Nonlinear normal mode-based study of synchronization in delay coupled limit cycle oscillators

Abstract

Nonlinear normal modes (NNMs) are defined as curved invariant structures (manifolds) in the phase-space of a nonlinear dynamical system, which confine the trajectories initiated on them to themselves and hence provide a reduced order subspace for the system to evolve. In this work, NNMs, obtained through graph style parameterization method, are used to study synchronization dynamics of two non-isochronous, frequency detuned limit cycle oscillators (LCOs) under displacement-based/reactive and velocity-based/dissipative coupling, also considering the effect of delayed interactions. The NNMs contain closed curves corresponding to either the in-phase or out-of-phase synchronized oscillations. The motion initiated on these NNMs is found to evolve strictly on them, unless it encounters a fixed point or closed orbit with an out-of-plane unstable direction, in which case it may even transition to the other manifold. In all other cases, the motion settles onto a stable fixed point or orbit on either manifold if existent, or on to a quasi-periodic orbit lying between the two manifolds in the absence of one. The NNMs are used to uncouple the governing equations of the system, directly giving the in-phase and out-of-phase synchronization frequencies and stability. Parametric study using averaged equations as well as the direct numerical integration-based evolution of the original system on the NNMs is used to identify various bifurcations. The properties of NNMs are tested for qualitatively different motions hence determined. NNMs for the delay coupled oscillators are obtained, by first expanding the delay terms in a Taylor series. It is found that even a small delay ($<1\%$ of LCOs time period) brings a significant change in the behavior of the system. The quantitative predictions made by NNM in this case are found to be good for finite delay times ($\approx 10\%$ of LCOs time period), while qualitative agreement was achieved for even larger delays. The results obtained here agree with prior work in the literature, at the same time extending them to hitherto unexplored parameter range and provide a new perspective on the same from the point of view of NNMs.

A Appendix: KBM averaging

The expected solution for the governing equations (4) and (5) is of the form [5].

$$\begin{aligned} x_{1,2}(t)&=A_{1,2}(t)( \cos {\omega {t}+\phi _{1,2}(t)}), \end{aligned}$$

(68)

$$\begin{aligned} \dot{x}_{1,2}(t)&= - \omega A_{1,2}(t) (\sin {\omega {t}+\phi _{1,2}(t)}), \end{aligned}$$

(69)

Equations (68) and (69) can be expressed in exponential form as shown below

$$\begin{aligned} x_{1,2} (t)&=\frac{1}{2}(a_{1,2} e^{i\omega t}+a_{1,2}^* e^{-i\omega t}), \end{aligned}$$

(70)

$$\begin{aligned} \dot{x}_{1,2} (t)&=\frac{i\omega }{2}(a_{1,2} e^{i\omega t}-a_{1,2}^* e^{-i\omega t}), \end{aligned}$$

(71)

where the complex amplitudes $a_{1,2}$ and their complex-conjugates $a_{1,2}^*$ can be expressed as $a_{1,2}=A_{1,2} e^{i\phi _{1,2} }$ and $a_{1,2}^*=A_{1,2} e^{-i\phi _{1,2} }$. Differentiating (70) w.r.t time and substituting (71) will give

$$\begin{aligned} \dot{a}_{1,2} e^{i\omega t}+\dot{a}_{1,2}^* e^{-i\omega t}=0. \end{aligned}$$

(72)

Differentiating (71) w.r.t time and substituting (72) will give

$$\begin{aligned} \ddot{x}_{1,2} (t)=i\omega \dot{a}_{1,2} e^{i\omega t}-\frac{\omega ^2}{2}(a_{1,2} e^{i\omega t}+a_{1,2}^* e^{-i\omega t}). \end{aligned}$$

(73)

Substituting Eqs. (70), (71), (73) in the first equation of motion of the system Eq. (4), then multiplying throughout by $\frac{e^(-i\omega t)}{i\omega }$ gives,

$$\begin{aligned}&\frac{1/8\,i \left( -{\textrm{e}^{2\,i\omega \,t}} {\textit{a}_{{\textit{1}}}}^{3}-3 \,{\textrm{e}^{-2\,i\omega \,t}} \textit{a}_{{\textit{1}}}\,{{\textit{a}_{{\textit{1}}}}^{*}}^{2} -{\textrm{e}^{-4\,i\omega \,t}}{{\textit{a}_{{\textit{1}}}}^{*}}^{3} -3\,{\textit{a}_{{\textit{1}}}}^{2}{\textit{a}_{{\textit{1}}}}^{*} \right) \beta }{\omega }\nonumber \\&\quad +1/8\,i \left( -i{\textrm{e}^{2\,i\omega \,t}}{\textit{a}_{{\textit{1}}}}^{3} +i{\textrm{e}^{-2\,i\omega \,t}}{} \textit{a}_{{\textit{1}}}\,{{\textit{a}_{{\textit{1}}}}^{*}}^{2} +i{\textrm{e}^{-4\,i \omega \,t}}{{\textit{a}_{{\textit{1}}}}^{*}}^{3}\right. \nonumber \\&\qquad \left. -i{\textit{a}_{{\textit{1}}}}^{2}{\textit{a}_{{\textit{1}}}}^{*} -4\,i{\textrm{e}^{-2\,i\omega \,t}}{\textit{a}_{{\textit{1}}}}^{*} +4\,i\textit{a}_{{\textit{1}}} \right) \lambda \nonumber \\&\quad +1/8\,i \left( 4\,i{\textrm{e}^{-2\,i\omega \,t}} {\textit{a}_{{\textit{1}}}}^{*}+4\,i{\textrm{e}^{-i\omega \,\tau }} \textit{a}_{{\textit{2}}}\right. \nonumber \\&\qquad \left. -4\,i{\textrm{e}^{-i\omega \, \left( 2\,t-\tau \right) }}{\textit{a}_{{\textit{2}}}}^{*} -4\,i\textit{a}_{{\textit{1}}} \right) \textit{B}_{{\textit{D}}}\nonumber \\&\quad +{\frac{1/8\,i \left( -4\,{\textrm{e}^{-2\,i \omega \,t}} {\textit{a}_{{\textit{1}}}}^{*}+4\,{\textrm{e}^{-i\omega \,\tau }} \textit{a}_{{\textit{2}}}+4\,{\textrm{e}^{-i\omega \, \left( 2\,t-\tau \right) }}{\textit{a}_{{\textit{2}}}}^{*} -4\,\textit{a}_{{\textit{1}}} \right) \textit{B}_{{\textit{R}}}}{\omega }}\nonumber \\&\quad {+}{\frac{1/8\,i \left( 4\,{\textrm{e}^{{-}2\,i\omega \,t} }{\textit{a}_{{\textit{1}}}}^{*}\,{\omega }^{2}{-}4 \,{\textrm{e}^{{-}2\,i\omega \,t}}{{\omega _1}}^{2} {\textit{a}_{{\textit{1}}}}^{*}{-}8\,i\dot{\textit{a}}_{{\textit{1}}} \,\omega {+}4\,\textit{a}_{{\textit{1}}}\,{\omega }^{2} {-}4\,{{ \omega _1}^2}{} \textit{a}_{{\textit{1}}} \right) }{\omega }}\nonumber \\&\quad {=}0. \end{aligned}$$

(74)

The a and $\dot{a}$ are slow functions of time compared to functions of form $e^{n i \omega t}$ and $e^{-n i \omega t}$. This means that they (slow functions) almost do not change during one period of fast oscillations with the frequency $\omega$. On averaging the whole equation over one period $T=2\pi /\omega$ the fast terms get removed [5]. On averaging the terms containing $e^{-i2\omega t},e^{-i4\omega t},e^{i2\omega t}, e^{i4\omega t}$,...etc. will turn out to be zero. After averaging substitution for a,$\dot{a}$ and $a^*$ gives,

$$\begin{aligned}&3\,i\pi \,{\textit{A}_{{\textit{1}}}}^{3}\beta -\pi \,{\textit{A}_{{\textit{1}}}}^{3}\lambda \,\omega -4\,i \pi \,\textit{A}_{{\textit{1}}}\,{\omega }^{2}\nonumber \\&\quad -8\,i\pi \,\textit{A}_{{\textit{1}}}\,\omega \,{\dot{\phi _1}} +4\,i\pi \,\textit{A}_{{\textit{1}}}\,{ \omega _1}^{2}\nonumber \\&\quad {-}4\,\pi \,\textit{B}_{{\textit{D}}}\,\textit{A}_{{\textit{1}}}\, \omega {+}4\,i\pi \,\textit{B}_{{\textit{R}}}\,\textit{A}_{{\textit{1}}} {+}4\,\pi \,\textit{A}_{{\textit{1}}}\,\lambda \, \omega {-}8\,\pi \,\dot{\textit{A}}_{{\textit{1}}}\,\omega \nonumber \\&\quad +4\,{\frac{\pi \,\textit{A2}\,\omega \,\textit{B}_{{\textit{D}}}\,{\textrm{e}^{i{ \phi _2}}}}{{\textrm{e}^{i\omega \,\tau }}{\textrm{e}^{i{\phi _1}}}}} -{\frac{4\,i\pi \,\textit{B}_{{\textit{R}}}\,\textit{A2} \,{\textrm{e}^{i{ \phi _2}}}}{{\textrm{e}^{i\omega \,\tau }} {\textrm{e}^{i \phi _1 }}}}=0. \end{aligned}$$

(75)

Then, separating the real part and the imaginary part gives,

$$\begin{aligned}&\dot{\textit{A}}_{{\textit{1}}}=-1/8 \, \left( -4\,\textit{A}_{{\textit{2}}}\,\cos \left( \omega \,\tau -\theta \right) +4\,\textit{A}_{{\textit{1}}}\right) \textit{B}_{{\textit{D}}}\nonumber \\&\quad -1/2\,{\frac{\textit{A}_{{\textit{2}}}\, \sin \left( \omega \,\tau -\theta \right) \textit{B}_{{\textit{R}}}}{\omega }}\nonumber \\&\quad -1/8\, \left( {\textit{A}_{{\textit{1}}}}^{3} -4\,\textit{A}_{{\textit{1}}} \right) \lambda , \end{aligned}$$

(76)

$$\begin{aligned} {\dot{\phi }_1}&{=}3/8\,{\frac{{\textit{A}_{{\textit{1}}}}^{2} \beta }{\omega }}{+}1/8\,{\frac{\left( {-}4\,\textit{A}_{{\textit{2}}} \,\cos \left( \omega \,\tau {-}\theta \right) {+}4\,\textit{A}_{{\textit{1}}} \right) \textit{B}_{{\textit{R}}}}{\textit{A}_{{\textit{1}}} \,\omega }}\nonumber \\&\quad {-}1/2\,{\frac{\textit{A}_{{\textit{2}}} \,\sin \left( \omega \,\tau -\theta \right) \textit{B}_{{\textit{D}}}}{\textit{A1}}}\nonumber \\&\quad + 1/8\,{\frac{-4\,\textit{A}_{{\textit{1}}} \,{\omega }^{2}+4\,\textit{A}_{{\textit{1}}} \,{\textit{w1}}^{2}}{\textit{A}_{{\textit{1}}}\,\omega }} \end{aligned}$$

(77)

Similarly, averaging the second governing equation (5) gives,

$$\begin{aligned} \dot{A}_2&=1/8\, \left( 4\,\textit{A}_{{\textit{1}}} \,\cos \left( \omega \,\tau +\theta \right) -4\,\textit{A2} \right) \textit{B}_{{\textit{D}}}\nonumber \\&\quad -1/2\,{\frac{\textit{A}_{{\textit{1}}}\,\sin \left( \omega \,\tau +\theta \right) \textit{B}_{{\textit{R}}}}{\omega }}\nonumber \\&\quad +1/8\, \left( -{\textit{A}_{{\textit{2}}}}^{3} +4\,\textit{A}_{{\textit{2}}} \right) \lambda , \end{aligned}$$

(78)

$$\begin{aligned} \dot{\phi }_2&=3/8\,{\frac{{\textit{A}_{{\textit{2}}}}^{2}\beta }{\omega }} -1/2\,{\frac{\textit{A}_{{\textit{1}}}\,\sin \left( \omega \,\tau +\theta \right) \textit{B}_{{\textit{D}}}}{\textit{A}_{{\textit{2}}}}}\nonumber \\&\quad -1/8\,{\frac{ \left( 4\,\textit{A}_{{\textit{1}}}\,\cos \left( \omega \,\tau +\theta \right) -4\,\textit{A}_{{\textit{2}}}\right) \textit{B}_{{\textit{R}}}}{\textit{A}_{{\textit{2}}}\,\omega }}\nonumber \\&\quad -1/8\,{\frac{4\,\textit{A}_{{\textit{2}}}\,{\omega }^{2} -4\,\textit{A}_{{\textit{2}}}\,{\omega _2}^{2}}{\textit{A}_{{\textit{2}}}\,\omega }}. \end{aligned}$$

(79)

where, $\theta =\phi _2-\phi _1$ is the phase difference between the oscillators. The averaged equations can again be reduced to a three dimensional form as,

$$\begin{aligned} \dot{A}_1&=\frac{\lambda }{8} (4A_1-A_1^3) -\frac{B_R}{2\omega }A_2 \sin {(\omega \tau -\theta )} \nonumber \\&\quad +\frac{B_D}{2}[A_2 \cos {(\omega \tau -\theta )}-A_1], \end{aligned}$$

(80)

$$\begin{aligned} \dot{A}_2&=\frac{\lambda }{8} (4A_2-A_2^3) -\frac{B_R}{2\omega }A_1 \sin {(\omega \tau +\theta )}\nonumber \\&\quad +\frac{B_D}{2}[A_1 \cos {(\omega \tau +\theta )}-A_2], \end{aligned}$$

(81)

$$\begin{aligned} \dot{\theta }&=\varDelta +\frac{3\beta }{8\omega } (A_2^2-A_1^2)\nonumber \\&\quad +\frac{B_R}{2\omega } \left[ \frac{A_2}{A_1}\cos {(\omega \tau -\theta )} -\frac{A_1}{A_2}\cos {(\omega \tau +\theta )}\right] ...\nonumber \\&\quad +\frac{B_D}{2} \left[ \frac{A_2}{A_1}\sin {(\omega \tau -\theta )} -\frac{A_1}{A_2}\sin {(\omega \tau +\theta )}\right] . \end{aligned}$$

(82)

where $\varDelta =\frac{\omega ^2_2-\omega ^2_1}{2\omega }\approx \omega _2-\omega _1$.