Embedding nonlinear systems with two or more harmonic phase terms near the Hopf–Hopf bifurcation

Abstract

Nonlinear problems involving phases occur ubiquitously throughout applied mathematics andphysics, ranging from neuronal models to the search for elementary particles. The phase variables present in such models usually enter as harmonic terms and, being unbounded, pose an open challenge for studying bifurcations in these systems through standard numerical continuation techniques. Here, we propose to transform and embed the original model equations involving phases into structurally stable generalized systems that are more suitable for analysis via standard predictor–corrector numerical continuation methods. The structural stability of the generalized system is achieved by replacing each harmonic term in the original system by a supercritical Hopf bifurcation normal form subsystem. As an illustration of this general approach, specific details are provided for the ac-driven, Stewart–McCumber model of a single Josephson junction. It is found that the dynamics of the junction is underpinned by a two-parameter Hopf–Hopf bifurcation, detected in the generalized system. The Hopf–Hopf bifurcation gives birth to an invariant torus through Neimark–Sacker bifurcation of limit cycles. Continuation of the Neimark–Sacker bifurcation of limit cycles in the two-parameter space provides a complete picture of the overlapping Arnold tongues (regions of frequency-locked periodic solutions), which are in precise agreement with the widths of the Shapiro steps that can be measured along the current–voltage characteristics of the junction at various fixed values of the ac-drive amplitude.

Appendices

Appendix A: transformation

The original system (1) can be transformed into system (5) by noting that

$$\begin{aligned} \begin{aligned} x\left( \varphi \right)&=\sqrt{p}\cos {\varphi } \\ y\left( \varphi \right)&=\sqrt{p}\sin {\varphi } \end{aligned} \end{aligned}$$

(7)

is the solution to

$$\begin{aligned} \begin{aligned} \frac{{\mathrm{d}}x}{{\mathrm{d}}\varphi }&=px-y-x\left( x^2+y^2\right) ,\\ \frac{{\mathrm{y}}x}{{\mathrm{y}}\varphi }&=x+py-y\left( x^2+y^2\right) , \end{aligned} \end{aligned}$$

(8)

for $p>0$ and with the initial condition

$$\begin{aligned} x\left( 0\right)&=\sqrt{p}, \, \, \, y\left( 0\right) =0. \end{aligned}$$

(9)

We apply the chain rule to the system (8), and using the first equation of (1), to find

$$\begin{aligned} \begin{aligned} \frac{{\mathrm{d}}x}{{\mathrm{d}}t}&=\frac{{\mathrm{d}}x}{{\mathrm{d}}\varphi }\frac{{\mathrm{d}}\varphi }{{\mathrm{d}}t}=\left( px-y-x\left( x^2+y^2\right) \right) V, \\ \frac{{\mathrm{d}}y}{{\mathrm{d}}t}&=\frac{{\mathrm{y}}x}{{\mathrm{y}}\varphi }\frac{{\mathrm{d}}\varphi }{{\mathrm{d}}t}=\left( x+py-y\left( x^2+y^2\right) \right) V. \end{aligned} \end{aligned}$$

(10)

The system (10) is a normal form of the Hopf bifurcation with parameter p that gives birth to a limit cycle with amplitude $\sqrt{p}$ for $p>0$. Hopf bifurcation from $p=0$ can be easily continued in a parametric space as well as the limit cycle with continuation software (e.g., MATCONT).

Similarly, we can replace the harmonic oscillations

$$\begin{aligned} \begin{aligned} u\left( \theta \right)&= \sqrt{a} \cos {\theta } \\ w\left( \theta \right)&= \sqrt{a} \sin {\theta } \end{aligned} \end{aligned}$$

(11)

with a system

$$\begin{aligned} \begin{aligned} \frac{{\mathrm{d}}u}{{\mathrm{d}}\theta }&= a u-w-u\left( u^2+w^2\right) \\ \frac{{\mathrm{d}}w}{{\mathrm{d}}\theta }&=u+ a w-w\left( u^2+w^2\right) \end{aligned} \end{aligned}$$

(12)

and the initial condition

$$\begin{aligned} \begin{aligned} u\left( 0\right)&=\sqrt{a} \, \, \, \, , w\left( 0\right) = 0, \end{aligned} \end{aligned}$$

(13)

where $\sqrt{a} = A$. Once again, we apply the chain rule to the system to obtain a generalized system in the Hopf bifurcation normal form

$$\begin{aligned} \begin{aligned} \frac{{\mathrm{d}}u}{{\mathrm{d}}t}&=\frac{{\mathrm{d}}u}{{\mathrm{d}}\theta }\frac{{\mathrm{d}}\theta }{{\mathrm{d}}t}=\left( au-w-u\left( u^2+w^2\right) \right) \omega \\ \frac{{\mathrm{d}}w}{{\mathrm{d}}t}&=\frac{{\mathrm{w}}\theta }{{\mathrm{d}}\theta }\frac{{\mathrm{d}}\theta }{{\mathrm{d}}t}=\left( u+aw-w\left( u^2+w^2\right) \right) \omega . \end{aligned} \end{aligned}$$

(14)

Equations (14) constitute the last two equations of the system (5) and can be interpreted as a master subsystem of the coupled oscillators. Similarly, (8) form part of the slave subsystem, consisting of the first three equations in (5). Since (8) and (14) each have their own Hopf bifurcation point (at $p=0$ and $a=0$, respectively), together they produce a Hopf–Hopf bifurcation in the overall system  (5), giving birth to a torus for $p>0$ and $a>0$.

Appendix B: generalizations of other coupled systems

As alluded to in the main text, the present technique may be applicable to a wide range of systems involving phase variables. As further examples of how the method may be applied to more complicated systems, we give generalizations for two additional systems related to Josephson junctions.

1.1 B.1 Generalization for RLC-shunted intrinsic Josephson junctions

For a stack of N intrinsic Josephson junctions, in parallel with resistive (R), inductive (L) and capacitive (C) shunting elements, the dimensionless form of the system equations can be written as [68,69,70],

$$\begin{aligned} \begin{aligned} \frac{\mathrm {d}\varphi _{\ell }}{\mathrm {d}t}&= V_{\ell }-\alpha (V_{\ell +1}+V_{\ell -1}-2V_{\ell }), \\ \frac{\mathrm {d}V_{\ell }}{\mathrm {d}t}&= I-\sin \varphi _{\ell }-\beta \frac{\mathrm {d}\varphi _{\ell }}{\mathrm {d}t}-CU, \\ \frac{\mathrm {d}u_{c}}{\mathrm {d}t}&= U, \\ \frac{\mathrm {d}U}{\mathrm {d}t}&= \frac{1}{LC}\left( \sum \limits _{i=1}^{N}V_{i}-u_{c}\right) -\frac{R}{L}U, \end{aligned} \end{aligned}$$

(15)

where $V_{l}$ is the voltage difference, $\varphi _{l}$ is the gauge-invariant phase difference [63], both between superconducting layers $l+1$ and l, with $\ell = 1,2,\ldots , N$. In this model, $u_{c}$ is the voltage across the shunt capacitance, and there is no ac-drive. Once again, we can transform the system into the Cartesian representation by setting

$$\begin{aligned} \begin{aligned} x_{\ell }&= \cos \varphi _{\ell }\text { and } y_{\ell } =\sin \varphi _{\ell }{.} \end{aligned} \end{aligned}$$

(16)

This produces

$$\begin{aligned} \begin{aligned} \frac{\mathrm {d}x_{\ell }}{\mathrm {d}t}&= -\sin \varphi _{\ell }\frac{ \mathrm {d}\varphi _{\ell }}{\mathrm {d}t}\\&=-y_{\ell }\left[ V_{\ell }-\alpha (V_{\ell +1}+V_{\ell -1}-2V_{\ell })\right] \\ \ \frac{\mathrm {d}y_{\ell }}{\mathrm {d}t}&= \cos \varphi _{\ell } \frac{\mathrm {d}\varphi _{\ell }}{\mathrm {d}t}\\&=x_{\ell }\left[ V_{\ell }-\alpha (V_{\ell +1}+V_{\ell -1}-2V_{\ell })\right] \end{aligned} \end{aligned}$$

(17)

from which it is easy to obtain the generalized system of $3N+2$ equations:

$$\begin{aligned} \frac{\mathrm {d}x_{\ell }}{\mathrm {d}t}&=\left( px_{\ell }-y_{\ell }-x_{\ell }\left( x_{\ell }^{2}+y_{\ell }^{2}\right) \right) \nonumber \\&\quad \times \left[ V_{\ell }-\alpha (V_{\ell +1}+V_{\ell -1}-2V_{\ell })\right] , \nonumber \\ \frac{\mathrm {d}y_{\ell }}{\mathrm {d}t}&=\left( x_{\ell }+py_{\ell }-y_{\ell }\left( x_{\ell }^{2}+y_{\ell }^{2}\right) \right) \nonumber \\&\quad \times \left[ V_{\ell }-\alpha (V_{\ell +1}+V_{\ell -1}-2V_{\ell })\right] , \nonumber \\ \frac{\mathrm {d}V_{\ell }}{\mathrm {d}t}&=I-y_{\ell }-\beta \frac{\mathrm {d} \varphi _{\ell }}{\mathrm {d}t}-CU, \nonumber \\ \frac{\mathrm {d}u_{c}}{\mathrm {d}t}&=U, \nonumber \\ \frac{\mathrm {d}U}{\mathrm {d}t}&=\frac{1}{LC}\left( \sum \limits _{i=1}^{N}V_{i}-u_{c}\right) -\frac{R}{L}U. \end{aligned}$$

(18)

The generalized representation (18) reduces to the original when $p=1$.

1.2 B.2 Generalization for the coupled axion-Josephson junction system

Consider the model for a classical axion field that is capacitively coupled into a Josephson junction [10]. In this model, $\varphi$ is the Josephson phase difference, $\theta$ the axion misalignment angle, and the interaction strength is proportional to $c (\ddot{\varphi }-\ddot{\theta })$. Since the coupling constant $c>0$, equations (1) of Ref. [10] can be expressed as a first-order system in the form

$$\begin{aligned} \begin{aligned} \dot{\varphi }&= u, \dot{\theta } = v, \\ \dot{u}&= d\left( ua_{1}+b_{1}\left( 1+c\right) \sin \varphi +cua_{1}+cva_{2}+cb_{2}\sin \theta \right) , \\ \dot{v}&= d\left( va_{2}+b_{2}\left( 1+c\right) \sin \theta +cua_{1}+cva_{2}+cb_{1}\sin \varphi \right) , \end{aligned} \end{aligned}$$

(19)

where $d = -1/(1+2c)$. By making the transformation

$$\begin{aligned} \begin{aligned} x_{1}&= \cos \varphi {, } y_{1} = \sin \varphi , \\ x_{2}&= \cos \theta {, }y_{2}=\sin \theta , \end{aligned} \end{aligned}$$

(20)

we eliminate the phases to obtain

$$\begin{aligned} \begin{aligned} \dot{x}_{1}&= -y_{1}u, \dot{y}_{1} = x_{1}u, \\ \dot{u}&= d\left( ua_{1}+b_{1}\left( 1+c\right) y_{1}+cua_{1}+cva_{2}+cb_{2}y_{2}\right) , \\ \dot{v}&= d\left( va_{2}+b_{2}\left( 1+c\right) y_{2}+cua_{1}+cva_{2}+cb_{1}y_{1}\right) , \\ \dot{x}_{2}&= -y_{2}v{, } \dot{y}_{2} = x_{2}v, \end{aligned} \end{aligned}$$

(21)

which easily generalizes to

$$\begin{aligned} \begin{aligned} \dot{x}_{1}&= \left[ px_{1}-y_{1}-x_{1}\left( x_{1}^{2}+y_{1}^{2}\right) \right] u, \\ \dot{y}_{1}&= \left[ x_{1}+py_{1}-y_{1}\left( x_{1}^{2}+y_{1}^{2}\right) \right] u, \\ \dot{u}&= d\left( ua_{1}+b_{1}\left( 1+c\right) y_{1}+cua_{1}+cva_{2}+cb_{2}y_{2}\right) , \\ \dot{v}&= d\left( va_{2}+b_{2}\left( 1+c\right) y_{2}+cua_{1}+cva_{2}+cb_{1}y_{1}\right) , \\ \dot{x}_{2}&= \left[ qx_{2}-y_{2}-x_{2}\left( x_{2}^{2}+y_{2}^{2}\right) \right] v{, } \\ \dot{y}_{2}&= \left[ x_{2}+qy_{2}-y_{2}\left( x_{2}^{2}+y_{2}^{2}\right) \right] v. \end{aligned} \end{aligned}$$

(22)

The generalized representation (22) reduces to the original when $p=q=1$. We note that a similar transformation may also be applied to a closely related system, namely, the Josephson junction neuron [22].