Non-linear Beatings as Non-stationary Synchronization of Weakly Coupled Autogenerators

Abstract

The present work represents a selective overview of recent advances in the area dealing with a new type of synchronization in systems of weakly coupled active oscillators. The description is focused on the evolution of non-stationary beat wise self-sustained oscillations developed as attractors and repellers when the system non-linearity and dissipation is varied.

Appendix: Symmetry Analysis

The conditions of the system invariance leads to partial differential equations for the constituents of operator (3) as follows

$$ \begin{aligned} & \eta_{\tau } + p_{1} \eta_{{R_{1} }} + p_{2} \eta_{{R_{2} }} + p_{3} \eta_{\varDelta } - p_{1} \xi_{\tau } - p_{1}^{2} \xi_{{R_{1} }} - p_{1} p_{2} \xi_{{R_{2} }} - p_{1} p_{3} \xi_{\varDelta } \\ & \quad = \xi \left[ { - \gamma p_{1} + 3bR_{1}^{2} p_{1} - 10dR_{1}^{4} p_{1} + \beta p_{2} \sin \varDelta + p_{3} \beta R_{2} \cos \varDelta } \right] \\ & \quad + \eta \left[ { - \gamma + 3bR_{1}^{2} - 10dR_{1}^{4} } \right] + \zeta \left[ {\beta \,\sin \varDelta } \right] + \varsigma \left[ {\beta R_{2} \cos \varDelta } \right] \\ \end{aligned} $$

$$ \begin{aligned} & \zeta_{\tau } + p_{1} \zeta_{{R_{1} }} + p_{2} \zeta_{{R_{2} }} + p_{3} \zeta_{\Delta } - p_{2} \xi_{\tau } - p_{1} p_{2} \xi_{{R_{1} }} - p_{2}^{2} \xi_{{R_{2} }} - p_{2} p_{3} \xi_{\Delta } \\ & \quad = \xi \left[ { - \gamma p_{2} + 3bR_{2}^{2} p_{2} - 10dR_{2}^{4} p_{2} + \beta p_{1} sin\Delta - p_{3} \beta R_{1} cos\Delta } \right] \\ & \quad + \eta \left[ {\beta \,sin\Delta } \right] + \zeta \left[ { - \gamma + 3bR_{2}^{2} - 10dR_{2}^{4} } \right] + \varsigma \left[ { - \beta R_{1} cos\Delta } \right] \\ \end{aligned} $$

(1A)

$$ \begin{aligned} & \varsigma_{\tau } + p_{1} \varsigma_{{R_{1} }} + p_{2} \varsigma_{{R_{2} }} + p_{3} \varsigma_{\varDelta } - p_{3} \xi_{\tau } - p_{1} p_{3} \xi_{{R_{1} }} - p_{1} p_{2} \xi_{{R_{2} }} - p_{3}^{2} \xi_{\varDelta } \\ & \quad = \xi \left[ { - 3\alpha \left( {2R_{2} p_{2} - 2R_{1} p_{1} } \right) - \beta \left( {\cos \varDelta \left( {R_{1} p_{2} - R_{2} p_{1} } \right)\frac{{\left( {R_{1}^{2} + R_{2}^{2} } \right)}}{{R_{1}^{2} R_{2}^{2} }} + \beta \,\sin \varDelta \frac{{\left( {R_{2}^{2} - R_{1}^{2} } \right)}}{{R_{1} R_{2} }}p_{3} } \right)} \right] \\ & \quad + \eta \left( {6\alpha R_{1}^{2} + \beta \,\cos \varDelta \frac{{\left( {R_{1}^{2} + R_{2}^{2} } \right)}}{{R_{1}^{2} R_{2} }}} \right) - \zeta \left( {6\alpha R_{2}^{2} + \beta \,\cos \varDelta \frac{{\left( {R_{1}^{2} + R_{2}^{2} } \right)}}{{R_{1} R_{2}^{2} }}} \right) + \varsigma \left( {\beta \,\sin \varDelta \frac{{\left( {R_{2}^{2} - R_{1}^{2} } \right)}}{{R_{1} R_{2} }}} \right) \\ \end{aligned} $$

Solving Eq. (1A), gives the Lie group operator

$$ \begin{aligned} X & = \frac{\partial }{{\partial \tau_{1} }} + ( - \gamma R_{1} + bR_{1}^{3} - 2dR_{1}^{5} + \beta R_{2} \sin \Delta )\frac{\partial }{{\partial R_{1} }} + ( - \gamma R_{2} + bR_{2}^{3} - 2dR_{2}^{5} - \beta R_{1} \sin \Delta )\frac{\partial }{{\partial R_{2} }} \\ & \quad + \left( { - 3\alpha (R_{2}^{2} - R_{1}^{2} ) - \beta \frac{{(R_{2}^{2} - R_{1}^{2} )}}{{R_{1} R_{2} }}\cos \Delta } \right)\frac{\partial }{\partial \Delta } + X_{1} \\ \end{aligned} $$

If exists, the corresponding invariant, say I, must satisfy the condition

$$ \begin{aligned} XI & \equiv \frac{\partial I}{{\partial \tau_{1} }} + ( - \gamma R_{1} + bR_{1}^{3} - 2dR_{1}^{5} + \beta R_{2} \sin \varDelta )\frac{\partial I}{{\partial R_{1} }} \\ & \quad + ( - \gamma R_{2} + bR_{2}^{3} - 2dR_{2}^{5} - \beta R_{1} \sin \varDelta )\frac{\partial }{{\partial R_{2} }} \\ & \quad + \left( { - 3\alpha (R_{2}^{2} - R_{1}^{2} ) - \beta \frac{{(R_{2}^{2} - R_{1}^{2} )}}{{R_{1} R_{2} }}\cos \varDelta } \right)\frac{\partial I}{\partial \varDelta } + X_{1} I = 0 \\ \end{aligned} $$

As mentioned in the main text, it is assumed that $I{\kern 1pt} = I\left( {R_{1} ,R_{2} } \right)$. In this case, the variables $R_{1} ,R_{2}$ satisfy the following ordinary differential equation, which can be also directly obtained from first two Eqs. (3),

$$ \frac{{dR_{1} }}{{( - \gamma R_{1} + bR_{1}^{3} - 2dR_{1}^{5} + \beta R_{2} \sin \varDelta )}} = \frac{{dR_{2} }}{{( - \gamma R_{2} + bR_{2}^{3} - 2dR_{2}^{5} - \beta R_{1} \sin \varDelta )}} $$

(2A)

The group operator becomes a rotation operator with the invariant $I{\kern 1pt} \equiv N = R_{1}^{2} + R_{2}^{2}$ if the parameters of the system (3) satisfy the relation $b^{2} = 9\gamma d/2$ while the initial conditions provide the excitation level $N = 2b/3d$. This can be easily proved using the combination of 1st equation of (3) multiplied by R₁ minus 2nd equation of (3) multiplied by R_2.