Periodic orbit analysis of three dynamical systems for a nonlinear electrical dissipative transmission network

Abstract

A dissipative nonlinear transmission network is studied in the regular regime using both analytical approach and numerical technique that allow us to obtain the stable and unstable periodic orbits of this network in a systematic way. Going from a one-dimensional Ginzburg-Landau equation that governs the dynamics of modulated waves in the network system under consideration and assuming variations in both the amplitude and phase of the network voltage signals, we have proposed an amplitude-phase model consisting of a simple three-dimensional continuous quadratic dynamical system that describes, in the situations of weak dissipation, the dynamics of voltage signals (filters) through the network. We investigate, through the use of dynamic analysis tools such as the perturbation theory, the phase portrait, and Hopf bifurcation theory, the rich dynamics of the proposed, which have some interesting characteristics for different network parameters and initial conditions. Explicit analytical results are obtained for the post-bifurcation periodic orbits and their stability, and the conditions under which the constructed 3D dynamical system undergoes supercritical and/or subcritical Hopf bifurcations are presented. Computer simulations and numerical results are carried out to provide numerical verification of the theoretical studies; our numerical analysis confirms the existence of various type of periodic orbits, stable node-foci, unstable saddle-foci, as well as stable limit cycles. Basins of attraction of equilibrium positions for the derived 3D dynamical system are built by means of the Lyapunov’s direct method.

Appendix

1.1 Appendix A: On the coefficients of Eq. ( 2c )

Coefficients P and Q of Eq. (2c) are defined as $P=P_{r}+iP_{i}$ and $Q=Q_{r}+iQ_{i},$ where $P_{r},$ $P_{i},$ $Q_{r},$ and $Q_{i}$ are four real parameters given in terms of the network parameters as follows:

$$\begin{aligned}&P_{r}=\frac{\omega _{cr}^{4}-\omega ^{4}}{8\omega ^{3}},\nonumber \\&P_{i}=\frac{3\sigma _{1}\omega ^{6}-\omega ^{4}\left[ \left( 2\sigma _{1}+\sigma _{2}\right) u_{0}^{2}-3\left( \sigma _{2}u_{0}^{2}-\sigma _{1}\omega _{0}^{2}\right) \right] -3\omega _{0}^{2}\left( 4u_{0}^{2}+\omega _{0}^{2}\right) \left[ \sigma _{2}u_{0}^{2}+\sigma _{1}\left( \omega ^{2}-\omega _{0}^{2}\right) \right] }{8\omega ^{5}}, \nonumber \\&Q_{r}=\frac{3\beta \omega \left( \omega ^{2}-\omega _{+}^{2}\right) \left( \omega ^{2}-\omega _{-}^{2}\right) }{2\alpha \left[ \left( \omega ^{2}-\omega _{0}^{2}\right) ^{2}+3u_{0}^{2}\omega _{0}^{2}\right] }\exp \left[ -2\chi n\right] , \nonumber \\&Q_{i}=Q_{r}\left[ \frac{\sigma _{2}u_{0}^{2}+\sigma _{1}\left( \omega ^{2}-\omega _{0}^{2}\right) }{u_{0}\omega }+\frac{32\alpha ^{2}\omega ^{3}}{ 3\beta u_{0}^{5}}\frac{\left( \sigma _{2}+2\sigma _{1}\right) \left( \omega ^{2}-\omega _{0}^{2}\right) -u_{0}^{2}\sigma _{2}}{\Delta }\right] , \end{aligned}$$

(A1)

with

(A2)

Here, n is the cell number, $\omega$ is the wavenumber of the carrier, while $\omega _{cr}^{2}=\omega _{0}\sqrt{\omega _{0}^{2}+4u_{0}^{2}},$ and $\chi$ is the linear dissipation coefficient [42].

1.2 Appendix B: Coefficients of Eqs. ( 10a ) and ( 13a )

Analytical expressions of coefficients $\delta _{1},$ $\delta _{2},$ and $\delta _{3}$ of Eq. (10a)

$$\begin{aligned} \left. \begin{array}{l} \delta _{1}=\frac{3\left( \gamma _{6}+\gamma _{1}\phi _{0}\right) }{\gamma _{3}+2\phi _{0}}-2\gamma _{1}, \\ \delta _{2}=\frac{\gamma _{1}^{2}\left( \gamma _{3}+2\phi _{0}\right) -2\gamma _{1}\left( \gamma _{6}+\gamma _{1}\phi _{0}\right) }{\gamma _{3}+2\phi _{0}}+4\phi _{0}\gamma _{3}+4\phi _{0}^{2}+\gamma _{3}^{2}, \\ \delta _{3}=\frac{\left( \gamma _{6}+\gamma _{1}\phi _{0}\right) \left[ 4\left( 4\gamma _{4}+\gamma _{1}^{2}-\gamma _{3}^{2}\right) \phi _{0}^{2}+4\gamma _{3}\left( \gamma _{1}^{2}+4\gamma _{4}-\gamma _{3}^{2}\right) \phi _{0}+\gamma _{3}^{2}\left( 4\gamma _{4}-\gamma _{1}^{2}-\gamma _{3}^{2}\right) +8\gamma _{6}\left( \gamma _{1}\gamma _{3}-\gamma _{6}\right) \right] }{\left( \gamma _{3}+2\phi _{0}\right) ^{3}}. \end{array} \right. \end{aligned}$$

(B1)

Analytical expressions of coefficients $\delta ^{(1)},$ $\delta ^{(2)},$ and $\delta ^{(3)}$ of Eq. (13a)

(B2)

1.3 Appendix C: On the stability analysis for non-zero amplitude wave equilibrium positions

$$\begin{aligned}&\frac{\partial \rho _{1}}{\partial \zeta _{0}}=\rho _{0}s_{1},\nonumber \\&\frac{\partial s_{1}}{\partial \zeta _{0}}=2\gamma _{2}\rho _{0}\rho _{1}+\frac{P_{i}}{\left| P\right| ^{2}}\upsilon _{0}s_{1}+\frac{2\left| P\right| ^{2}\phi _{0}-\upsilon _{0}P_{r}}{\left| P\right| ^{2}}\phi _{1}, \nonumber \\&\frac{\partial \phi _{1}}{\partial \zeta _{0}}=2\rho _{0}\gamma _{5}\rho _{1}+\frac{P_{r}\upsilon _{0}-2\left| P\right| ^{2}\phi _{0}}{ \left| P\right| ^{2}}s_{1}+\frac{P_{i}\upsilon _{0}}{\left| P\right| ^{2}}\phi _{1}; \nonumber \\\end{aligned}$$

(C1)

$$\begin{aligned}&-\rho _{0}s_{2}+\frac{\partial \rho _{2}}{\partial \zeta _{0}}=s_{1}\rho _{1}-\frac{\partial \rho _{1}}{\partial \zeta _{1}}, \nonumber \\&-2\gamma _{2}\rho _{0}\rho _{2}-\frac{P_{i}}{\left| P\right| ^{2}} \upsilon _{0}s_{2}\nonumber \\&\quad +\frac{P_{r}\upsilon _{0}-2\left| P\right| ^{2}\phi _{0}}{\left| P\right| ^{2}}\phi _{2}+\frac{\partial s_{2}}{ \partial \zeta _{0}}=\nonumber \\&\quad -\frac{\upsilon _{2}P_{r}}{\left| P\right| ^{2}} \phi _{0}+\gamma _{2}\rho _{1}^{2}-s_{1}^{2}+\phi _{1}^{2}-\frac{\partial s_{1}}{\partial \zeta _{1}}, \nonumber \\&-2\gamma _{5}\rho _{0}\rho _{2}+\frac{2\left| P\right| ^{2}\phi _{0}-P_{r}\upsilon _{0}}{\left| P\right| ^{2}}s_{2}\nonumber \\&-\quad \frac{P_{i}\upsilon _{0}}{\left| P\right| ^{2}}\phi _{2}+\frac{\partial \phi _{2}}{\partial \zeta _{0}}=\frac{\upsilon _{2}P_{i}}{\left| P\right| ^{2}}\phi _{0}\nonumber \\&\quad +\gamma _{5}\rho _{1}^{2}-2s_{1}\phi _{1}-\frac{ \partial \phi _{1}}{\partial \zeta _{1}}; \end{aligned}$$

(C2)

$$\begin{aligned}&\frac{\partial \rho _{3}}{\partial \zeta _{0}}=\rho _{1}s_{2}+s_{1}\rho _{2}+\rho _{0}s_{3}\nonumber \\&\quad -\frac{\partial \rho _{1}}{\partial \zeta _{2}}-\frac{ \partial \rho _{2}}{\partial \zeta _{1}}, \nonumber \\&-2\gamma _{2}\rho _{0}\rho _{3}-\frac{P_{i}\upsilon _{0}}{\left| P\right| ^{2}}s_{3}\nonumber \\&\quad -\frac{2\left| P\right| ^{2}\phi _{0}-\upsilon P_{r0}}{\left| P\right| ^{2}}\phi _{3}+\frac{\partial s_{3}}{\partial \zeta _{0}}\nonumber \\&\quad =\frac{P_{i}\upsilon _{2}}{\left| P\right| ^{2}}s_{1}-\frac{P_{r}\upsilon _{2}}{\left| P\right| ^{2}}\phi _{1}+2\gamma _{2}\rho _{1}\rho _{2}-2s_{1}s_{2}+2\phi _{1}\phi _{2}\nonumber \\&\quad -\frac{\partial s_{1}}{\partial \zeta _{2}}-\frac{\partial s_{2}}{ \partial \zeta _{1}}, \nonumber \\&-2\gamma _{5}\rho _{0}\rho _{3}-\frac{P_{r}\upsilon _{0}-2\left| P\right| ^{2}\phi _{0}}{\left| P\right| ^{2}}s_{3}\nonumber \\&\quad -\frac{ P_{i}\upsilon _{0}}{\left| P\right| ^{2}}\phi _{3}+\frac{\partial \phi _{3}}{\partial \zeta _{0}}\nonumber \\&\quad =\frac{P_{r}\upsilon _{2}}{\left| P\right| ^{2}}s_{1}+\frac{P_{i}\upsilon _{2}}{\left| P\right| ^{2}}\phi _{1}\nonumber \\&\quad +2\gamma _{5}\rho _{1}\rho _{2}-2s_{1}\phi _{2}-2\phi _{1}s_{2}\nonumber \\&\quad -\frac{\partial \phi _{1}}{\partial \zeta _{2}}-\frac{\partial \phi _{2}}{\partial \zeta _{1}}. \end{aligned}$$

(C3)

1.4 Appendix D: Different parameters appearing in Eq. ( 20b )

$$\begin{aligned} \phi _{230}= & {} \frac{2i\left| P\right| ^{2}k_{1}-P_{i}\upsilon _{0}}{ \left| P\right| ^{2}}+\frac{\gamma _{5}\rho _{0}^{2}\left( 2\left| P\right| ^{2}\phi _{0}-\upsilon _{0}P_{r}\right) }{ \left| P\right| ^{2}\left( 2k_{1}^{2}+\gamma _{2}\rho _{0}^{2}\right) +ik_{1}P_{i}\upsilon _{0}}-\frac{k_{1}\left( P_{r}\upsilon _{0}-2\left| P\right| ^{2}\phi _{0}\right) ^{2}}{\left| P\right| ^{2}\left[ k_{1}P_{i}\upsilon _{0}-i\left| P\right| ^{2}\left( 2k_{1}^{2}+\gamma _{2}\rho _{0}^{2}\right) \right] }, \\ \phi _{231}= & {} \frac{1}{\phi _{230}}\left\{ \gamma _{5}-2s_{10}\phi _{10}+ \frac{\gamma _{5}\rho _{0}\left[ P_{i}s_{10}\upsilon _{0}-\left| P\right| ^{2}\gamma _{2}\rho _{0}-2is_{10}\left| P\right| ^{2}k_{1}\right] }{\left| P\right| ^{2}\left( 2k_{1}^{2}+\gamma _{2}\rho _{0}^{2}\right) +ik_{1}P_{i}\upsilon _{0}}\right. \\&\left. +\frac{2\left| P\right| ^{2}\phi _{0}-P_{r}\upsilon _{0}}{ \left| P\right| ^{2}\rho _{0}}\left[ s_{10}+\frac{k_{1}\left[ \left| P\right| ^{2}\gamma _{2}\rho _{0}-s_{10}P_{i}\upsilon _{0}+2is_{10}\left| P\right| ^{2}k_{1}\right] }{\left[ k_{1}P_{i}\upsilon _{0}-i\left| P\right| ^{2}\left( 2k_{1}^{2}+\gamma _{2}\rho _{0}^{2}\right) \right] }\right] \right\} , \\ \rho _{230}= & {} \frac{\rho _{0}\left( \upsilon _{0}P_{r}-2\left| P\right| ^{2}\phi _{0}\right) }{2\left[ \left| P\right| ^{2}\left( 2k_{1}^{2}+\gamma _{2}\rho _{0}^{2}\right) +ik_{1}P_{i}\upsilon _{0}\right] }\phi _{231}+\frac{P_{i}s_{10}\upsilon _{0}-\left| P\right| ^{2}\gamma _{2}\rho _{0}-2is_{10}\left| P\right| ^{2}k_{1}}{2\left[ \left| P\right| ^{2}\left( 2k_{1}^{2}+\gamma _{2}\rho _{0}^{2}\right) +ik_{1}P_{i}\upsilon _{0}\right] }, \\ s_{230}= & {} \frac{k_{1}\left( \upsilon _{0}P_{r}-2\left| P\right| ^{2}\phi _{0}\right) }{k_{1}P_{i}\upsilon _{0}-i\left| P\right| ^{2}\left( 2k_{1}^{2}+\gamma _{2}\rho _{0}^{2}\right) }\phi _{231}-\frac{1}{ \rho _{0}}\left[ s_{10}+\frac{k_{1}\left[ \left| P\right| ^{2}\gamma _{2}\rho _{0}-P_{i}s_{10}\upsilon _{0}+2is_{10}\left| P\right| ^{2}k_{1}\right] }{k_{1}P_{i}\upsilon _{0}-i\left| P\right| ^{2}\left( 2k_{1}^{2}+\gamma _{2}\rho _{0}^{2}\right) }\right] ; \end{aligned}$$

1.5 Appendix E: Initial conditions appearing in the phase portrait analysis

$$\begin{aligned}&I_{1}=\left( -0.0373,0,0.00236\right) ,\nonumber \\&I_{2}=\left( -0.023,0.0,-0.0013\right) ,\nonumber \\&I_{3}=\left( 0.023,0.0,-0.0013\right) , \\&I_{4}=\left( 0.0373,0,0.00236\right) . \end{aligned}$$

(E1)

$$\begin{aligned}&I_{11}=\left( -0.01,0,0.002\right) , I_{41}=\left( 0.01,0,0.002\right) , \nonumber \\&I_{12}=\left( -0.01,0,-0.002\right) ,\nonumber \\&I_{42}=\left( 0.01,0,-0.002\right) . \end{aligned}$$

(E2)

$$\begin{aligned}&\left. \begin{array}{l} I_{13}=\left( -0.0005,0.003,0.002\right) ,\nonumber \\ I_{21}=\left( -0.023,0.001,-0.0013\right) , \nonumber \\ I_{31}=\left( 0.023,0.001,-0.0013\right) ,\nonumber \\ I_{43}=\left( 0.000\,5,0.003,0.002\right) , \nonumber \\ I_{14}=\left( -0.0005,0.003,-0.002\right) ,\nonumber \\ I_{22}=\left( -0.023,0.001,-0.0013\right) ,\nonumber \\ I_{32}=\left( 0.023,0.001,-0.0013\right) ,\nonumber \\ I_{44}=\left( 0.0005,0.003,-0.002\right) . \end{array} \right. \end{aligned}$$

(E3)

$$\begin{aligned}&i_{1}:I_{11}\left( -0.004,0.00002,-0.000169\right) , \nonumber \\&I_{12}\left( -0.00194,0.00005,-0.000169\right) , \nonumber \\&\quad I_{13}\left( 0.00194,0.00005,-0.000169\right) ,\nonumber \\&I_{14}\left( 0.004,0.00002,-0.000169\right) ; \nonumber \\&i_{2}:I_{21}\left( -0.004,0.0000,-0.000169\right) ,\nonumber \\&I_{22}\left( -0.004,0.00002,-0.000065\right) , \nonumber \\&\quad I_{23}\left( 0.004,0.00002,-0.000065\right) ,\nonumber \\&I_{24}\left( 0.004,0.0000,-0.000169\right) . \end{aligned}$$

(E4)