Analysis and synthesis of oscillator systems described by a perturbed double-well Duffing equation

Abstract

This paper presents an investigation of limit cycles in oscillator systems described by a perturbed double-well Duffing equation. The analysis of limit cycles is made by the Melnikov theory. Expressing the solutions of the unperturbed Duffing equation by Jacobi elliptic functions allows us to calculate explicitly the Melnikov function, whereupon the final result is a function involving the complete elliptic integrals. The Melnikov function is analyzed with the aid of the Picard–Fuchs and Riccati equations. It has been proved that the considered oscillator system can have two small hyperbolic limit cycles located symmetrically with respect to the y-axis, or one large hyperbolic limit cycle, or two large hyperbolic limit cycles, or one large limit cycle of multiplicity 2. Moreover, we have obtained the conditions under which each of these limit cycles arises. The present work gives the conditions for the arising of limit cycles around the homoclinic trajectory. In this connection, an alternative approach is proposed for obtaining a series expansion of the Melnikov function near the homoclinic trajectory. This approach uses the series expansion of the complete elliptic integrals as the elliptic modulus tends to 1. It is shown that a jumping phenomenon may occur between limit cycles in the analyzed oscillator system. The conditions for the occurrence of this jumping phenomenon are given. A method for the synthesis of an oscillator system with a preliminary assigned limit cycle is also presented in the article. The obtained analytical results are illustrated and confirmed by numerical simulations.

Appendix

In this section we give some properties and identities concerning the complete elliptic integrals of the first and second kind and the Jacobi elliptic function, which are used in the presentation. Let us recall that the complete elliptic integrals of the first and second kind, $\mathbf{K}$ and $\mathbf{E}$, are regarded as functions of m, where $m=k^{2}$ and k is a modulus. In this case, we have $\mathbf{K}=\mathbf{K}(m)$ and $\mathbf{E}=\mathbf{E}(m)$.

The derivatives of the complete elliptic integrals $\mathbf{K}$ and $\mathbf{E}$ with respect to m are

$$\begin{aligned} \frac{\hbox {d}{} \mathbf{K}(m)}{\hbox {d}m}= & {} \frac{\mathbf{E}(m)-(1-m)\mathbf{K}(m)}{2m(1-m)}, \quad \frac{\hbox {d}{} \mathbf{E}(m)}{\hbox {d}m}\nonumber \\= & {} \frac{\mathbf{E}(m)-\mathbf{K}(m)}{2m} \end{aligned}$$

(A.1)

The following limit holds

$$\begin{aligned} \mathop {\lim }\limits _{m\rightarrow 1-} (1-m)\mathbf{K}(m)=0 \end{aligned}$$

(A.2)

The Legendre’s Relation

$$\begin{aligned}&\mathbf{E}(m)\mathbf{K}(1-m)+\mathbf{E}(1-m)\mathbf{K}(m)\nonumber \\&\quad -\mathbf{K}(m)\mathbf{K}(1-m)=\pi /2 \end{aligned}$$

(A.3)

at $m=1/2$ give the inequalities

$$\begin{aligned}&[2\mathbf{E}(1/2)-\mathbf{K}(1/2)]>0,\hbox { or }[\mathbf{E}(1/2)\nonumber \\&\quad -(1/2)\mathbf{K}(1/2)]>0. \end{aligned}$$

(A.4)

For small values of m ($m\ll 1$, or $m\rightarrow 0+)$ the following series expansion of $\mathbf{K}$ and $\mathbf{E}$ are valid:

$$\begin{aligned} \mathbf{K}(m)= & {} \frac{\pi }{2}\left\{ {1+\sum _{n=1}^\infty {\left[ {\frac{(2n-1)!!}{(2n)!!}} \right] ^{2}m^{n}} } \right\} =\nonumber \\= & {} \frac{\pi }{2}\left( 1+\frac{1}{4}m+\frac{9}{64}m^{2}+\frac{25}{256}m^{3}\right. \nonumber \\&\left. +\,\frac{1225}{16384}m^{4}+\cdots \right) , \end{aligned}$$

(A.5a)

$$\begin{aligned} \mathbf{E}(m)= & {} \frac{\pi }{2}\left\{ {1-\sum _{n=1}^\infty {\left[ {\frac{(2n-1)!!}{(2n)!!}} \right] ^{2}\frac{1}{2n-1}m^{n}} } \right\} =\nonumber \\= & {} \frac{\pi }{2}\left( 1-\frac{1}{4}m-\frac{3}{64}m^{2}-\frac{5}{256}m^{3}\right. \nonumber \\&\left. -\,\frac{175}{16384}m^{4}+\cdots \right) . \end{aligned}$$

(A.5b)

For values of m close to 1 ($m\rightarrow 1-)$ the following series expansion of $\mathbf{K}$ and $\mathbf{E}$ are valid:

$$\begin{aligned}&{} \mathbf{K}(m)=\ln \frac{4}{\sqrt{1-m}}+\sum _{n=1}^\infty \left\{ \left( {\frac{(2n-1)!!}{(2n)!!}} \right) ^{2}\right. \nonumber \\&\quad \left. \left[ {\ln \frac{4}{\sqrt{1-m}}-2\sum _{j=1}^n {\frac{1}{(2j-1)(2j)}} } \right] \right\} (1-m)^{n} \nonumber \\&\quad =\ln \frac{4}{\sqrt{1-m}}+\frac{1}{4}\left( {\ln \frac{4}{\sqrt{1-m}}-1} \right) (1-m)\nonumber \\&\quad +\frac{9}{64}\left( {\ln \frac{4}{\sqrt{1-m}}-\frac{7}{6}} \right) (1-m)^{2}\nonumber \\&\quad +\frac{25}{256}\left( {\ln \frac{4}{\sqrt{1-m}}-\frac{37}{30}} \right) (1-m)^{3}+\cdots ,\nonumber \\ \end{aligned}$$

(A.6a)

$$\begin{aligned}&{} \mathbf{E}(m)=1+\sum _{n=1}^\infty \left\{ \left( {\frac{(2n-1)!!}{(2n)!!}} \right) ^{2}\frac{2n}{2n-1}\right. \nonumber \\&\quad \left. \left[ \ln \frac{4}{\sqrt{1-m}}+\frac{1}{(2n-1)(2n)}\right. \right. \nonumber \\&\quad \left. \left. -\,2\sum _{j=1}^n {\frac{1}{(2j-1)(2j)}} \right] \right\} (1-m)^{n}\nonumber \\&\quad =1+\frac{1}{2}\left( {\ln \frac{4}{\sqrt{1-m}}-\frac{1}{2}} \right) (1-m)\nonumber \\&\quad +\,\frac{3}{16}\left( {\ln \frac{4}{\sqrt{1-m}}-\frac{13}{12}} \right) (1-m)^{2}\nonumber \\&\quad +\,\frac{15}{128}\left( {\ln \frac{4}{\sqrt{1-m}}-\frac{6}{5}} \right) (1-m)^{3}+\cdots \end{aligned}$$

(A.6b)

The following integral relations are valid [36] (for shorter expressions the argument of elliptic functions of Jacobi is dropped):

$$\begin{aligned} \int \limits _0^{2\mathbf{K}} {\hbox {dn}^{2}\hbox {d}z}= & {} 2\mathbf{E}, \end{aligned}$$

(A.7a)

$$\begin{aligned} \int \limits _0^{2\mathbf{K}} {\hbox {dn}^{4}\hbox {d}z}= & {} \frac{2}{3}\left[ {2(2-k^{2})\mathbf{E}-(1-k^{2})\mathbf{K}} \right] , \end{aligned}$$

(A.7b)

$$\begin{aligned} \int \limits _0^{2\mathbf{K}} {\hbox {dn}^{6}\hbox {d}z}= & {} \frac{2}{15}\left[ (8k^{4}-23k^{2}+23)\mathbf{E}\right. \nonumber \\&\left. +(-4k^{4}+12k^{2}-8)\mathbf{K} \right] , \end{aligned}$$

(A.7c)

$$\begin{aligned} \int \limits _0^{4\mathbf{K}} {\hbox {cn}^{2}\hbox {d}z}= & {} \frac{4}{k^{2}}[\mathbf{E}-(1-k^{2}) \mathbf{K}], \end{aligned}$$

(A.8a)

$$\begin{aligned} \int \limits _0^{4\mathbf{K}} {\hbox {cn}^{4}\hbox {d}z}= & {} \frac{4}{3k^{4}}\left[ {2(2k^{2}-1)\mathbf{E}+(2-5k^{2}+3k^{4})\mathbf{K}} \right] \nonumber \\= & {} \frac{4}{3k^{4}}\left[ {2(2k^{2}{-}1)\mathbf{E}{+}(2{-}3k^{2})(1{-}k^{2})\mathbf{K}} \right] ,\nonumber \\ \end{aligned}$$

(A.8b)

$$\begin{aligned} \int \limits _0^{4\mathbf{K}} {\hbox {cn}^{6}\hbox {d}z}= & {} \frac{4}{15k^{6}}\left[ (8-23k^{2}+23k^{4})\mathbf{E}\right. \nonumber \\&\left. -(8-27k^{2}+34k^{4}-15k^{6})\mathbf{K} \right] \nonumber \\= & {} \frac{4}{15k^{6}}\left[ (8-23k^{2}+23k^{4})\mathbf{E}\right. \nonumber \\&\left. -(8-19k^{2}+15k^{4})(1-k^{2})\mathbf{K} \right] . \end{aligned}$$

(A.8c)

Further let us define the functions

$$\begin{aligned} f(m)= & {} \left[ { (2-m)\mathbf{E}(m)-2 (1-m)\mathbf{K}(m)} \right] , \end{aligned}$$

(A.9)

$$\begin{aligned} g(m)= & {} [2\mathbf{E}(m)-\mathbf{K}(m)], \end{aligned}$$

(A.10)

$$\begin{aligned} F(m)= & {} [\mathbf{E}(m)-(1-m)\mathbf{K}(m)], \end{aligned}$$

(A.11)

$$\begin{aligned} G(m)= & {} 8m(1-m)[2\mathbf{E}(m)-\mathbf{K}(m)]. \end{aligned}$$

(A.12)

The properties of these functions, used in the presentation, are given in the following lemma.

Lemma A1

The following statements are valid:

• (a) $f(0)=0$, $f(1)=1$,

  $$\begin{aligned} f(m)>0\hbox { and }{f}'(m)=(3/2)(\mathbf{K}-\mathbf{E})>0,\nonumber \\ \end{aligned}$$

  (A.13)

  i.e. the function f(m) is positive and monotonically increasing for $m\in (0 , 1)$.

• (b) $g(1/2)=[2\mathbf{E}(1/2)-\mathbf{K}(1/2)]>0$, $g(1)=-\infty$ and

  $$\begin{aligned} {g}'(m)= & {} \frac{-1}{2m(1-m)}[(2m-1)\mathbf{E}(m)\nonumber \\&+(1-m)\mathbf{K}(m)]<0, \hbox {for }m\in (1/2 , 1),\nonumber \\ \end{aligned}$$

  (A.14)

i.e. the function g(m) is monotone decreasing for $m\in (1/2 , 1)$. Moreover, there exists unique value $m_{ss}$, $(1/2)<m_{ss} <1$, for which we have

$$\begin{aligned}&g(m)>0\hbox { for }m\in (1/2 , m_{ss} ), \end{aligned}$$

(A.15a)

$$\begin{aligned}&g(m_{ss} )=[2\mathbf{E}(m_{ss} )-\mathbf{K}(m_{ss} )]=0, \end{aligned}$$

(A.15b)

$$\begin{aligned}&g(m)<0\hbox { for }m\in (m_{ss} , 1). \end{aligned}$$

(A.15c)

(c) $F(0)=0$, $F(1/2)=[\mathbf{E}(1/2)-(1/2)\mathbf{K}(1/2)]>0$, $F(1)=1$,

$$\begin{aligned} {F}'(m)=(1/2)\mathbf{K}(m)>0, \end{aligned}$$

(A.16)

i.e. the function F(m) is positive and monotone increasing for $m\in (0 , 1)$.

(d) $G(1/2)=2[2\mathbf{E}(1/2)-\mathbf{K}(1/2)]>0$, $G(m_{ss} )=0$, $G(1)=0$ and

$$\begin{aligned} {G}'(m)<0\hbox { for }m\in (1/2 , m_{ss} ), \end{aligned}$$

(A.17)

i.e. the function G(m) is monotone decreasing for $m\in (1/2 , m_{ss} )$.

(e) $G(1/2)=4F(1/2)>0$, $G(m_{ss} )=0$, $F(m_{ss} )>0$ and there exists unique value $m_1$, $(1/2)<m_1 <m_{ss}$, such that

$$\begin{aligned} G(m_1 )=F(m_1 )\hbox { for }m_1 \in (1/2 , m_{ss} ). \end{aligned}$$

(A.18)

Proof

The above statements are proved by using the basic properties of the complete elliptic integrals and the facts already proven in the present lemma. A little more complicated is the proof of the inequality in (A.17) and we will outline it now. Using (A.1) we obtain

$$\begin{aligned} {G}'(m)=4[(5-10m)\mathbf{E}(m)-(3-5m)\mathbf{K}(m)]. \end{aligned}$$

The resulting expression is transformed appropriately, for example

${G}'(m)=4\left[ {(2-5m)(2\mathbf{E}-\mathbf{K})+(\mathbf{E}-\mathbf{K})} \right] <0$ for $m\in (1/2 , m_{ss} )$.

This proves (A.17). $\square$

The graphs of Functions F(m) and G(m) are shown in Fig. 13.

Fig. 13

Graph of the Functions F(m) and G(m)