mathcal {D}_\infty \cup \mathcal {H}$ , where

$$ \begin{align*}\mathcal H := \{v_1=0\}\cup\{v_2=0\}.\end{align*} $$

This section $\mathcal H$ , called a ‘horizontal line’ in the following, by a small abuse of common terminology, is topologically equivalent to the formal sum $L+\mathcal {D}_\infty $ in $\mathrm {H}_2(\mathbb {F}_1,\mathbb {Z})$ . In particular, its self-intersection number is given by $ \mathcal H\cdot \mathcal H=L\cdot L+\mathcal {D}_\infty \cdot \mathcal {D}_\infty +2L\cdot \mathcal {D}_\infty =-1+0+2=+1$ , where the dot $\cdot $ denotes the intersection form of divisor classes in the Picard group of the surface. In each chart, the vector field, respectively, becomes

$$ \begin{align*} & \begin{cases} \dot{u} &= \dfrac{u (u -1)(u -x )}{x-1}\left(2v -\dfrac{\theta_0}{u }-\dfrac{\theta_1}{u -1}-\dfrac{\theta_x-1}{u -x }\right),\\ \dot{v} &= -\dfrac{3u^2-2(x+1)u +x}{x-1}v^2+2\dfrac{\theta+\overline{\theta} }{x-1}u v -\left( \dfrac{x\theta_0}{ x-1}+ \theta_1 +\dfrac{\theta+\overline{\theta}}{x-1} \right)v -\dfrac{\theta \overline{\theta} }{x-1}\, , \end{cases} \\ &\begin{cases} \dot{u}_1 &= \dfrac{u_1 (u_1 -1)(u_1 -x)}{x-1}\left(2\dfrac{1}{v_1} -\dfrac{\theta_0}{u_1 }-\dfrac{\theta_1}{u_1 -1}-\dfrac{\theta_x -1}{u_1 -x}\right),\\ \dot{v}_1 &= \dfrac{3u_1^2-2(x+1)u_1 +x}{x-1}-2\dfrac{\theta+\overline{\theta} }{x-1}u_1v_1 +\left( \dfrac{x\theta_0}{ x-1}+ \theta_1+\dfrac{\theta+\overline{\theta}}{x-1} \right)v_1 +\dfrac{ \theta\overline{\theta}}{x-1}v_1^2 , \end{cases}\\ & \begin{cases} \dot u_2 &= 2\dfrac{(u_2-1)(xu_2-1) }{(1-x)v_2}+\dfrac{(u_2-1)(\theta+\overline{\theta}-x\theta_0u_2) }{1-x}- \theta_1 u_2 , \\ \dot v_2 &= -\dfrac{(\theta v_2-1)(\overline{\theta}v_2-1)}{(1-x)u_2}-\dfrac{x}{1-x}(\theta_0v_2-1)u_2 , \end{cases} \\ & \begin{cases} \dot u_3 &= -\dfrac{(u_3-1)(2v_3-(\theta+\overline{\theta}))}{1-x}- \theta_1 u_3+\dfrac{x}{1-x}u_3(u_3-1)(2v_3-\theta_0),\\ \dot v_3 &=\dfrac{(v_3-\theta)(v_3-\overline{\theta})}{(1-x)u_3}-x\dfrac{(v_3-\theta_0)u_3v_3}{1-x}. \end{cases} \end{align*} $$

One realises that the vector field is infinite on $\mathcal {H}:\{v_1=0\}\cup \{ v_2=0\}$ . More precisely, it is infinite or undetermined precisely there. We use the term base point for points where the vector field becomes undetermined. For example, the point $(u_1,v_1)=(0,0)$ in the coordinate chart $\mathbb {C}^2_{u_1,v_1}$ is a base point because the equation for $\dot {u}_1$ approaches $0/0$ as $(u_1, v_1)\to (0,0)$ . In total, we find the following five base points in $\mathbb {F}_1$ , possibly visible in several charts. This initial situation is summarised in Table 1 and Figure 2. Where needed in figures, we indicate the self-intersection number n of an exceptional divisor by annotating it by $(n)$ .

Table 1

Five base points and the charts in which they are visible. The chart $( u_0, v_0)$ is omitted because no base points are visible in this chart.

Figure 2

The surface $\mathbb {F}_1$ with its coordinates and the base point configuration. The numbers in parentheses indicate self-intersection numbers.

Okamoto’s procedure consists in resolving the vector field by successively blowing up the base points until the vector field becomes determined. Since later on we need a well-defined compact surface if we set $x=0$ , we may not blow up $\beta _0$ and $\beta _x$ simultaneously. As detailed in Appendix B.1.1, the blow-up of $\beta _0,\beta _x,\beta _1$ with $\beta _x$ after $\beta _0$ consists of replacing the charts $\mathbb {C}^2_{u_1,v_1}$ and $\mathbb {C}^2_{u_2,v_2}$ by the following five $\mathbb {C}^2$ -charts, endowed with the obvious rational transition maps,

$$ \begin{align*}\begin{array}{rclcrcl}(\tilde u_{1},\tilde v_{1})&:=&\left(u_1, \frac{v_1}{u_1(u_1-1)(u_1-x)} \right) &\quad &(\tilde u_{2},\tilde v_{2})&:=&\left(u_2, \frac{v_2}{(1-u_2)(1-xu_2)} \right) \\ (u_{02},v_{02})&:=&\left(\frac{u_1}{v_1}, {v_1}\right)&\quad &(u_{12},v_{12})&:=&\left(\frac{u_1-1}{v_1}, {v_1}\right)\\ (u_{x2},v_{x2})&:=&\left(\frac{u_1(u_1-x)}{v_1}, \frac{v_1}{u_1}\right) \end{array}\end{align*} $$

For each $i\in \{0,1,x\}$ , what formerly was the point $\beta _i$ is now replaced by an exceptional line

$$\begin{align*}\mathcal{D}_i:\{\tilde{u}_{1}=i\} \cup \{v_{i2}=0\}\end{align*}$$

of self-intersection $-1$ . The strict transform of $\mathcal {H}$ , that is the closure of $\mathcal {H}\setminus \{\beta _0,\beta _x,\beta _1\}$ after blow-up is given by

$$\begin{align*}\mathcal{H}^*:= \{\tilde{v}_{1}=0\}\cup \{\tilde{v}_{2}=0\} \, .\end{align*}$$

As a general fact, each time we blow up a point on a curve, the self-intersection number of the strict transformation of the curve is the former self-intersection number decreased by unity. Since here we have blown up three points, $\mathcal {H}^*$ has self-intersection number $(-2)$ . The blow-up of $\beta _\infty $ consists of removing the point $(0,1/\theta )$ (corresponding to $\beta _\infty $ ) from the chart $\mathbb {C}^2_{\tilde {u}_2,\tilde {v}_2}$ and replacing the chart $\mathbb {C}^2_{{u}_3, {v}_3}$ by the following pair of $\mathbb {C}^2$ -charts.

$$ \begin{align*}\begin{array}{rclcrcl} (\tilde u_3, \tilde v_3)&:=&\left(u_3,- \frac{v_3-\theta}{u_3}\right) &\quad &( u_{\infty 2}, v_{\infty 2})&:=&\left(\frac{u_3}{v_3-\theta}, v_3-\theta\right) \end{array}\end{align*} $$

Again, we obtain an exceptional line $\mathcal {E}_\infty $ and a strict transform $ \mathcal {D}_\infty ^*$ , such that $\mathcal {D}_\infty = \mathcal {E}_\infty \cup \mathcal {D}_\infty ^*$ , where

$$ \begin{align*}\mathcal{E}_\infty:=\{ \tilde u_3 =0\}\cup \{ v_{\infty 2}=0\}\, , \quad \mathcal{D}_\infty^*=\{\tilde{u}_2=0\}\cup \{ u_{\infty 2}=0\}\, .\end{align*} $$

In each of the seven new charts that we have to add to $\mathbb {C}^2_{u,v}$ in order to fully describe the surface resulting of $\mathbb {F}_1$ after this first sequence of blow-ups, we again look at the resulting vector field (see Section B.1.2) and find the following base points, including the still unresolved $\beta _\infty ^-$ . The situation is summarised in Table 2.

Table 2

Base points remaining after blowing up $\beta _0,\beta _x,\beta _1$ and $\beta _\infty $ . The chart $( \tilde {u}_3, \tilde {v}_3)$ is ommitted as there is no base point remaining in this chart.

In Figure 3, the notation ‘ $(n)$ ’ again indicates ‘self-intersection number equal to n’. Moreover, as a visual guideline, we again included the strict transforms $\mathcal {V}_i^*:= \{u_{i2}=0\} \cup \{u_0=i\} $ of the former vertical lines $\mathcal {V}_i.$ Those have self-intersection $(-1)$ .

Figure 3

The surface $\mathbb {F}_1$ after the first sequence of blow-ups and the new base point configuration.

Figure 4

The space of initial values of the resolved Painlevé VI vector field for $x\neq 0$ .

Figure 5

The Dynkin diagram with nodes representing $(-2)$ -lines in Okamoto’s space, for $x\neq 0,1$ , is equivalent to that for $D_4^{(1)}$ .

We blow-up the remaining base points by replacing each chart $\mathbb {C}^2_{u_{i2},v_{i2}}$ by a pair of $\mathbb {C}^2$ -charts with corresponding index as follows, and then removing the already blown up base points that are still visible from other charts.

$$ \begin{align*}\begin{array}{rclcrcl}(u_{03},v_{03})&:=& \left(u_{02}-\theta_0,\frac{v_{02}}{u_{02}-\theta_0} \right) &\quad &(u_{04},v_{04})&:=& \left(\frac{u_{02}-\theta_0}{v_{02}}, v_{02}\right) \\ (u_{x3},v_{x3})&:=& \left(u_{x2}-x(\theta_x-1),\frac{v_{x2}}{u_{x2}-x(\theta_x-1)} \right) &\quad &(u_{x4},v_{x4})&:=& \left(\frac{u_{x2}-x(\theta_x-1)}{v_{x2}}, v_{x2}\right) \\ (u_{13},v_{13})&:=& \left(u_{12}-\theta_1,\frac{v_{12}}{u_{12}-\theta_1} \right) &\quad &(u_{14},v_{14})&:=& \left(\frac{u_{12}-\theta_1}{v_{12}}, v_{12}\right) \\ (u_{\infty 3},v_{\infty 3})&:=& \left(u_{ \infty 2} ,\frac{v_{\infty 2}+\theta_\infty}{u_{\infty 2}} \right) &\quad &(u_{\infty 4},v_{\infty 4})&:=& \left(\frac{u_{\infty 2} }{v_{\infty 2}+\theta_\infty}, v_{\infty 2}\right) \end{array}\end{align*} $$

We obtain the following new exceptional lines, for $i\in \{0,1,x\}$ .

$$ \begin{align*}\mathcal{E}_i:=\{u_{i3}=0\}\cup \{v_{i4}=0\} \, , \quad \mathcal{E}_\infty^-:=\{u_{\infty 3}=0\}\cup \{v_{\infty 4}=0\}.\end{align*} $$

Moreover, we have the following new strict transforms, for $i\in \{0,1,x\}$ .

$$ \begin{align*}\mathcal{D}_i^*:=\{v_{i3}=0\}\cup \{\tilde{u}_1=i\} \, , \quad \mathcal{D}_\infty^{**}:=\{\tilde u_2=0 \}\cup \{u_{\infty 4}=0\}.\end{align*} $$

The above charts of the Hirzebruch surface blown up in our eight base points are detailed in the Appendix Section B.2. As we can see from the equations there, the vector field is now free of base points. We say that the initial value space is resolved or regularised. Moreover, the function E is well-defined there, that is when resolving the base points of the vector field, we also resolved the indeterminacy points of E. For each of the new coordinate charts $(u_{mn}, v_{mn})$ , we also define the Jacobian

$$\begin{align*}\omega_{mn} =\frac{\partial u_{mn}}{\partial u}\, \frac{\partial v_{mn}}{\partial v}\,-\,\frac{\partial u_{mn}}{\partial v}\,\frac{\partial v_{mn}}{\partial u}.\end{align*}$$

Figure 4 illustrates a schematic drawing of the resultant collection of exceptional lines, $\mathcal H^*$ and $D_\infty ^{**}$ and their intersections in the resolved space, as well as the coordinates that will be most important in the following. For each $x=x_0\not = 0, 1$ , this regularised space will be denoted as $\mathcal {S}(x_0)$ . Moreover, we define $\mathcal {S}(0)$ to be the result of the blow-up procedure for $x=0$ . Its relation to the vector field is studied in the next section. The union of $S(x_0)$ forms a fibre bundle

$$\begin{align*}\mathcal{S}:=\bigcup_{x_0\in \mathbb{C}\setminus\{1\}} \mathcal{S}(x_0). \end{align*}$$

From the detailed charts in the Appendix Section B.2, one sees that for $x_0\neq 0$ , the Painlevé vector field is ‘vertical’ or tangent to the lines $\mathcal H^*$ , $\mathcal D_\infty ^{**}$ , $\mathcal D_0^*$ , $\mathcal D_x^*$ , $\mathcal D_1^*$ , which each have self-intersection $-2$ . For this reason, such curves are often referred to as ‘vertical leaves’ in Okamoto’s construction. For each $x=x_0\not =0, 1$ , we define

$$ \begin{align*}\mathcal{I}(x_0):=\mathcal H^*\cup \mathcal D_\infty^{**}\cup \mathcal D_0^*\cup \mathcal D_x^*\cup \mathcal D_1^*\end{align*} $$

the infinity set, corresponding to the black part of the diagram shown in Figure 4. Okamoto’s space of initial values for $x_0\not =0, 1$ is $\mathrm {Oka}(x_0):=\mathcal {S}(x_0)\setminus \mathcal {I}(x_0)$ .

Note that the strict transforms $\mathcal H^*$ , $\mathcal D_\infty ^{**}$ , $\mathcal D_0^*$ , $\mathcal D_x^*$ , $\mathcal D_1^*$ each have self-intersection $-2$ . The corresponding Dynkin diagram reflecting their intersections, given in Figure 5, is equivalent to that for $D_4^{(1)}$ .

4 The vector field in the limit space

When $x\to 0$ (or, more precisely, $\Re (t)\to -\infty $ ), we get the autonomous limiting system

(4.1) $$ \begin{align} \begin{cases} \dot u = -u \{ (u-1)\left(2uv-2\theta+\theta_\infty \right)-\theta_1\},\\ \dot v = uv\left((3u-2)v-4\theta+2\theta_\infty\right)+(2\theta-\theta_\infty-\theta_1)v + \theta (\theta-\theta_\infty), \end{cases} \end{align} $$

where $\dot u=\partial E_0/\partial v$ , $\dot v=-\,\partial E_0/\partial u$ , with

$$ \begin{align*}E_0:=-u \{ (u-1)v\left(uv-2\theta+\theta_\infty\right)-\theta_1v + \theta (\theta-\theta_\infty) \}. \end{align*} $$

We can solve this Hamiltonian system completely: if the values of the $\theta _i$ ’s are generic, that is if they belong to an open dense subset of the set of all possible values of those parameters, we obtain a one-parameter family of solutions that lies on the line $\{u=0\}$ . Again for generic $\theta _i$ values, no solutions lie on the line $\{u=1\}$ . Let us assume $u\not \equiv 0,1$ . Then the Hamiltonian system (4.1) yields

$$ \begin{align*} v =&- \frac{\dot u}{2u^2(u-1)}+\frac{\theta_1}{2u(u-1)}+\frac{\theta+\overline{\theta }}{2u}, \end{align*} $$

leading to

$$ \begin{align*} \ddot u =& \frac{3u-2}{2u(u-1)}\dot u^2+\frac{\theta_\infty^2}{2} (u-1)u^2-\frac{\theta_1^2u^2}{2(u-1)}. \end{align*} $$

Note that if $(u(t),v(t))$ is a solution of the autonomous Hamiltonian system, then $\eta _0:=E_0(u(t),v(t))$ is constant. Setting $u_3:=1/u$ , the autonomous differential equation for u yields $(\dot u_3)^2=\alpha u_3^2+\beta u_3+\gamma $ , where $\alpha = 4\eta _0+( \theta +\overline {\theta }-\theta _1)^2\, , \beta = \theta _1^2-\theta _\infty ^2 -\alpha $ and $\gamma =\theta _\infty ^2$ . This integrates as

$$ \begin{align*}u_3(t)=\begin{cases} \frac{\sqrt{4\gamma \alpha-\beta^2}\mathrm{sinh}(\sqrt{\alpha}\, t+\eta_1)-\beta}{2\alpha} &\text{if}\quad \alpha \neq 0 \\ \left(\frac{\sqrt{ \beta}}{2}\, t+\eta_1\right)^2-\frac{\gamma }{ \beta} &\text{if}\quad \alpha = 0\, , \end{cases} \end{align*} $$

where $\eta _1$ is an arbitrary integration constant. In particular, we find the following list of equilibrium points (trajectories reduced to one point) of the autonomous Hamiltonian system for generic $\theta _i$ ’s:

$$ \begin{align*}\left\{\begin{array}{rcllll} (u,v)&=& \left(\frac{\theta_\infty-\theta_1}{ \theta_\infty}\, , \frac{\theta_\infty\overline{\theta}}{\theta_\infty-\theta_1}\right) &\mathrm{for} & \eta_0=\frac{(\theta_1-\theta_\infty)^2-(\theta_0+\theta_x-1)}{4}\\ (u,v)&=& \left(\frac{\theta_\infty+\theta_1}{ \theta_\infty}\, , \frac{\theta_\infty \theta}{\theta_\infty+\theta_1}\right) &\mathrm{for} & \eta_0=\frac{(\theta_1+\theta_\infty)^2-(\theta_0+\theta_x-1)}{4} \,. \end{array} \right.\end{align*} $$

We may now compactify the space of initial values $\mathbb {C}^2_{u,v}$ to $\mathcal {S}(0)$ . Figure 6 contains a schematic drawing of how the limits of the components of the infinity set and the exceptional lines arrange in this space. Here, as usual, red lines have self-intersection $(-1)$ . The notable differences with the configuration in $\mathcal {S}(x)$ with $x=0$ are the following, where we use the superscript ‘ $0$ ’ when convenient to indicate particularities for the $x=0$ case:

• • After blow-up of $\beta _0:(u_1,v_1)=(0,0)$ , the point $\beta _x^0:(u_{01},v_{01})=(0,0)$ which has to be blown up lies on the intersection of (the strict transform) of $\mathcal {H}$ and the exceptional line $\mathcal {D}_0=\mathcal {D}_0^0$ .

• • As a result, we still have $\mathcal {H}^*=\{\tilde v_1=0\}\cup \{\tilde v_2=0\}$ , but $\mathcal {D}_0^0=\mathcal {D}_0^{0*}\cup \mathcal {D}_x^0$ .

• • Moreover, the point $\gamma _x^0:(u_{x2},v_{x2})=(0,0)$ now corresponds to the intersection $\mathcal {D}_0^{0*}\cap \mathcal {D}_x^0$ .

• • As a result, we still have $\mathcal {D}_x^{0*}=\{v_{x3}=0\}\cup \{\tilde u_1=0\}$ , but $\mathcal {D}_0^{0*}=\mathcal {D}_0^{0**}\cup \mathcal {E}_x^0$ , where $\mathcal {E}_x^0:\{u_{x3}=0\}\cup \{v_{x4}=0\}. $

• • Finally, the blow-up of $\gamma _0^0:(u_{02},v_{02})=(\theta _0,0)$ yields the strict transform $\mathcal {D}_0^{0***}:\{v_{03}=0\}\cup \{u_{x4}=0\}$ of self-intersection $(-4)$ .

Figure 6

The limit space for $x=0$ of the space of initial values for $x\neq 0$ .

The resulting autonomous vector field in $\mathcal {S}(0)$ is obtained from the one in the Appendix Section B.2 by systematically setting $x=0$ . For convenience of the reader, the formulae are given in the Appendix Section D.

We use the term elliptic base points for a point where the induced autonomous vector field in $\mathcal {S}(0)$ is undetermined. There is one such elliptic base point, given by

$$ \begin{align*}\mathfrak{u}: (u_{x4},v_{x4})=(0,0) \in \mathcal{D}_0^{***}\cap \mathcal{E}_x^0\, .\end{align*} $$

This elliptic base point cannot be resolved by blow-ups!Footnote ¹ Note, however, that the autonomous energy function $E_0$ is well-defined (and infinite) at $\mathfrak {u}$ .

Let us denote $\mathcal {I}^0$ the subset of $\mathcal {S}(0)$ where the autonomous vector field is infinite or undefined. We find

$$ \begin{align*}\mathcal{I}^0=\mathcal{E}_x^{0}\cup \mathcal{D}_x^{0*}\cup\mathcal{H}^*\cup \mathcal{D}_1^*\cup \mathcal{D}_\infty^{**}.\end{align*} $$

This set corresponds precisely to the points where the autonomous energy function $E_0$ is infinite. As explained above, we have

$$ \begin{align*}\lim_{x\to 0} \mathcal{I}(x)=\mathcal{D}_0^{0***}\cup \mathcal{I}^0\, .\end{align*} $$

In order to complete the description of the autonomous vector field in $\mathcal {S}(0)\setminus \{\mathfrak {u} \}$ , it remains to investigate trajectories that might be contained in $\mathcal {S}(0)\setminus \left (\mathcal {I}^0\cup \mathbb {C}^2_{u_0,v_0} \right )$ . We find the following, where, as usual, we assume the values of the $\theta _i$ ’s to be generic:

• • There is no trajectory contained in any of the following:

  • – $\mathcal {E}_\infty \setminus \mathcal {D}_\infty ^{**}:\{\tilde u_3=0\}$ ,

  • – $\mathcal {E}_\infty ^-\setminus \mathcal {D}_\infty ^{**}:\{u_{\infty 3}=0\}$ ,

  • – $\mathcal {E}_1\setminus \mathcal {D}_1^*:\{v_{14}=0\}$ .

• • The line $ \mathcal {E}_0^{0}\setminus \mathcal {D}_0^{0***}:\{v_{04}=0\}$ is the union of one trajectory and one equilibrium point, given by

  $$ \begin{align*}u_{04}=-\frac{ ( \theta_0-\theta)( \theta_0-\overline{\theta}) \theta_0}{2\theta_0+\theta_1-(\theta+\overline{\theta})}\end{align*} $$
  with energy $\eta _0=- (\theta _x-1)\theta _0$ .

• • Every point of $\mathcal {D}_0^{0***}\setminus \mathcal {E}_x^{0}:\{v_{03}=0\}$ is an equilibrium point of the autonomous vector field, with energy $\eta _0 =(u_{03}-(\theta _x-1))(u_{03}+\theta _0)$ .

5 Movable singularities in the Okamoto’s space

In this section, we will consider neighbourhoods of exceptional lines where the Painlevé vector field becomes unbounded. The construction given in Appendix B shows that these are given by the lines $\mathcal {E}_0$ , $\mathcal {E}_x$ , $\mathcal {E}_1$ , $\mathcal {E}_{\infty }$ , $\mathcal {E}_{\infty }^-$ .

5.1 Points where u has a zero and v a pole

The set $\mathcal {E}_0\setminus \mathcal {I}$ is given by $\{v_{04}=0\}$ , in the $(u_{04},v_{04})$ chart (see Section B.2.6). Suppose $u_{04}(\tau )=B$ , $v_{04}(\tau )=0$ , for some complex numbers $\tau $ , B. From the system of differential equations in Section B.2.6, we get:

$$ \begin{align*}\begin{aligned} v_{04}(t) =\ & \frac{e^\tau}{e^\tau-1}(t-\tau)-e^{\tau}\frac{\theta_0(1+e^{\tau}) -\theta_x -\theta_1e^{\tau}+2} {2(e^{\tau}-1)^2} (t-\tau)^2 \\ & +\left( \frac{2B e^{2\tau}(1+e^{\tau})}{3(1-e^{\tau})^3} + F_1(\tau) \right) (t-\tau)^3 + O((t-\tau)^4), \end{aligned} \end{align*} $$

with

$$ \begin{align*}\begin{aligned} F_1(\tau)=\ & - \frac{e^{\tau}}{6(1-e^{\tau})^3} \Bigl[ 3( \theta_0-\theta_x+1) + ( \theta_0-\theta_x)^2 + 3 e^{\tau} - 8 \theta_0 (\theta_x+\theta_1) e^{\tau} +13\theta_0e^{\tau} - 2 \theta_x e^{\tau} \\ &\qquad\qquad \qquad +\theta_1 ( 2 \theta_x-5 ) e^{\tau} +2(\theta+\theta_0)(\bar \theta+\theta_0) e^{\tau} + ( \theta_0-\theta_1)^2 e^{2\tau} \Bigr]\,. \end{aligned} \end{align*} $$

Since (see Section B.2.6)

$$ \begin{align*}u=u_{04}v_{04}^2+\theta_0v_{04}, \quad v=\frac{1}{v_{04}}, \end{align*} $$

we obtain the series expansions for $(u,v)$ :

$$ \begin{align*}\left\{\begin{array}{rcl} u(t)&=&\ \frac{\theta_0 e^\tau}{e^\tau-1}(t-\tau) + \left( B\frac{e^{2\tau}}{(e^{\tau}-1)^2}- \theta_0e^{\tau}\frac{\theta_0(1+e^{\tau})-\theta_x -\theta_1e^{\tau}+2}{2(e^{\tau}-1)^2} \right) (t-\tau)^2 + O((t-\tau)^3)\, , \\ v(t)&=&\ \frac{e^{\tau}-1}{e^{\tau}}\frac{1}{t-\tau} + \frac{ \theta_0(1+e^{\tau})-\theta_x -\theta_1e^{\tau}+2} {2e^{\tau}} +\left(\frac{2B(e^{\tau}+1)}{3(e^{\tau}-1)}+F_2(t)\right)(t-\tau) +O((t-\tau)^2)\, , \end{array} \right. \end{align*} $$

with

$$ \begin{align*}\begin{aligned} F_2(\tau)=\ &\frac{1}{12e^{\tau}(e^{\tau}-1)} \Bigl[ (\theta_0-\theta_x +3)^2 -3 -4( \theta-\theta_0)(\bar \theta-\theta_0) e^{\tau} +4\theta_0^2 e^{\tau} - 2(2\theta_0+1)^2 e^{\tau} \\ & \qquad \qquad\qquad + 2(\theta_0+\theta_x-1 )(\theta_0+ \theta_1+2) e^{\tau} + (\theta_0-\theta_1)^2 e^{2\tau} \Bigr]. \end{aligned} \end{align*} $$

Note that u has a simple zero at $t=\tau $ and v a simple pole with residue $1-e^{-\tau }$ .

5.2 Points where $u=1$ and v has a pole

The set $\mathcal {E}_1\setminus \mathcal {I}$ is given by $\{v_{14}=0\}$ , in the $(u_{14},v_{14})$ chart (see Section B.2.10). Suppose $u_{14}(\tau )=B$ , $v_{14}(\tau )=0$ , for some complex numbers $\tau $ , B. From the system of differential equations in Section B.2.10, we get:

$$ \begin{align*} v_{14}(t) = -(t-\tau)+\frac{\theta+\bar\theta+\theta_1(e^{\tau}-3) - \theta_0e^{\tau}}{2(e^{\tau}-1)}(t-\tau)^2 + \left( \frac{2B(e^{\tau}-2)}{3(1-e^{\tau})}+F_3(\tau) \right) (t-\tau)^3 +O((t-\tau)^4), \end{align*} $$

with

$$ \begin{align*}\begin{aligned} F_3(\tau)=-\frac{1}{6(e^{\tau}-1)^2}&\Bigl[ ( \theta+\bar\theta-5\theta_1)^2 + 2( \theta \bar\theta -5 \theta_1^2)+ ( \theta_0 -\theta_1)^2e^{2\tau} -2( \theta_0( \theta+ \bar\theta+2) + \theta \bar\theta) e^{\tau} \\ & \ + 2(3\theta_1+1)( \theta + \bar\theta + \theta_0-2\theta_1 ) e^{\tau} \ \Bigr]. \end{aligned} \end{align*} $$

Since (see Section B.2.10)

$$ \begin{align*}u=u_{14}v_{14}^2+\theta_1v_{14}+1, \quad v=\frac{1}{v_{14}}, \end{align*} $$

we obtain the series expansions for $(u,v)$ :

$$ \begin{align*}\left\{\begin{array}{rcl} u(t)&=&\ 1- \theta_1(t-\tau) + \left(B+ \theta_1 \frac{\theta+\bar\theta+\theta_1(e^{\tau}-3) - \theta_0e^{\tau}}{2(e^{\tau}-1)} \right)(t-\tau)^2 +O((t-\tau)^3), \\ v(t)&=&\ -\frac{1}{t-\tau} - \frac{\theta+\bar\theta+\theta_1(e^{\tau}-3)-\theta_0e^{\tau}}{2(e^{\tau}-1)} - \left( \frac{2B(e^{\tau}-2)}{3(1-e^{\tau})}+F_4(\tau) \right)(t-\tau) +O((t-\tau)^2), \end{array} \right. \end{align*} $$

with

$$ \begin{align*}F_4(\tau)=F_3(\tau)+\left(\frac{\theta+\bar\theta+\theta_1(e^{\tau}-3) - \theta_0e^{\tau}}{2(e^{\tau}-1)}\right)^2. \end{align*} $$

At $t=\tau $ , $u-1$ has a simple zero, while v has a simple pole with residue $-1$ .

5.3 Points where $u(\tau )=e^{\tau }$ and v has a pole

The set $\mathcal {E}_x\setminus \mathcal {I}$ is given by $\{v_{x4}=0\}$ , in the $(u_{x4},v_{x4})$ chart (see Section B.2.8). Suppose $u_{x4}(\tau )=B$ , $v_{x4}(\tau )=0$ , for some complex number B. From the system of differential equations in Section B.2.8, we get:

$$ \begin{align*}\begin{aligned} v_{x4}=&\ (t-\tau) + \frac{\theta_0 +\theta_x - (\theta +\bar \theta -\theta_x) e^{\tau} }{2(e^{\tau}-1)}(t-\tau)^2 +\left(\frac{ B (e^{2\tau}-1 )}{3e^{\tau}(e^{\tau}-1)^2} + F_5(\tau) \right)(t-\tau)^3 +O((t-\tau)^4), \end{aligned} \end{align*} $$

with

$$ \begin{align*}F_5(\tau)=\frac{ \Bigl[( \theta+\bar \theta-2\theta_x)^2 -3 \theta_{x}^2 +2\theta\bar \theta\Bigr] e^{2\tau} +( \theta_{0}+2\theta_x)^2- \theta_{x}^2 -2 \Bigl[1+\theta_{0}(\theta+\bar \theta)+\theta_{x} -\theta_1+\theta \bar \theta\Bigr]e^{\tau} } {6 (e^{\tau}-1)^2}. \end{align*} $$

Since, as calculated in Section B.2.8:

$$ \begin{align*}u=(u_{x4}v_{x4}+e^{t}\theta_x)v_{x4}+e^{t}, \quad v=\frac{1}{((u_{x4}v_{x4}+e^{t}\theta_x)v_{x4}+e^{t})v_{x4}}, \end{align*} $$

we obtain:

$$ \begin{align*}\left\{ \begin{array}{rcl} u(t)&=&\ e^{\tau}+e^{\tau}(\theta_x+1)(t-\tau) +\left(B+\frac{e^{\tau}}{2}+ e^{\tau}\theta_x\frac{\theta_0 +\theta_x-2 - (\theta_0+\theta_1-3) e^{\tau}}{2(e^{\tau}-1)}\right)(t-\tau)^2+O((t-\tau)^3)\, , \\ v(t)&=&\ \frac{e^{-\tau}}{t-\tau} - e^{-\tau} \frac{\theta_0 -\theta_x-2 - (\theta +\bar \theta -3\theta_x-2) e^{\tau}}{2(e^{\tau}-1)} +e^{-\tau}\left(B\cdot\frac{2-4e^{\tau}}{3e^{\tau}(e^{\tau}-1)}+F_6(\tau)\right)(t-\tau) +O((t-\tau)^2)\, , \end{array} \right. \end{align*} $$

with

$$ \begin{align*}\begin{aligned} F_6(\tau)=-\frac{1}{12(e^\tau -1)^2}\Bigl\{&\left[\theta_\infty^2+2(\theta+\bar \theta+2\theta_x)(\theta_x-3)+(3\theta_x+5)^2-19 \right]e^{2\tau}\\ & -\left[\theta_\infty^2-(\theta_1-2\theta_x)^2+2(\theta_x-12)+(5\theta_x+1)^2 +(\theta_0-6)^2\right]e^\tau \\&+\left[8\theta_x^2+(\theta_0-\theta_x-3)^2-2\right]\Bigr\}. \end{aligned} \end{align*} $$

At $t=\tau $ , obviously, v has a simple pole with residue $e^{-\tau }$ , while $u(t)-e^{\tau }$ has a simple zero.

5.4 Points where u has a pole and v a zero

Such points belong to $\mathcal {E}_{\infty }$ and $\mathcal {E}_{\infty }^-$ , which are obtained by blowing up the points $\beta _{\infty }$ and $\beta _{\infty }^-$ on $\mathcal {D}_{\infty }$ . We notice that the initial vector field (see Section B.2.1) does not depend on the sign of $\theta _{\infty }$ . Moreover, if we replace $\theta _\infty $ by $-\theta _\infty $ , the roles of $\beta _{\infty }$ and $\beta _{\infty }^-$ are interchanged. Because of that symmetry, we may consider only the case when the solution intersects $\mathcal {E}_{\infty }$ .

The set $\mathcal {E}_{\infty }\setminus \mathcal {I}$ is given by $\{\tilde u_3=0\}$ in the $(\tilde u_3,\tilde v_3)$ chart (see Section B.2.4). Suppose $\tilde u_3(\tau )=0$ , $\tilde v_3(\tau )=B$ . From the differential equations in Section B.2.4, we get:

$$ \begin{align*}\begin{aligned} \tilde u_3(t)=&\ \frac{\theta_{\infty}}{1-e^{\tau}}(t-\tau) -\theta_{\infty} \frac{ (\theta_{\infty}+\theta_{x}-2 )e^{\tau}+\theta_{\infty}+\theta_1 +2 B }{2(e^{\tau}-1)^2}(t-\tau)^2 + O((t-\tau)^3). \end{aligned} \end{align*} $$

Then, using the relations:

$$ \begin{align*}u=\frac{1}{\tilde u_3}, \quad v=(\theta -\tilde u_3\tilde v_3)\tilde u_3, \end{align*} $$

we get:

$$ \begin{align*}\left\{ \begin{array}{rcl} u(t)&=&\frac{1-e^{\tau}}{\theta_{\infty}(t-\tau)} + \frac{ (\theta_{\infty}+\theta_{x}-2 )e^{\tau}+\theta_{\infty}+\theta_1 +2 B}{2\theta_\infty} + O(t-\tau), \\ v(t)&= & \frac{\theta\theta_{\infty}}{(1-e^{\tau})}(t-\tau) - \theta_{\infty}\frac{ \theta(\theta_{\infty}+\theta_{x}-2)e^{\tau}+\theta (\theta_{\infty} +\theta_1 )+2(\theta+\theta_{\infty})B}{ 2(e^{\tau}-1)^2} (t-\tau)^2 +O((t-\tau)^3). \end{array} \right. \end{align*} $$

Note that u has a simple pole with residue $(1-e^{\tau })/\theta _{\infty }$ , while v has a simple zero. In the intersection points with $\mathcal {E}_{\infty }^-$ , u has a simple pole with residue $-(1-e^{\tau })/\theta _{\infty }$ and v a simple zero.

6 Estimates and the main result

In this section, we estimate the distance of the vector field from each vertical leaf, for sufficiently small x. These estimates allow us to describe the domain of each solution in $\mathcal S\backslash \mathcal I$ , which is Okamoto’s space of initial values. The results will be used to prove properties of the limit set of each solution.

Given $0<\epsilon <1$ , $\epsilon \in \mathbb R$ , define a disk $R=R_{\epsilon }=\{x\in \mathbb C \mid |x|<\epsilon \}$ . Letting $\xi \in R$ , $r<|\xi |<\epsilon $ , define a disk $D=D_r(\xi )=\{x\in R \bigm | |x-\xi |<r\}$ that lies in the interior of R. Defining a new time coordinate $t=\ln x$ , we have corresponding domains $R_t$ and $D_t$ in the t-plane. Note that $D_t$ is no longer a circular disk but lies inside a rectangular region in the left half of the t-plane (see Figure 7).

Figure 7

Domains $R_{\epsilon }$ and $D_r$ . $R_{\epsilon }$ is the disk centred at the origin with radius $0< \epsilon <1$ . $D_r$ is a disk within $R_{\epsilon }$ and does not contain the origin. The image of $R_{\epsilon }$ by the logarithmic function is the half-plane placed on the left to the boundary $\Re t=\ln \epsilon $ . On the left side of the figure, notice a curvilinear ‘quadrangle’ consisting of two circular arcs centred at the origin and two segments placed on radii of $R_{\epsilon }$ , such that it is circumscribed about $D_r$ . That ‘quadrangle’ is mapped to the red rectangle on the right side, which thus will contain the image of $D_r$ .

The reader may find it useful to consult Figure 4 in the proofs of the following results.

Lemma 6.1. Given $x\in \mathbb C\backslash \{0\}$ , there exists a continuous complex valued function d in a neighbourhood of the infinity set $\mathcal I$ in Okamoto’s space, such that

$$\begin{align*}d=\begin{cases} \frac{1}{E}& \ \textrm{in a neighbourhood of}\ \mathcal{H}^*\cup \mathcal{D}_1^*\cup \mathcal{D}_\infty^{**} \setminus ( \mathcal{D}_0^*\cup\mathcal{D}_x^*) ,\\ -\frac{x-1}{x}\omega_{03}& \ \textrm{in a neighbourhood of}\ \mathcal{D}_0^*\backslash \mathcal{H}^*,\\ -\omega_{x3}& \ \textrm{in a neighbourhood of}\ \mathcal{D}_x^*\backslash \mathcal{H}^*.\\ \end{cases} \end{align*}$$

Note that d vanishes on $\mathcal I$ and that d is not defined at $x=0$ .

Proof. From Section B.2.5, the set $\mathcal {D}_0^*\backslash \mathcal {H}^*$ is given by $v_{03}=0$ in the $(u_{03},v_{03})$ chart. As we approach $\mathcal {D}_0^*$ , we have:

$$ \begin{align*} E\omega_{03}\sim-\frac{x}{x-1},\quad \text{as}\quad v_{03}\to0. \end{align*} $$

From Section B.2.7, the set $\mathcal {D}_x^*\backslash \mathcal {H}^*$ is given by $v_{x3}=0$ in the $(u_{x3},v_{x3})$ chart. As we approach $\mathcal {D}_x^*$ , we have:

$$ \begin{align*} E\omega_{x3}\sim-1-\frac{x}{u_{x3}}\quad\text{as}\quad v_{x3}\to0. \end{align*} $$

Thus, as we approach $\mathcal {H}^*$ : $u_{x3}\to \infty $ , we have that $-\omega _{x3}\sim 1/E$ .

Lemma 6.2. For every $\epsilon>0$ , there exists a neighbourhood U of $\mathcal {D}_{\infty }^{**}$ , such that

$$ \begin{align*} \left| \frac{\dot{E}}{E}+\frac{e^{t}}{e^{t}-1} \right| <\epsilon. \end{align*} $$

Lemma 6.3. For every compact subset K of $\mathcal D_\infty ^{**}\cup \mathcal D_1^{*}\cup \mathcal H^*\backslash \mathcal {D}_x^*$ , there exists a neighbourhood V of K and a constant $C>0$ , such that

$$\begin{align*}\left|e^{-t}\,\frac{\dot E}{E}\right|<C \end{align*}$$

in V for all t, such that $e^t$ is bounded away from $1$ .

Proof. Note that $\mathcal H^*=\{\tilde v_1=0\}\cup \{\tilde v_2=0\}$ is parametrised by $\tilde u_1$ and $\tilde u_2=1/\tilde u_1$ , respectively. Moreover, $ \mathcal {D}_x^*$ is given in these charts by $\{\tilde u_1=x\}$ and $ \{x\tilde u_2=1\}$ . In the respective coordinate charts $(\tilde u_1,\tilde v_1)$ , $(\tilde u_2, \tilde v_2)$ (see Sections B.2.2 and B.2.3), we have

$$\begin{align*}e^{-t}\,\frac{\dot E}{E}=\begin{cases} \frac{\tilde u_1-1}{\left(e^t-1\right) \left(e^t-\tilde u_1\right)}+\frac{ \left(\theta _{x}-1\right)\tilde u_1(\tilde u_1-1)}{e^t-\tilde u_1}\tilde v_1+{O}\left(\tilde v_1^2\right),\\ -\frac{ \tilde u_2-1}{\left(e^t-1\right) \left(e^t \tilde u_2-1\right)}-\frac{ \left(\theta _{x}-1\right)\tilde u_2(\tilde u_2-1)}{ e^t \tilde u_2-1 }\tilde v_2 +{O}\left(\tilde v_2^2\right).\\ \end{cases} \end{align*}$$

So as long as we consider compact subsets of $ \mathcal H^*\backslash \mathcal {D}_x^*$ , the values of $\frac {1}{\tilde u_1-x}$ and $\frac {1}{x\tilde u_2-1}$ are bounded. We have now proven the desired result in a neighbourhood of any compact subset of $\mathcal {H}^*\setminus \mathcal {D}_x^*$ . Since $\mathcal D_\infty ^{**}$ intersects with $\mathcal {H}^*\setminus \mathcal {D}_x^*$ , the result holds in a neighbourhood of $\mathcal D_\infty ^{**}\cap \mathcal H^*$ .

On the other hand, near $\mathcal D_\infty ^{**}\setminus \mathcal H^*$ , given by $\{u_{\infty 4}=0\}$ , we may consider only bounded values of $v_{\infty 4}$ , and so we have (see Section B.2.11)

$$\begin{align*}e^{-t}\,\frac{\dot E}{E}=-\frac{1}{e^t-1}+\left[\theta v_{\infty 4}-\left(v_{\infty 4}+\overline{\theta}\right)\left(v_{\infty 4}+\theta_x-1\right) \right]u_{\infty 4}+O(u_{\infty 4}^2) \, .\end{align*}$$

Hence, the result holds in a neighbourhood of the compact set $\mathcal D_\infty ^{**}=\{u_{\infty 4}=0\}\cup \{\tilde u_2=0\}$ . Similarly, near $\mathcal D_1^{**}\setminus \mathcal H^*$ , given by $\{v_{13}=0\}$ , where we may consider only bounded values of $u_{13}$ , we have (see Section B.2.9)

$$\begin{align*}e^{-t}\,\frac{\dot E}{E}=\frac{(u_{13}+\theta_1)(u_{13}-\theta_x+1)}{(e^t-1)^2}v_{13}+O(v_{13}^2) \, .\end{align*}$$

Hence, the result holds for any compact subset K of $\mathcal D_\infty ^{**}\cup \mathcal D_1^{*}\cup \mathcal H^*\backslash \mathcal {D}_x^*$ and any t as long as $\frac {1}{e^t-1}$ is bounded.

Proof. For the study of the solutions near $\mathcal {D}_x^*\setminus \mathcal {H}^*$ , we use coordinates $(u_{x3},v_{x3})$ (see Section B.2.7). In this chart, the set $\mathcal {D}_x^*\setminus \mathcal {H}^*$ is given by $v_{x3}=0$ and parametrised by $u_{x3}\in \mathbb {C}$ . Moreover, $\mathcal {E}_x$ is given by $\{u_{x3}=0\}$ and parametrised by the variable $v_{x3}$ . From Lemma 6.1, we recall that $d=-\omega _{x3}$ in this chart.

Asymptotically, for $v_{x3}\to 0$ , bounded $u_{x3}$ and $x=e^t$ bounded away from $0$ and $1$ , we have:

(6.1a) $$ \begin{align} \dot u_{x3}&\sim \frac{1}{v_{x3}}, \end{align} $$

(6.1b) $$ \begin{align} \dot v_{x3}&\sim \left(\frac{2u_{x3}}{x(x-1)}-(\theta_0+\theta_x)-\frac{x\theta_1-\theta_x-1}{x-1}\right)v_{x3}, \end{align} $$

(6.1c) $$ \begin{align} \omega_{x3}&\sim -x v_{x3}, \end{align} $$

(6.1d) $$ \begin{align} \frac{\dot \omega_{x3}}{\omega_{x3}}&\sim(1-\theta_0-\theta_1)\frac{x}{x-1}+\frac{2u_{x3}+\theta_0+\theta_x }{x-1}, \end{align} $$

(6.1e) $$ \begin{align} E\omega_{x3}&\sim -1-\frac{x}{u_{x3}}. \end{align} $$

Note that integrating Equation (6.1d) from $t_0$ to $t_1$ , where $t_0, t_1\in D_t$ (see Figure 7) leads to

$$\begin{align*}\log\left(\frac{\omega_{x3}(t_1)}{\omega_{x3}(t_0)}\right)\sim (1-\theta_0-\theta_1)\log\left(\frac{1-e^{t_1}}{1-e^{t_0}}\right)+\int_{t_0}^{t_1}\frac{2u_{x3}+\theta_0+\theta_x }{e^t-1}\,dt. \end{align*}$$

Therefore, if for all t on the line segment from $t_0$ to $t_1$ , we have $|e^{t}-e^{t_0}|\ll |e^{t_0}|$ and $|u_{x3}(t)|$ is bounded, then $\omega _{x3}(t)/\omega _{x3}(t_0)\sim \left ((1-e^{t})/(1-e^{t_0})\right )^{1-\theta _0-\theta _1}$ , where the right side is upperbounded by $e^{t_0}$ . In view of this situation, Equation (6.1c) shows that $v_{x3}$ is approximately given by a small constant. We take $t_0=\tau $ in the following analysis. From (6.1a), it follows that:

$$ \begin{align*} u_{x3}\sim u_{x3}(\tau)+\frac{t-\tau}{v_{x3}(\tau)}. \end{align*} $$

Thus, if t runs over an approximate disk $\Delta $ centred at $\tau $ with radius $|v_{x3}|R$ , then $u_{x3}$ fills an approximate disk centred at $u_{x3}(\tau )$ with radius R. Therefore, if $|v_{x3}|\ll 1/|\tau |$ , the solution has the following properties for $t\in \Delta $ :

$$ \begin{align*} \frac{v_{x3}(t)}{v_{x3}(\tau)}\sim1, \end{align*} $$

and $u_{x3}$ is a complex analytic diffeomorphism from $\Delta $ onto an approximate disk with centre $u_{x3}(\tau )$ and radius R. If R is sufficiently large, we will have $0\in u_{x3}(\Delta )$ , that is the solution of the Painlevé equation will have a pole at a unique point in $\Delta $ . Now, it is possible to take $\tau $ to be the pole point. We have:

$$ \begin{align*} u_{x3}(t)\sim \frac{t-\tau}{v_{x3}(\tau)}\sim -\frac{(t-\tau)e^{\tau}}{d(\tau)}. \end{align*} $$

Let $R_x$ be a large positive real number. Then the equation $|u_{x3}(t)|=R_x$ corresponds to $|t-\tau |\sim |e^{-\tau }d(\tau )|R_x$ , which is still small compared to $|\tau |$ if $|d(\tau )|$ is sufficiently small. It follows that the connected component $\Delta _x$ of the set of all $t\in \mathbb {C}$ , such that $\{t\mid |u_{x3}(t)|\le R_x\}$ is an approximate disk with centre $\tau $ and radius $|d(\tau )e^{-\tau }|R_x$ . More precisely, $u_{x3}$ is a complex analytic diffeomorphism from $\Delta _x$ onto $\{u\in \mathbb {C}\mid |u|\le R_x\}$ , and $ \frac {d(t)}{d(\tau )}\sim 1$ for all $t\in \Delta _x. $

Proof. For the study of the solutions near $\mathcal {D}_0^*\setminus \mathcal {H}^*$ , we use coordinates $(u_{03},v_{03})$ (see Section B.2.5). In this chart, the set $\mathcal {D}_0^*\setminus \mathcal {H}^*$ is given by $v_{03}=0$ and parametrised by $u_{03}\in \mathbb {C}$ . Moreover, $\mathcal {E}_0$ is given by $u_{03}=0$ and parametrised by $v_{03}$ .

Asymptotically, for $v_{03}\to 0$ , bounded $u_{03}$ and $x=e^t$ bounded away from $0$ and $1$ , we have:

(6.2a) $$ \begin{align} &\dot u_{03}\sim -\frac{x}{(1-x)v_{03}}, \end{align} $$

(6.2b) $$ \begin{align} &\dot v_{03}\sim -\frac{(x+1)(2u_{03}+\theta_0)-\theta_x+1-x\theta_1}{x-1}v_{03}, \end{align} $$

(6.2c) $$ \begin{align} &\omega_{03}=-v_{03}, \end{align} $$

(6.2d) $$ \begin{align} &\frac{\dot\omega_{03}}{\omega_{03}} \sim 2u_{03}+\theta_0-\theta_1+\frac{4u_{03}+\theta_0-\theta_1+1}{x-1}, \end{align} $$

(6.2e) $$ \begin{align} &E\omega_{03}\sim -\frac{x}{x-1}. \end{align} $$

Arguments similar to those in the proof of Lemma 6.6 show that $v_{03}$ is approximately equal to a small constant, and from (6.2a), it follows that:

$$ \begin{align*} u_{03}\sim u_{03}(\tau)-\frac{e^t-e^{\tau}}{v_{03}(\tau)}. \end{align*} $$

Thus, if t runs over an approximate disk $\Delta $ centred at $\tau $ with radius $|v_{03}|\log R$ , then $u_{03}$ fills an approximate disk centred at $u_{03}(\tau )$ with radius R. Therefore, if $|v_{03}|\ll e^{-|\tau |}$ , the solution has the following properties for $t\in \Delta $ :

$$ \begin{align*} \frac{v_{03}(t)}{v_{03}(\tau)}\sim1, \end{align*} $$

and $u_{03}$ is a complex analytic diffeomorphism from $\Delta $ onto an approximate disk with centre $u_{03}(\tau )$ and radius R. If R is sufficiently large, we will have $0\in u_{03}(\Delta )$ , that is the solution of the Painlevé equation will vanish at a unique point in $\Delta $ . Now, it is possible to take $\tau $ to be that point. We have:

$$ \begin{align*} u_{03}(t)\sim - \frac{e^t-e^{\tau}}{v_{03}(\tau)}\sim -\frac{(e^t-e^\tau)e^{\tau}}{(e^{\tau}-1)d(\tau)}. \end{align*} $$

Let $R_0$ be a large positive real number. Then the equation $|u_{03}(t)|=R_0$ corresponds to $|1-e^{t-\tau }|\sim |e^{-2\tau }(e^{\tau }-1)d(\tau )|R_0$ , which is still small compared to $|e^{\tau }|$ if $|d(\tau )|$ is sufficiently small. It follows that the connected component $\Delta _0$ of the set of all $t\in \mathbb {C}$ , such that $\{t\mid |u_{03}(t)|\le R_0\}$ is an approximate disk with centre $\tau $ and radius $|d(\tau )(e^{-\tau }+e^{\tau })|R_0$ . More precisely, $u_{03}$ is a complex analytic diffeomorphism from $\Delta _0$ onto $\{u\in \mathbb {C}\mid |u|\le R_0\}$ , and $ \frac {d(t)}{d(\tau )}\sim 1$ for all $ t\in \Delta _0. $

Theorem 6.9. Let $\epsilon _1$ , $\epsilon _2$ , $\epsilon _3$ be given, such that $0<\epsilon _1<1$ , $0<\epsilon _2<1$ , $0<\epsilon _3<1$ . Then there exists $\delta>0$ , such that if $|e^{t_0}|<\epsilon _1$ and $|d(t_0)|<\delta $ , it follows that

$$ \begin{align*} \rho=\inf\{r<|e^{t_0}|\ \text{, such that}\ |d(t)|<\delta\ \text{whenever}\ |e^{t_0}|\ge|e^t|\ge r\} \end{align*} $$

satisfies:

1. (i) $\rho>0$ and is bounded below by the relation:

   $$\begin{align*}\delta\ge|d(t_0)|\left((1-\rho)/|1-e^{t_0}|\right)^{1-\epsilon_2}(1-\epsilon_3);\end{align*}$$

2. (ii) if $|e^{t_0}|\ge |e^t|\ge \rho $ , then

   $$ \begin{align*} d(t)=d(t_0)\left(\frac{1-e^{t}}{1-e^{t_0}}\right)^{1+\varepsilon_2(t)}(1+\varepsilon_3(t)), \end{align*} $$
   where $|\varepsilon _2(t)|\le \epsilon _2$ and $|\varepsilon _3(t)|\le \epsilon _3$ ; and,

3. (iii) if $|e^t|$ is less than $\rho $ , but still sufficiently close to $\rho $ , then $|d(t)|\ge \delta (1-\epsilon _3)$ .

Proof. Suppose a solution of the system (3.1) is close to the infinity set at times $t_0$ and $t_1$ . If follows from Lemmas 6.6 and 6.8 that for every solution close to $\mathcal {I}$ , the set of complex times t, such that the solution is not close to $\mathcal {I}\setminus (\mathcal {D}_0^*\cup \mathcal {D}_x^*)$ is the union of approximate disks of radius $\sim |d|$ . Hence, if the solution is near $\mathcal {I}$ for all complex times t, such that $|e^{t_0}|\ge |e^t|\ge |e^{t_1}|$ , then there exists a path $\mathcal {P}$ from $t_0$ to $t_1$ , such that the solution is close to $\mathcal {I}\setminus (\mathcal {D}_0^*\cup \mathcal {D}_x^*)$ for all $t\in \mathcal {P}$ and $\mathcal {P}$ is $C^1$ -close to the path: $s\mapsto t_1^st_0^{1-s}$ , $s\in [0,1]$ .

Then Lemma 6.2 implies that near $\mathcal {D}_{\infty }^{**}$ :

$$ \begin{align*} \frac{E(t)}{E(t_0)}\sim \frac{1-e^{t_0}}{1-e^t}, \end{align*} $$

and by using Lemma 6.1, we find:

(6.3) $$ \begin{align} d(t)\sim d(t_0)\frac{1-e^{t}}{1-e^{t_0}}. \end{align} $$

For the first statement of the theorem, we have:

$$ \begin{align*} \delta>|d(t)| \ge |d(t_0)|\left(\frac{1-|e^{t}|}{|1-e^{t_0}|}\right)^{1-\epsilon_2}(1-\epsilon_3), \end{align*} $$

and the desired result follows from $\rho \le e^t$ . For $|e^t|\le |e^{t_0}|$ , the second statement follows from (6.3) and the third one from the definition of $\rho $ . The symmetries of the sixth Painlevé equation show that the same statements follow near other lines of the infinity set $\mathcal I$ .

As a consequence of Theorem 6.9, we can prove the repelling property of the set $\mathcal {I}$ .

7 The limit set

Our definition of the limit set is the extension of the standard concept of limit sets in dynamical systems to complex-valued solutions.

Definition 7.1. Let $(u(t),v(t))$ be a solution of (3.1). The limit set $\Omega _{u,v}$ of $(u(t),v(t))$ is the set of all $s\in \mathcal {S}(0)\setminus \mathcal {I}(0)$ , such that there exists a sequence $t_n\in \mathbf {C}$ satisfying:

$$ \begin{align*} \lim_{n\to\infty}\Re(t_n)=-\infty \quad\text{and}\quad \lim_{n\to\infty}(u(t_n),z(v_n))=s. \end{align*} $$

Proof. For any positive numbers $\eta $ , r, let $K_{\eta ,r}$ denote the set of all $s\in \mathcal {S}(x)$ , such that $|x|\le r$ and $|d(s)|\ge \eta $ . Since $\mathcal {S}(x)$ is a complex analytic family over $\mathbb {P}^1\setminus \{0,1\}$ of compact surfaces $\mathcal {S}(x)$ , $K_{\eta ,r}$ is also compact. Furthermore, $K_{\eta ,r}$ is a compact subset of the Okamoto’s space $\mathcal {S}\setminus \mathcal {I}(0)$ . When r approaches $0$ , the sets $K_{\eta ,r}$ shrink to the compact set:

$$ \begin{align*} K_{\eta,0} = \{s\in\mathcal{S}(0)\mid |d(s)\ge\eta \} \subset\mathcal{S}(0)\setminus\mathcal{I}(0). \end{align*} $$

If follows from Theorem 6.9 that there is $\eta>0$ , such that for every solution $(u,v)$ , there exists $r_0>0$ with the following property:

$$ \begin{align*} (u(t),v(t))\in K_{\eta,r_0} \text{ for every } t, \text{ such that } |e^t|\le r_0. \end{align*} $$

Hereafter, we take $r\le r_0$ , when it follows that $(u(t),v(t))\in K_{\eta ,r}$ whenever $|e^t|\le r$ .

Let $T_r=\{t\in \mathbb {C}\mid |e^t|\le r\}$ , and let $\Omega _{(u,v),r}$ denote the closure of the image set $\bigl (u(T_r), v(T_r)\bigr )$ in $\mathcal {S}$ . Since $T_r$ is connected and $(u,v)$ is continuous, $\Omega _{(u,v),r}$ is also connected. Since $\bigl (u(T_r), v(T_r)\bigr )$ is contained in the compact set $K_{\eta , r}$ , its closure $\Omega _{(u,v),r}$ is also contained in $K_{\eta , r}$ , and, therefore, $\Omega _{(u,v),r}$ is a nonempty compact and connected subset of $\mathcal {S}\setminus \mathcal {S}(0)$ . The intersection of a decreasing sequence of nonempty, compact and connected sets is a nonempty, compact and connected. Therefore, as $\Omega _{(u,v),r}$ decreases to $\Omega _{(u,v)}$ as r approaches zero, it follows that $\Omega _{(u,v)}$ is a nonempty, compact and connected subset of $\mathcal {S}$ . Since $\Omega _{(u,v),r}\subset K_{\eta ,r}$ , for all $r\le r_0$ , and the sets $K_{\eta ,r}$ shrink to the compact subset $K_{\eta ,0}$ of $\mathcal {S}(0)\setminus \mathcal {I}(0)$ as r decreases to zero, it follows that $\Omega _{(u,v)}\subset K_{\eta ,0}$ . This proves the first statement of the theorem with $K=K_{\eta ,0}$ .

Since $\Omega _{(u,v)}$ is the intersection of the decreasing family of compact sets $\Omega _{(u,v),r}$ , there exists for every neighbourhood A of $\Omega _{(u,v)}$ in $\mathcal {S}$ , an $r>0$ , such that $\Omega _{(u,v),r}\subset A$ . Hence, $(u(t),v(t))\in A$ for every $t\in \mathbb {C}$ , such that $|e^t|\le r$ . If $\{t_j\}$ is any sequence in $\mathbb {C}\setminus \{0\}$ , such that $|t_j|\to 0$ , then the compactness of $K_{\eta ,r}$ , in combination with $\bigl (u(T_r), v(T_r)\bigr )\subset K_{\eta ,r}$ , implies that there is a subsequence $j=j(k)\to \infty $ as $k\to \infty $ and an $s\in K_{\eta ,r}$ , such that:

$$ \begin{align*} (u(t_{j(k)}),v(t_{j(k)}))\to s\ \text{as}\ k\to\infty. \end{align*} $$

It follows, therefore, that $s\in \Omega _{(u,v)}$ . Next, we prove that $\Omega _{(u,v)}$ is invariant under the flow $\Phi ^{\tau }$ of the autonomous Hamiltonian system. Let $s\in \Omega _{(u,v)}$ and $t_j$ be a sequence in $\mathbb {C}\setminus \{0\}$ , such that $e^{t_j}\to 0$ and $(u(t_j),v(t_j))\to s$ . Since the t-dependent vector field of the Painlevé system converges in $C^1$ to the vector field of the autonomous Hamiltonian system as $e^t\to 0$ , it follows from the continuous dependence on initial data and parameters that the distance between $(u(t_j+\tau ),v(t_j+\tau ))$ and $\Phi ^{\tau }(u(t_j),v(t_j))$ converges to zero as $j\to \infty $ . Since $\Phi ^{\tau }(u(t_j),v(t_j))\to \Phi ^{\tau }(s)$ and $|e^t_j|\to 0$ as $j\to \infty $ , it follows that $(u(t_j+\tau ),v(t_j+\tau ))\to \Phi ^{\tau }(s)$ and $e^{t_j+\tau }\to 0$ as $j\to \infty $ , hence, $\Phi ^{\tau }(s)\in \Omega _{(u,v)}$ .

Proof. For conciseness, we refer to the solution $(u(t),v(t))$ of the system as the Painlevé vector field and denote the vector field near each of the five exceptional lines $\mathcal {E}_0$ , $\mathcal {E}_x$ , $\mathcal {E}_1$ , $\mathcal {E}_{\infty }$ , $\mathcal {E}_{\infty }^-$ by $(U(t), V(t))$ . Furthermore, let

$$\begin{align*}\mathscr E=\mathcal{E}_0\cup\mathcal{E}_x\cup\mathcal{E}_1\cup\mathcal{E}_{\infty}\cup\mathcal{E}_{\infty}^-. \end{align*}$$

Now suppose that $(U(t),V(t))$ intersects $\mathscr E$ only finitely many times. According to Theorem 7.2, the limit set $\Omega _{(u,v)}$ is a compact set in $\mathcal {S}(0)\setminus \mathcal {I}(0)$ . If $\Omega _{(u,v)}$ intersects one the five exceptional lines $\mathcal {E}_0$ , $\mathcal {E}_x$ , $\mathcal {E}_1$ , $\mathcal {E}_{\infty }$ , $\mathcal {E}_{\infty }^-$ at a point p, then there exists a t, such that $e^t$ is arbitrarily close to zero and the Painlevé vector field is arbitrarily close to p, when the transversality of the vector field to the exceptional line implies that $(U(\tau ),V(\tau ))\in \mathscr E$ for a unique $\tau $ ( $\not = t$ ) near t. This is a contradiction to our assumption, as it follows that $(U(t),V(t))$ intersects $\mathscr E$ infinitely many times. Therefore, we must have that $\Omega _{(u,v)}$ is a compact subset of $\mathcal {S}(0) \setminus (\mathcal {I}(0)\cup \mathscr E) $ .

However, $\mathscr E$ is equal to the set of all points in $\mathcal {S}(0)\setminus \mathcal {I}(0)$ , which project (blow-down) to the line $\mathcal D_\infty \cup \mathcal {H}^*$ , and, therefore, $\mathcal {S}(0) \setminus (\mathcal {I}(0)\cup \mathscr E) $ is the affine $(u,v)$ -coordinate chart, of which $\Omega _{(u,v)}$ is a compact subset, which implies that $u(t)$ and $v(t)$ remain bounded for small $|e^t|$ . $|e^t|\to 0$ . From there, $x=0$ is not an essential singularity.

Proof. At the intersection points with $\mathcal {E}_0$ , $\mathcal {E}_1$ , $\mathcal {E}_{\infty }$ , $\mathcal {E}_{\infty }^-$ , the solution will have zeroes, 1s and poles, as explained in detail in Section 5. Thus, the statement for an essential singularity at $x=0$ follows from Proposition 7.3. If $y(x)$ is a solution of (1.1), observe that the following Bäcklund transformations:

$$ \begin{gather*} \mathcal{S}_1: y_1(x_1)=\frac{y}x, \quad x_1=\frac1x, \quad (\theta_{\infty,1},\theta_{0,1},\theta_{1,1},\theta_{x,1}) = \left(\theta_{\infty},\theta_0,\sqrt{\theta_x^2+\frac12},\sqrt{\theta_1^2+\frac12}\right) \\ \mathcal{S}_2: y_2(x_2)=1-y, \quad x_2=1-x, \quad (\theta_{\infty,2},\theta_{0,2},\theta_{1,2},\theta_{x,2}) = \left(\theta_{\infty},i\theta_1,i\theta_0,\theta_x\right) \end{gather*} $$

give the solutions $y_1(x_1)$ and $y_2(x_2)$ of the sixth Painlevé equation with respective parameters $(\theta _{\infty ,1},\theta _{0,1},\theta _{1,1},\theta _{x,1})$ and $(\theta _{\infty ,2},\theta _{0,2},\theta _{1,2},\theta _{x,2})$ , [Reference Olver, Daalhuis, Lozier, Schneider, Boisvert, Clark, Miller, Saunders, Cohl and McClain24, Section 32.7(vii)]. Transformation $\mathcal {S}_1$ maps point $x=\infty $ to $x_1=0$ , while $\mathcal {S}_2$ maps point $x=1$ to $x_2=0$ , thus, the statement will also hold for essential singularities at $x=1$ and $x=\infty $ .