The number of limit cycles of Josephson equation

Xiangqin Yu; Hebai Chen; Changjian Liu

doi:10.3934/dcdsb.2023208

Article Contents

2024, Volume 29, Issue 7: 2947-2971. Doi: 10.3934/dcdsb.2023208

This issue Previous Article Differential-spectral approximation based on reduced-dimension scheme for fourth-order parabolic equation Next Article A multigroup approach to delayed prion production

The number of limit cycles of Josephson equation

1.
School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, 519082, China
2.
School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, 410083, China

^*Corresponding author: Changjian Liu
^*Corresponding author: Changjian Liu

Received: June 2023

Revised: October 2023

Early access: December 2023

Published: July 2024

Abstract / Introduction Full Text(HTML) Figure(9) / Table(3) Related Papers Cited by

Abstract

In this paper, the existence and number of non-contractible limit cycles of the Josephson equation $ \beta \frac{d^{2}\Phi}{dt^{2}}+(1+\gamma \cos \Phi)\frac{d\Phi}{dt}+\sin \Phi = \alpha $ are studied, where $ \phi\in \mathbb S^{1} $ and $ (\alpha,\beta,\gamma)\in \mathbb R^{3} $. Concretely, by using some appropriate transformations, we prove that such type of limit cycles are changed to limit cycles of some Abel equation. By developing the methods on limit cycles of Abel equation, we prove that there are at most two non-contractible limit cycles, and the upper bound is sharp. At last, combining with the results of the paper (Chen and Tang, J. Differential Equations, 2020), we show that the sum of the number of contractible and non-contractible limit cycles of the Josephson equation is also at most two, and give the possible configurations of limit cycles when two limit cycles appear.

Keywords:

Josephson equation,
Abel equation,
limit cycle,
Hopf bifurcation,
monotonic family of differential equations.

Mathematics Subject Classification: 34C07; 34A34; 34C25.

Citation:

Full Text(HTML)

Figure 1. Cross-sections of the bifurcation diagram of system (3)

Download: Full-size image PowerPoint slide

Figure 2. Phase portrait of $ (a,b,c)\in S_{3} $

Download: Full-size image PowerPoint slide

Figure 3. Stability at infinity for $ (a,b,c)\in S_{1} $

Download: Full-size image PowerPoint slide

Figure 4. Stability at infinity for $ (a,b,c)\in S_{5} $

Download: Full-size image PowerPoint slide

Figure 5. Relationship between the functions $ b = \psi_{1}(a,c),b = \lvert c\sqrt{1-a^2}\rvert $ and $ b = \lvert c\rvert $, where "$ 0 $" and "$ 1 $" denote the number of limit cycles in the corresponding region

Download: Full-size image PowerPoint slide

Figure 6. Solution curve of equation (4) when $ x\in[\frac{\pi}{2},\frac{5\pi}{2}] $

Download: Full-size image PowerPoint slide

Figure 7. Solution curve of equation (4) when $ x\in[x^{*},x^{*}+2\pi] $

Download: Full-size image PowerPoint slide

Figure 8. Solution curve of equation (26) when $ x\in[\frac{\pi}{2},\frac{5\pi}{2}] $

Download: Full-size image PowerPoint slide

Figure 9. Numerical limit cycle of equation (1)

Download: Full-size image PowerPoint slide

Table 1. Stability of the zero solution of equation (4)

Classification of parameters	Stability of the zero solution
$ b>0 $	upper unstable and lower stable
$ b=0, a>0 $	upper stable and lower stable
$ a=b=0, c>0 $	upper unstable and lower stable
$ a=b=0, c<0 $	upper stable and lower unstable

| Show Table

DownLoad: CSV

Table 2. Stability of system (4) at infinity

Region	Classification of parameters	Stability of $ y=+\infty $	Stability of $ y=-\infty $
$ S_{1} $	$ 0<b<\psi_{2}(a,c),0<a<a_{*},c<0 $	unstable	stable
$ S_{2} $	$\begin{gathered}\psi_{2}(a,c)<b<\varphi(a,c),0<a<a_{},c<0 \\ 0<b<\varphi(a,c),a_{}\leq a<1,c<0\end{gathered}$	unstable	unstable
$ S_{3} $	$\begin{gathered}\varphi(a,c)<b<\psi_{1}(a,c),0<a<a^{},c<0 \\ \varphi(a,c)<b<-c\sqrt{1-a^{2}},a^{}\leq a<1,c<0\end{gathered}$	unstable	unstable
$ S_{4} $	$ \psi_{1}(a,c)<b<-c\sqrt{1-a^{2}},0<a<a^{*},c<0 $	stable	unstable
$ S_{5} $	$\begin{gathered}b\geq-c\sqrt{1-a^{2}},0<a<a^{},c<0 \\ b>\psi_{1}(a,c),a^{}\leq a<1,c<0 \\ b> \max \{\psi_{1}(a,c),0\},0<a<1,c\geq0\end{gathered}$	stable	unstable
$ S_{6} $	$\begin{gathered}-c\sqrt{1-a^{2}}\leq b<\psi_{1}(a,c),a^{*}<a<1,c<0 \\ 0<b<\psi_{1}(a,c),0<a<1,c\geq0\end{gathered}$	unstable	unstable
$ S_{7} $	$ a>1 $	unstable	unstable
$ HL $	$ b=\varphi(a,c),0<a<1,c<0 $	unstable	unstable
$ SC_{11} $	$\begin{gathered}b=\psi_{1}(a,c),a^{*}\leq a<1,c<0 \\ b=\psi_{1}(a,c),0<a<1,c\geq0\end{gathered}$	-	unstable
$ SC_{12} $	$ b=\psi_{1}(a,c),0<a<a^{*},c<0 $	-	unstable
$ SC_{2} $	$ b=\psi_{2}(a,c),0<a<a_{*}<1,c<0 $	unstable	-
$ P_{1} $	$ a=1,b=\psi_{1}(1,c),c<0 $	-	unstable
$ SN_{1} $	$ a=1,b>\psi_{1}(1,c) $	stable	unstable
$ SN_{2} $	$ a=1,0<b<\psi_{1}(1,c) $	unstable	unstable
$ BT $	$ a=1,b=0,c<0 $	unstable	unstable
$ HLC $	$ 0<a<1,b=c=0 $	unstable	unstable
$ HE $	$ a=0,b=\varphi(0,c),c<0 $	-	-

| Show Table

DownLoad: CSV

Table 3. Sign of $ \frac{dy}{dx} $ in different parts

	Ⅰ	Ⅱ	Ⅲ	Ⅳ	Ⅴ
$ \sin x-a $	+	+	-	-	+
$ y+\frac{f(x)}{g(x)} $	+	-	+	-	+
$ \frac{dy}{dx} $	+	-	-	+	+

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	A. Álvarez, J. Bravo and M. Fernández, Limit cycles of Abel equations of the first kind, J. Math. Anal. Appl., 423 (2015), 734-745. doi: 10.1016/j.jmaa.2014.10.019.
[2]	M. Álvarez, A. Gasull and H. Giacomini, A new uniqueness criterion for the number of periodic orbits of Abel equations, J. Differ. Equ., 234 (2007), 161-176. doi: 10.1016/j.jde.2006.11.004.
[3]	J. Bravo, M. Fernández and A. Gasull, Limit cycles for some Abel equations having coefficients without fixed signs, Int. J. Bifurc. Chaos, 19 (2009), 3869-3876. doi: 10.1142/S0218127409025195.
[4]	J. Bravo, M. Fernández and A. Gasull, Stability of singular limit cycles for Abel equations, Disc. Cont. Dyn. Syst., 35 (2015), 1873-1890. doi: 10.3934/dcds.2015.35.1873.
[5]	J. Bravo, M. Fernández and I. Ojeda, Stability of singular limit cycles for Abel equations revisited, J. Differ. Equ., 379 (2024), 1-25. doi: 10.1016/j.jde.2023.10.003.
[6]	J. Bravo and J. Torregrosa, Abel-like differential equations with no periodic solutions, J. Math. Anal. Appl., 342 (2008), 931-942. doi: 10.1016/j.jmaa.2007.12.060.
[7]	H. Chen and Y. Tang, Global dynamics of the Josephson equation in $TS^{1}$, J. Differ. Equ., 269 (2020), 4884-4913. doi: 10.1016/j.jde.2020.03.048.
[8]	L. A. Cherkas, Number of limit cycles of an autonomous second-order system, Differencial?nye Uravnenija, 12 (1976), 944-946.
[9]	G. F. D. Duff, Limit-cycles and rotated vector fields, Ann. Math., 57 (1953), 15-31. doi: 10.2307/1969724.
[10]	A. Gasull, Some open problems in low dimensional dynamical systems, SeMA J., 78 (2021), 233-269. doi: 10.1007/s40324-021-00244-3.
[11]	A. Gasull, A. Geyer and F. Mañosas, On the number of limit cycles for perturbed pendulum equations, J. Differ. Equ., 261 (2016), 2141-2167. doi: 10.1016/j.jde.2016.04.025.
[12]	A. Gasull, A. Geyer and F. Mañosas, A Chebyshev criterion with applications, J. Differ. Equ., 269 (2020), 6641-6655. doi: 10.1016/j.jde.2020.05.015.
[13]	A. Gasull and A. Guillamon, Limit cycles for generalized Abel equations, Int. J. Bifurc. Chaos, 16 (2006), 3737-3745. doi: 10.1142/S0218127406017130.
[14]	A. Gasull and J. Libre, Limit cycles for a class of Abel equations, SIAM J. Math. Anal., 21 (1990), 1235-1244. doi: 10.1137/0521068.
[15]	M. Han, X. Hou, L. Sheng and C. Wang, Theory of rotated equations and applications to a population model, Discrete Contin. Dyn. Syst., 38 (2018), 2171-2185. doi: 10.3934/dcds.2018089.
[16]	T. Harko and M. K. Mak, Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach, Math. Biosci. Eng., 12 (2015), 41-69. doi: 10.3934/mbe.2015.12.41.
[17]	J. Huang and H. Liang, Estimate for the number of limit cycles of Abel equation via a geometric criterion on three curves, NoDEA. Nonlinear. Differ. Equ. Appl., 24 (2017), 1-31. doi: 10.1007/s00030-017-0469-3.
[18]	J. Huang and H. Liang, A geometric criterion for equation $\dot{x} = \sum_{i = 0}^{m}a_{i}(t)x^{i}$ having at most $m$ isolated periodic solutions, J. Differ. Equ., 268 (2020), 6230-6250. doi: 10.1016/j.jde.2019.11.032.
[19]	J. Huang and Y. Zhao, Periodic solutions for equation $\dot{x} = A(t)x^{m}+B(t)x^{n}+C(t)x^{l}$ with $A(t)$ and $B(t)$ changing signs, J. Differ. Equ., 253 (2012), 73-99. doi: 10.1016/j.jde.2012.03.021.
[20]	Y. Ilyashenko, Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions, Nonlinearity, 13 (2000), 1337-1342. doi: 10.1088/0951-7715/13/4/319.
[21]	R. L. Kautz, On a proposed Josephson-effect voltage standard at zero current bias, Appl. Phys. Lett., 36 (1980), 386-388.
[22]	J. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Int. J. Bifurc. Chaos, 13 (2003), 47-106. doi: 10.1142/S0218127403006352.
[23]	A. Lins-Neto, On the number of solutions of the equation $\frac{dx}{dt} = \sum\limits_{j = 0}^{n} a_{j}(t)x^{j}, 0\leq t\leq 1$, for which $x(0) = x(1)$, Invent. Math., 59 (1980), 67-76. doi: 10.1007/BF01390315.
[24]	N. G. Lloyd, The number of periodic solutions of the equation $\dot{z} = z^{N}+p_{1}(t)z^{N-1}+...+p_{N}(t)$, Proc. London. Math. Soc., 27 (1973), 667-700. doi: 10.1112/plms/s3-27.4.667.
[25]	N. G. Lloyd, A note on the number of limit cycles in certain two-dimensional systems, J. London. Math. Soc., 20 (1979), 277-286. doi: 10.1112/jlms/s2-20.2.277.
[26]	N. G. Lloyd, Small amplitude limit cycles of polynomial differential equations, Lect. Notes. Math., Springer-Verlag, Berlin, 1032 (1983), 346-357. doi: 10.1007/BFb0076806.
[27]	A. A. Panov, On the diversity of Poincaré mappings for cubic equations with variable coefficients, Funct. Anal. Appl., 33 (1999), 310-312. doi: 10.1007/BF02467118.
[28]	L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 2001. doi: 10.1007/978-1-4613-0003-8.
[29]	J. A. Sanders and R. Cushman, Limit cycles in the Josephson equation, SIAM J. Math. Anal., 17 (1986), 495-511. doi: 10.1137/0517039.
[30]	R. J. Soulen and R. P. Giffard, Impedance and noise of an rf-biased RSQUID operated in a nonhysteretic regime, Appl. Phys. Lett., 32 (1978), 770.