Minkowski Dimension and Slow–Fast Polynomial Liénard Equations Near Infinity

Abstract

In planar slow–fast systems, fractal analysis of (bounded) sequences in ${\mathbb {R}}$ has proved important for detection of the first non-zero Lyapunov quantity in singular Hopf bifurcations, determination of the maximum number of limit cycles produced by slow–fast cycles, defined in the finite plane, etc. One uses the notion of Minkowski dimension of sequences generated by slow relation function. Following a similar approach, together with Poincaré–Lyapunov compactification, in this paper we focus on a fractal analysis near infinity of the slow–fast generalized Liénard equations $\dot{x}=y-\sum _{k=0}^{n+1} B_kx^k,\ \dot{y}=-\epsilon \sum _{k=0}^{m}A_kx^k$. We extend the definition of the Minkowski dimension to unbounded sequences. This helps us better understand the fractal nature of slow–fast cycles that are detected inside the slow–fast Liénard equations and contain a part at infinity.

A Family Blow-Up Near Infinity for \(m>2n+1\)

m is odd. To desingularize (23), we use the family blow-up (24) at the origin in $(r,{\bar{y}},\epsilon )$-space. We use different charts. In the family chart $\{{\tilde{\epsilon }}=1\}$ we have

$$\begin{aligned} (r,{\bar{y}},\epsilon )=(v{\tilde{r}},v^{\frac{m+1}{2}-n-1}{\tilde{y}},v^{m-2n-1}) \end{aligned}$$

where $({\tilde{r}},{\tilde{y}})$ is kept in a large compact set. System (23) changes, after division by $v^{\frac{m+1}{2}-n-1}$ and $v\rightarrow 0$, into

$$\begin{aligned} \left\{ \begin{array}{rcl} \dot{{\tilde{r}}} &{}=&{} -{\tilde{r}}\left( {\tilde{y}}-\tilde{r}^{\frac{m+1}{2}-n-1}\right) \\ \dot{{\tilde{y}}} &{}=&{}-A-\frac{m+1}{2}{\tilde{y}} \left( {\tilde{y}}-\tilde{r}^{\frac{m+1}{2}-n-1}\right) . \end{array}\right. \end{aligned}$$

(76)

When $A=1$, system (76) has no singularities. When $A=-1$, (76) has an attracting node at $({\tilde{r}},{\tilde{y}})=(0, \sqrt{\frac{2}{m+1}})$ with eigenvalues $(-\sqrt{\frac{2}{m+1}},-\sqrt{2(m+1)})$ and a repelling node at $({\tilde{r}},{\tilde{y}})=(0, -\sqrt{\frac{2}{m+1}})$ with eigenvalues $(\sqrt{\frac{2}{m+1}},\sqrt{2(m+1)})$.

In the phase directional chart $\{{\tilde{y}}=1\}$ we have

$$\begin{aligned} (r,{\bar{y}},\epsilon )=(v{\tilde{r}},v^{\frac{m+1}{2}-n-1},v^{m-2n-1}{\tilde{\epsilon }}). \end{aligned}$$

System (23) changes, after dividing by $v^{\frac{m+1}{2}-n-1}$, into

$$\begin{aligned} \left\{ \begin{array}{rcl} \dot{{\tilde{r}}} &{}=&{} \frac{2}{m-2n-1}{\tilde{r}} \left( {\tilde{\epsilon }} A+O({\tilde{r}} v {\tilde{\epsilon }})\right) +\frac{2(n+1)}{m-2n-1}\tilde{r} \Psi ({\tilde{r}},v) \\ \dot{v} &{}=&{}-\frac{2}{m-2n-1}v\left( \tilde{\epsilon }A+O({\tilde{r}} v {\tilde{\epsilon }})+\frac{m+1}{2}\Psi (\tilde{r},v)\right) \\ \dot{{\tilde{\epsilon }}} &{}=&{}2{\tilde{\epsilon }}\left( \tilde{\epsilon }A+O({\tilde{r}} v {\tilde{\epsilon }})+\frac{m+1}{2}\Psi (\tilde{r},v)\right) , \end{array}\right. \end{aligned}$$

(77)

where $\Psi ({\tilde{r}},v)=1-{\tilde{r}}^{\frac{m+1}{2}-n-1}(1+O({\tilde{r}} v))$. When $v={\tilde{\epsilon }}=0$, (77) has a hyperbolic saddle at ${\tilde{r}}=0$ with eigenvalues $(\frac{2(n+1)}{m-2n-1},-\frac{m+1}{m-2n-1},m+1)$ and a semi-hyperbolic singularity at ${\tilde{r}}=1$ with the stable manifold $\{v={\tilde{\epsilon }}=0 \}$ and a two-dimensional center manifold transverse to the stable manifold. Using asymptotic expansions in ${\tilde{\epsilon }}$ and the fact that the curve of singularities of (77) is $\{\Psi (\tilde{r},v)=0,{\tilde{\epsilon }}=0\}$, the dynamics inside center manifolds is given by $\{\dot{v}=v{\tilde{\epsilon }}\left( \frac{A}{n+1}+O(v,{\tilde{\epsilon }})\right) ,\dot{{\tilde{\epsilon }}}=-{\tilde{\epsilon }}^2\left( \frac{A(m-2n-1)}{n+1}+O(v,{\tilde{\epsilon }})\right) \}$.

In the phase directional chart $\{{\tilde{y}}=-1\}$ we have

$$\begin{aligned} (r,{\bar{y}},\epsilon )=(v{\tilde{r}},-v^{\frac{m+1}{2}-n-1},v^{m-2n-1}{\tilde{\epsilon }}). \end{aligned}$$

System (23) changes, after dividing by $v^{\frac{m+1}{2}-n-1}$, into

$$\begin{aligned} \left\{ \begin{array}{rcl} \dot{{\tilde{r}}} &{}=&{}- \frac{2}{m-2n-1}{\tilde{r}} \left( {\tilde{\epsilon }} A+O({\tilde{r}} v {\tilde{\epsilon }})\right) -\frac{2(n+1)}{m-2n-1}\tilde{r} \Psi ({\tilde{r}},v) \\ \dot{v} &{}=&{}\frac{2}{m-2n-1}v\left( {\tilde{\epsilon }} A+O({\tilde{r}} v {\tilde{\epsilon }})+\frac{m+1}{2}\Psi ({\tilde{r}},v)\right) \\ \dot{{\tilde{\epsilon }}} &{}=&{}-2{\tilde{\epsilon }}\left( {\tilde{\epsilon }} A+O({\tilde{r}} v {\tilde{\epsilon }})+\frac{m+1}{2}\Psi (\tilde{r},v)\right) , \end{array}\right. \end{aligned}$$

(78)

where $\Psi ({\tilde{r}},v)=1+{\tilde{r}}^{\frac{m+1}{2}-n-1}(1+O({\tilde{r}} v))$. When $v={\tilde{\epsilon }}=0$, (78) has a hyperbolic saddle at ${\tilde{r}}=0$ with eigenvalues $(-\frac{2(n+1)}{m-2n-1},\frac{m+1}{m-2n-1},-(m+1))$.

We find one extra singularity in the phase directional chart $\{{\tilde{r}}=1\}$

$$\begin{aligned} (r,{\bar{y}},\epsilon )=(v,v^{\frac{m+1}{2}-n-1}\tilde{y},v^{m-2n-1}{\tilde{\epsilon }}). \end{aligned}$$

System (23) changes, after dividing by $v^{\frac{m+1}{2}-n-1}$, into

$$\begin{aligned} \left\{ \begin{array}{rcl} \dot{{\tilde{y}}} &{}=&{}-{\tilde{\epsilon }} A+O( v {\tilde{\epsilon }}) -(n+1){\tilde{y}}\left( {\tilde{y}}-1+O(v)\right) \\ \dot{v} &{}=&{}-v\left( {\tilde{y}}-1+O(v)\right) \\ \dot{{\tilde{\epsilon }}} &{}=&{}(m-2n-1){\tilde{\epsilon }}\left( {\tilde{y}}-1+O(v)\right) . \end{array}\right. \end{aligned}$$

(79)

When $v={\tilde{\epsilon }}=0$, (79) has a hyperbolic saddle at ${\tilde{y}}=0$ with eigenvalues $(n+1,1,2n+1-m)$.

To desingularize (26), we use the family blow-up (24) at the origin in $(r,{\bar{y}},\epsilon )$-space. As usual we work with different charts. In the family chart $\{\tilde{\epsilon }=1\}$ system (26) changes, after division by $v^{\frac{m+1}{2}-n-1}$ and $v\rightarrow 0$, into

$$\begin{aligned} \left\{ \begin{array}{rcl} \dot{{\tilde{r}}} &{}=&{} {\tilde{r}}\left( {\tilde{y}}+(-1)^n\tilde{r}^{\frac{m+1}{2}-n-1}\right) \\ \dot{{\tilde{y}}} &{}=&{}A+\frac{m+1}{2}{\tilde{y}} \left( \tilde{y}+(-1)^n{\tilde{r}}^{\frac{m+1}{2}-n-1}\right) . \end{array}\right. \end{aligned}$$

(80)

When $A=1$, system (80) has no singularities. When $A=-1$, (80) has a repelling node at $({\tilde{r}},{\tilde{y}})=(0, \sqrt{\frac{2}{m+1}})$ with the eigenvalues $\sqrt{\frac{2}{m+1}}(1,m+1)$ and an attracting node at $({\tilde{r}},{\tilde{y}})=(0, -\sqrt{\frac{2}{m+1}})$ with the eigenvalues $-\sqrt{\frac{2}{m+1}}(1,m+1)$.

In the phase directional chart $\{{\tilde{y}}=1\}$ system (26) changes, after dividing by $v^{\frac{m+1}{2}-n-1}$, into

$$\begin{aligned} \left\{ \begin{array}{rcl} \dot{{\tilde{r}}} &{}=&{} -\frac{2}{m-2n-1}{\tilde{r}} \left( {\tilde{\epsilon }} A+O({\tilde{r}} v {\tilde{\epsilon }})\right) -\frac{2(n+1)}{m-2n-1}\tilde{r} \Psi ({\tilde{r}},v) \\ \dot{v} &{}=&{}\frac{2}{m-2n-1}v\left( {\tilde{\epsilon }} A+O({\tilde{r}} v {\tilde{\epsilon }})+\frac{m+1}{2}\Psi ({\tilde{r}},v)\right) \\ \dot{{\tilde{\epsilon }}} &{}=&{}-2{\tilde{\epsilon }}\left( {\tilde{\epsilon }} A+O({\tilde{r}} v {\tilde{\epsilon }})+\frac{m+1}{2}\Psi (\tilde{r},v)\right) , \end{array}\right. \end{aligned}$$

(81)

where $\Psi ({\tilde{r}},v)=1+\tilde{r}^{\frac{m+1}{2}-n-1}((-1)^n+O({\tilde{r}} v))$. When $v={\tilde{\epsilon }}=0$, (81) has a hyperbolic saddle at ${\tilde{r}}=0$ with eigenvalues $(-\frac{2(n+1)}{m-2n-1},\frac{m+1}{m-2n-1},-(m+1))$ and, if n is odd, a semi-hyperbolic singularity at ${\tilde{r}}=1$ with the unstable manifold $\{v={\tilde{\epsilon }}=0 \}$ and a two-dimensional center manifold transverse to the unstable manifold. The dynamics inside center manifolds is given by $\{\dot{v}=v{\tilde{\epsilon }}\left( -\frac{A}{n+1}+O(v,{\tilde{\epsilon }})\right) ,\dot{{\tilde{\epsilon }}}={\tilde{\epsilon }}^2\left( \frac{A(m-2n-1)}{n+1}+O(v,{\tilde{\epsilon }})\right) \}$.

In the phase directional chart $\{{\tilde{y}}=-1\}$ system (26) changes, after dividing by $v^{\frac{m+1}{2}-n-1}$, into

$$\begin{aligned} \left\{ \begin{array}{rcl} \dot{{\tilde{r}}} &{}=&{} \frac{2}{m-2n-1}{\tilde{r}} \left( {\tilde{\epsilon }} A+O({\tilde{r}} v {\tilde{\epsilon }})\right) +\frac{2(n+1)}{m-2n-1}\tilde{r} \Psi ({\tilde{r}},v) \\ \dot{v} &{}=&{}-\frac{2}{m-2n-1}v\left( {\tilde{\epsilon }} A+O({\tilde{r}} v {\tilde{\epsilon }})+\frac{m+1}{2}\Psi ({\tilde{r}},v)\right) \\ \dot{{\tilde{\epsilon }}} &{}=&{}2{\tilde{\epsilon }}\left( {\tilde{\epsilon }} A+O({\tilde{r}} v {\tilde{\epsilon }})+\frac{m+1}{2}\Psi (\tilde{r},v)\right) , \end{array}\right. \end{aligned}$$

(82)

where $\Psi ({\tilde{r}},v)=1-\tilde{r}^{\frac{m+1}{2}-n-1}((-1)^n+O({\tilde{r}} v))$. When $v={\tilde{\epsilon }}=0$, (82) has a hyperbolic saddle at ${\tilde{r}}=0$ with eigenvalues $(\frac{2(n+1)}{m-2n-1},-\frac{m+1}{m-2n-1},m+1)$ and, if n is even, a semi-hyperbolic singularity at ${\tilde{r}}=1$ with the stable manifold $\{v={\tilde{\epsilon }}=0 \}$ and a two-dimensional center manifold transverse to the stable manifold. The dynamics inside center manifolds is given by $\{\dot{v}=v{\tilde{\epsilon }}\left( \frac{A}{n+1}+O(v,{\tilde{\epsilon }})\right) ,\dot{{\tilde{\epsilon }}}={\tilde{\epsilon }}^2\left( -\frac{A(m-2n-1)}{n+1}+O(v,{\tilde{\epsilon }})\right) \}$.

We find one extra singularity in the phase directional chart $\{{\tilde{r}}=1\}$. System (26) changes, after dividing by $v^{\frac{m+1}{2}-n-1}$, into

$$\begin{aligned} \left\{ \begin{array}{rcl} \dot{{\tilde{y}}} &{}=&{}{\tilde{\epsilon }} A+O( v {\tilde{\epsilon }}) +(n+1){\tilde{y}}\left( {\tilde{y}}+(-1)^n+O(v)\right) \\ \dot{v} &{}=&{}v\left( {\tilde{y}}+(-1)^n+O(v)\right) \\ \dot{{\tilde{\epsilon }}} &{}=&{}-(m-2n-1){\tilde{\epsilon }}\left( \tilde{y}+(-1)^n+O(v)\right) . \end{array}\right. \end{aligned}$$

(83)

When $v={\tilde{\epsilon }}=0$, (83) has a hyperbolic saddle at ${\tilde{y}}=0$ with eigenvalues $(-1)^n(n+1,1,2n+1-m)$.

m is even. To desingularize (28), we use the family blow-up (29) at the origin in $(r,{\bar{y}},\epsilon )$-space. In the family chart $\{{\tilde{\epsilon }}=1\}$ system (28) changes, after division by $v^{m-2n-1}$ and $v\rightarrow 0$, into

$$\begin{aligned} \left\{ \begin{array}{rcl} \dot{{\tilde{r}}} &{}=&{} -\frac{1}{2}{\tilde{r}}\left( {\tilde{y}}-\tilde{r}^{m-2n-1}\right) \\ \dot{{\tilde{y}}} &{}=&{}-A-\frac{m+1}{2}{\tilde{y}} \left( {\tilde{y}}-\tilde{r}^{m-2n-1}\right) . \end{array}\right. \end{aligned}$$

(84)

System (84) has no singularities because $A=1$.

In the phase directional chart $\{{\tilde{y}}=1\}$ (28) changes, after dividing by $v^{m-2n-1}$, into

$$\begin{aligned} \left\{ \begin{array}{rcl} \dot{{\tilde{r}}} &{}=&{} \frac{1}{m-2n-1}{\tilde{r}} \left( {\tilde{\epsilon }} A+O({\tilde{r}} v {\tilde{\epsilon }})\right) +\frac{n+1}{m-2n-1}{\tilde{r}} \Psi ({\tilde{r}},v) \\ \dot{v} &{}=&{}-\frac{1}{m-2n-1}v\left( {\tilde{\epsilon }} A+O({\tilde{r}} v {\tilde{\epsilon }})+\frac{m+1}{2}\Psi ({\tilde{r}},v)\right) \\ \dot{{\tilde{\epsilon }}} &{}=&{}2{\tilde{\epsilon }}\left( {\tilde{\epsilon }} A+O({\tilde{r}} v {\tilde{\epsilon }})+\frac{m+1}{2}\Psi (\tilde{r},v)\right) , \end{array}\right. \end{aligned}$$

(85)

where $\Psi ({\tilde{r}},v)=1-{\tilde{r}}^{m-2n-1}(1+O({\tilde{r}} v))$. When $v={\tilde{\epsilon }}=0$, (85) has a hyperbolic saddle at ${\tilde{r}}=0$ with eigenvalues $(\frac{n+1}{m-2n-1},-\frac{m+1}{2(m-2n-1)},m+1)$ and a semi-hyperbolic singularity at ${\tilde{r}}=1$ with the stable manifold $\{v={\tilde{\epsilon }}=0 \}$ and a two-dimensional center manifold transverse to the stable manifold. The dynamics inside center manifolds is given by $\{\dot{v}=v{\tilde{\epsilon }}\left( \frac{A}{2(n+1)}+O(v,{\tilde{\epsilon }})\right) ,\dot{{\tilde{\epsilon }}}=-{\tilde{\epsilon }}^2\left( \frac{A(m-2n-1)}{n+1}+O(v,{\tilde{\epsilon }})\right) \}$.

Since $m-2n-1$ is odd, we can cover the phase directional chart $\{{\tilde{y}}=-1\}$ by applying $(t,{\tilde{r}},v)\mapsto (-t,-\tilde{r},-v)$ to (85). When $v={\tilde{\epsilon }}=0$, we find a hyperbolic saddle at ${\tilde{r}}=0$ with eigenvalues $(-\frac{n+1}{m-2n-1},\frac{m+1}{2(m-2n-1)},-(m+1))$.

We find one extra singularity in the phase directional chart $\{{\tilde{r}}=1\}$ in which system (28) changes, after dividing by $v^{m-2n-1}$, into

$$\begin{aligned} \left\{ \begin{array}{rcl} \dot{{\tilde{y}}} &{}=&{}-{\tilde{\epsilon }} A+O( v {\tilde{\epsilon }}) -(n+1){\tilde{y}}\left( {\tilde{y}}-1+O(v)\right) \\ \dot{v} &{}=&{}-\frac{1}{2}v\left( {\tilde{y}}-1+O(v)\right) \\ \dot{{\tilde{\epsilon }}} &{}=&{}(m-2n-1){\tilde{\epsilon }}\left( \tilde{y}-1+O(v)\right) . \end{array}\right. \end{aligned}$$

(86)

When $v={\tilde{\epsilon }}=0$, (86) has a hyperbolic saddle at ${\tilde{y}}=0$ with eigenvalues $(n+1,\frac{1}{2},2n+1-m)$.

To desingularize (30), we use the family blow-up (29) at the origin in $(r,\bar{y},\epsilon )$-space. In the family chart $\{{\tilde{\epsilon }}=1\}$ system (30) changes, after division by $v^{m-2n-1}$ and $v\rightarrow 0$, into

$$\begin{aligned} \left\{ \begin{array}{rcl} \dot{{\tilde{r}}} &{}=&{}\frac{1}{2} {\tilde{r}}\left( {\tilde{y}}+(-1)^n\tilde{r}^{m-2n-1}\right) \\ \dot{{\tilde{y}}} &{}=&{}-A+\frac{m+1}{2}{\tilde{y}} \left( \tilde{y}+(-1)^n{\tilde{r}}^{m-2n-1}\right) . \end{array}\right. \end{aligned}$$

(87)

Since $A=1$, system (87) has a repelling node at $({\tilde{r}},{\tilde{y}})=(0, \sqrt{\frac{2}{m+1}})$ with the eigenvalues $\sqrt{\frac{2}{m+1}}(\frac{1}{2},m+1)$ and an attracting node at $({\tilde{r}},{\tilde{y}})=(0, -\sqrt{\frac{2}{m+1}})$ with the eigenvalues $-\sqrt{\frac{2}{m+1}}(\frac{1}{2},m+1)$.

In the phase directional chart $\{{\tilde{y}}=1\}$ system (30) changes, after dividing by $v^{m-2n-1}$, into

$$\begin{aligned} \left\{ \begin{array}{rcl} \dot{{\tilde{r}}} &{}=&{} \frac{1}{m-2n-1}{\tilde{r}} \left( {\tilde{\epsilon }} A+O({\tilde{r}} v {\tilde{\epsilon }})\right) -\frac{n+1}{m-2n-1}{\tilde{r}} \Psi ({\tilde{r}},v) \\ \dot{v} &{}=&{}\frac{1}{m-2n-1}v\left( -{\tilde{\epsilon }} A+O({\tilde{r}} v {\tilde{\epsilon }})+\frac{m+1}{2}\Psi ({\tilde{r}},v)\right) \\ \dot{{\tilde{\epsilon }}} &{}=&{}-2{\tilde{\epsilon }}\left( -{\tilde{\epsilon }} A+O({\tilde{r}} v {\tilde{\epsilon }})+\frac{m+1}{2}\Psi (\tilde{r},v)\right) , \end{array}\right. \end{aligned}$$

(88)

where $\Psi ({\tilde{r}},v)=1+{\tilde{r}}^{m-2n-1}((-1)^n+O({\tilde{r}} v))$. When $v={\tilde{\epsilon }}=0$, (88) has a hyperbolic saddle at ${\tilde{r}}=0$ with eigenvalues $(-\frac{n+1}{m-2n-1},\frac{m+1}{2(m-2n-1)},-(m+1))$ and, if n is odd, a semi-hyperbolic singularity at ${\tilde{r}}=1$ with the unstable manifold $\{v={\tilde{\epsilon }}=0 \}$ and a two-dimensional center manifold transverse to the unstable manifold. The dynamics inside center manifolds is given by $\{\dot{v}=v{\tilde{\epsilon }}\left( \frac{A}{2(n+1)}+O(v,{\tilde{\epsilon }})\right) ,\dot{{\tilde{\epsilon }}}=-{\tilde{\epsilon }}^2\left( \frac{A(m-2n-1)}{n+1}+O(v,{\tilde{\epsilon }})\right) \}$.

We cover the phase directional chart $\{{\tilde{y}}=-1\}$ by applying $(t,{\tilde{r}},v)\mapsto (-t,-{\tilde{r}},-v)$ to (88). When $v={\tilde{\epsilon }}=0$, we find a hyperbolic saddle at ${\tilde{r}}=0$ with eigenvalues $(\frac{n+1}{m-2n-1},-\frac{m+1}{2(m-2n-1)},m+1)$ and, if n is even, a semi-hyperbolic singularity at ${\tilde{r}}=1$ with the stable manifold $\{v={\tilde{\epsilon }}=0 \}$ and a two-dimensional center manifold transverse to the stable manifold. The dynamics inside center manifolds is given by $\{\dot{v}=-v{\tilde{\epsilon }}\left( \frac{A}{2(n+1)}+O(v,{\tilde{\epsilon }})\right) ,\dot{{\tilde{\epsilon }}}={\tilde{\epsilon }}^2\left( \frac{A(m-2n-1)}{n+1}+O(v,{\tilde{\epsilon }})\right) \}$.

We find one extra singularity in the phase directional chart $\{{\tilde{r}}=1\}$. System (30) changes, after dividing by $v^{m-2n-1}$, into

$$\begin{aligned} \left\{ \begin{array}{rcl} \dot{{\tilde{y}}} &{}=&{}-{\tilde{\epsilon }} A+O( v {\tilde{\epsilon }}) +(n+1){\tilde{y}}\left( {\tilde{y}}+(-1)^n+O(v)\right) \\ \dot{v} &{}=&{}\frac{1}{2}v\left( {\tilde{y}}+(-1)^n+O(v)\right) \\ \dot{{\tilde{\epsilon }}} &{}=&{}-(m-2n-1){\tilde{\epsilon }}\left( {\tilde{y}}+(-1)^n+O(v)\right) . \end{array}\right. \end{aligned}$$

(89)

When $v={\tilde{\epsilon }}=0$, (89) has a hyperbolic saddle at ${\tilde{y}}=0$ with eigenvalues $(-1)^n(n+1,\frac{1}{2},2n+1-m)$.