Nonexistence of Periodic Orbits for Forced–Damped Potential Systems in Bounded Domains

Abstract

We prove $L^r$-estimates on periodic solutions of periodically forced, linearly damped mechanical systems with polynomially bounded potentials. The estimates are applied to obtain a nonexistence result of periodic solutions in bounded domains, depending on an upper bound on the gradient of the potential. The results are illustrated on examples.

Appendix: An Estimate on the Positive Root of Certain Polynomials

In this section, we prove an upper bound on the roots of certain polynomial equations, appearing in the estimates for nonlinearly softening potentials. They are of theoretical interest, as, for the analytical use of the estimates with nonlinearities greater than four, no explicit solution formula for the polynomial roots exists.

Lemma 4

Let A, B and C be positive real numbers and let $s\ge 2$ be a real number. The unique positive root $y^*$ of the polynomial

$$\begin{aligned} P(y)=Ay^s-By-C, \end{aligned}$$

(75)

can be bounded from above as

$$\begin{aligned} y^*\le \overline{y}+\sqrt{2\frac{|P(\overline{y})|}{P''(\overline{y})}}, \end{aligned}$$

(76)

where

$$\begin{aligned} \overline{y}=\left( \frac{B}{sA}\right) ^{\frac{1}{s-1}}. \end{aligned}$$

(77)

Proof

First, we note that the derivative of the polynomial P,

$$\begin{aligned} P'(y)=sAy^{s-1}-B, \end{aligned}$$

(78)

has a unique positive zero, which we denote as

$$\begin{aligned} \overline{y}=\left( \frac{B}{sA}\right) ^{\frac{1}{s-1}}. \end{aligned}$$

(79)

In particular,

$$\begin{aligned} \begin{aligned} P(\overline{y})&=A\left( \frac{B}{sA}\right) ^{\frac{s}{s-1}}-B\left( \frac{B}{sA}\right) ^{\frac{1}{s-1}}-C=\frac{(AB)^{\frac{s}{s-1}}}{(As)^{\frac{s+1}{s-1}}}\left( s^{\frac{1}{s-1}}-s^{\frac{s}{s-1}}\right) -C\\ {}&<0, \end{aligned} \end{aligned}$$

(80)

which follows from $s^{s-1}>1$, by assumption.

Since $y\mapsto P(y)$ is convex for $y>0$,

$$\begin{aligned} P''(y)=s(s-1)Ay^{s-2}> 0, \end{aligned}$$

(81)

by the assumption $s\ge 2$, it follows that, indeed, P has a unique positive solution $y^*$ with $\overline{y}\le y^*$.

By (79), we can write

$$\begin{aligned} P(y)=\int _{\overline{y}}^{y}\int _{\overline{y}}^{\eta }p''(\xi )\, \mathrm{d}\xi \, \mathrm{d}\eta + P(\overline{y}). \end{aligned}$$

(82)

Since, again by the assumption that $s\ge 2$, the function $y\mapsto P''(y)$ is monotonically increasing,

$$\begin{aligned} P'''(y)=s(s-1)(s-2)y^{s-3}\ge 0, \end{aligned}$$

(83)

for $y>0$, and it follows that $\inf _{\xi \in [\overline{y},\eta ]}P''(\xi )=P''(\overline{y})$, for any $\eta \ge \overline{y}$. Therefore, we can bound (82) as

$$\begin{aligned} \begin{aligned} P(y)&=\int _{\overline{y}}^{y}\int _{\overline{y}}^{\eta }P''(\xi )\, \mathrm{d}\xi \, \mathrm{d}\eta + P(\overline{y})\\&\ge \int _{\overline{y}}^{y}\int _{\overline{y}}^{\eta }P''(\overline{y})\, \mathrm{d}\xi \, \mathrm{d}\eta + P(\overline{y})=P''(\overline{y})\int _{\overline{y}}^{y}(\eta -\overline{y})\, \mathrm{d}\eta + P(\overline{y})\\ {}&=\frac{P''(\overline{y})}{2}y^2-\overline{y}P''(\overline{y})y+\frac{P''(\overline{y})\overline{y}^2}{2}+P(\overline{y}), \end{aligned} \end{aligned}$$

(84)

for $y\ge \overline{y}$.

In particular, the unique positive zero of P can be bounded from above by the positive root of the right-hand side in (84), i.e., by

$$\begin{aligned} y^{+}=\overline{y}+\sqrt{2\frac{|P(\overline{y})|}{P''(\overline{y})}}, \end{aligned}$$

(85)

where we have used (80). This proves the claim. $\square$

Remark 8

The quadratic function $y\mapsto \frac{P''(\overline{y})}{2}y^2-\overline{y}p''(\overline{y})y+\frac{P''(\overline{y})\overline{y}^2}{2}+P(\overline{y})$ in (84) defines a parabola with crest at $\overline{y}$. Since the global minimum of P is attained at $\overline{y}$ as well and since the growth rate of P is greater than or equal to the growth rate of the parabola by the assumption $p\ge 2$, we can, indeed, bound the zero of P from above by the positive zero of the parabola, c.f. Fig. 6.

We note that—thanks to the special structure of the polynomial, bound (76) improves classical, general a priori bounds on the zeros of polynomials. We compare (76), e.g., with the Lagrange bound (Lagrange and Poinsot 1826),

$$\begin{aligned} y^*\le \max \left\{ 1,\sum _{k=0}^{n-1}{\left| \frac{P_k}{P_n}\right| }\right\} , \end{aligned}$$

(86)

for the polynomial $P(y)=\sum _{k=1}^{n}P_ky^k$. For the polynomial $P(y)=y^5-y-1$, for which $y^*=1.1673$, we find that (86) predicts $y^*\le 2$, while the parabolic bound (76) predicts $y^*\le 1.38516$.

Fig. 6

The polynomial functions $P(y)=y^5-y-1$ and parabola (84) with $\overline{y}=0.66874$. The positive zero of the parabola is attained at 1.38516, while the unique positive zero of P is attained at 1.1673