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Symmetry breaking-induced state-dependent aging and chimera-like death state

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Abstract

Aging transition refers to the manifestation of the homogeneous steady state beyond a critical ratio of the inactive oscillators in a network of active and inactive oscillators. In contrast, symmetry breaking in coupled dynamical systems always results in heterogeneous states. In this work, surprisingly, we find that the symmetry breaking coupling facilitates the onset of the homogeneous steady state among the population at the critical proportion of the inactive oscillators despite the presence of a large number of the active oscillators. Further, increase in the natural frequency of the oscillation and the number of the inactive oscillators are conducive to the onset of the aging transition. Interestingly, chimera-like death state is observed in the study related to the aging transition for the first time in the literature in addition to the actual chimera death states among the active oscillator for the random initial conditions. The critical curves corresponding to the stability of the aging transition and the chimera-like death states are deduced, which agrees perfectly with the simulation results. Even a feeble decrease in the feedback factor in the coupling destabilizes the stable aging state thereby facilitating the oscillatory state in the entire parameter space despite the presence of a large proportion of the inactive oscillators increasing the robustness of the network against disorders.

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Acknowledgements

IG wishes to thank SASTRA Deemed University for research fund and extending infrastructure support to carry out this work. KS and DVS is supported by the CSIR EMR Grant No. 03(1400)/17/EMR-II. The work of VKC forms part of a research project sponsored by CSIR Project under Grant No. 03(1444)/18/EMR-II.

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Authors and Affiliations

  1. Centre for Nonlinear Science and Engineering, School of Electrical and Electronics Engineering, SASTRA Deemed University, Thanjavur, 613 401, India

    I. Gowthaman & V. K. Chandrasekar

  2. School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram, Kerala, 695551, India

    K. Sathiyadevi & D. V. Senthilkumar

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  1. I. Gowthaman
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  2. K. Sathiyadevi
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Appendices

Appendix A: Aging state

The coefficients \(\beta _0,~\beta _1,~\beta _2,\) and \(\beta _3\) for the above characteristic eigenvalue equation (3) are given by

$$\begin{aligned} \beta _0= & {} (a^2+\omega ^2)\beta _{01}+ak\beta _{02}, \nonumber \\ \beta _1= & {} b^2k(p-1+\alpha )+a^2\beta _{11}+\varepsilon \beta _{12}\omega ^2+b\beta _{13}, \nonumber \\ \beta _2= & {} a^2+b^2-\varepsilon ^2\alpha (1-\alpha )+a \beta _{21}+\beta _{22}, \nonumber \\ \beta _3= & {} 2(b-a)+\varepsilon (2\alpha -1), \end{aligned}$$
(11)

where

$$\begin{aligned} \beta _{01}= & {} b^2+b\varepsilon (\alpha -p)+\omega ^2, \\ \beta _{02}= & {} a\varepsilon (b\varepsilon \alpha (\alpha -1)+(b^2+1)(p-1+\alpha )),\\ \beta _{11}= & {} (2 b + \varepsilon (-p + \alpha )),\\ \beta _{12}= & {} (2\alpha -1),\\ \beta _{13}= & {} \varepsilon ^2 (-1 + \alpha ) \alpha + 2 \omega ^2,\\ \beta _{21}= & {} -4b+\varepsilon (1+p-3\alpha ), \\ \beta _{22}= & {} b\varepsilon (p-2+3\alpha )+3\omega ^2. \end{aligned}$$

Appendix B: Chimeralike state

The coefficients \(\gamma _0,~\gamma _1,~\gamma _2,~\gamma _3,~\gamma _4,\) and \(\gamma _5\) for the above characteristic eigenvalue equation (3) are given by

$$\begin{aligned} \gamma _0= & {} \gamma _{01}+(b\varepsilon \gamma _{02}+a^2\gamma _{03}+(b^2+\omega ^2)\gamma _{04}-\gamma _{05}) , \nonumber \\ \gamma _1= & {} a^4\gamma _{11}+a^3 \gamma _{12} +a^2 \gamma _{13}-a\gamma _{14}+b^2\gamma _{15}-b\gamma _{16}+\gamma _{17}, \nonumber \\ \gamma _2= & {} a^4+a^3\gamma _{21} + a^2\gamma _{22}-a\gamma _{23}+b^2\gamma _{24}+b\gamma _{25}+\gamma _{26}, \nonumber \\ \gamma _3= & {} 4a^3-3a^2 \gamma _{31}+a \gamma _{32}-b^2\gamma _{33}-b\gamma _{34}+\gamma _{35},\nonumber \\ \gamma _4= & {} 6 a^2+a \gamma _{41}+\gamma _{42}, \nonumber \\ \gamma _5= & {} 8 r^* -2(2 a + b) - \varepsilon + 3 \varepsilon \alpha , \end{aligned}$$
(12)

where

$$\begin{aligned} \gamma _{01}= & {} a^2+\omega ^2+3r^{*2}+\varepsilon \alpha ({A_1^{(1)}}^2+3{A_1^{(2)}}^2)-a(4r^*_\varepsilon \alpha ), \\ \gamma _{02}= & {} (\alpha -p) (\omega ^2 + 3 r^{*2}) + \varepsilon ({A_1^{(1)}}^2+3{A_1^{(2)}}^2) (\alpha -1)\alpha , \\ \gamma _{03}= & {} (b^2 + \omega ^2 + b \varepsilon ( \alpha -p)), \\ \gamma _{04}= & {} (\omega ^2 + 3 r^{*2} + \varepsilon ({A_1^{(1)}}^2+3{A_1^{(2)}}^2) ( p + \alpha -1)), \\ \gamma _{05}= & {} a(4r^* \gamma _{03}+\varepsilon (\alpha -1)\alpha +(\omega ^2+b^2)(p-1+\alpha )), \\ \gamma _{11}= & {} 2 b + \varepsilon (\alpha -p), \\ \gamma _{12}= & {} -4 b^2 - 4 \omega ^2 + 2 b (-8 r^* + \varepsilon (1 + p - 4 \alpha )) + \\&\varepsilon \alpha (\varepsilon - 8 r^* - 2 \varepsilon \alpha ) + \varepsilon p (8r^* + \varepsilon \alpha ), \\ \gamma _{13}= & {} 24 \omega ^2 r^* - 22 \varepsilon r^{*2} (p - \alpha ) - \varepsilon ^2 (5 {A_1^{(1)}}^2 + 7 {A_1^{(2)}}^2)\\&(1 + p - 2 \alpha ) \alpha + \varepsilon ^3 (-1 + \alpha ) \alpha ^2 + \varepsilon \omega ^2 (p-3 \\&+ 8 \alpha ) + 3 b^2 (8 r^* + \varepsilon (-1 + p + 2 \alpha )) +b (4 \omega ^2 \\&+ 44 r^{*2} - 2 \varepsilon ({A_1^{(2)}}^2 (7 + 5 p - 26 \alpha ) + {A_1^{(1)}}^2 (5 + 7 p \\&- 22 \alpha )) + \varepsilon ^2 \alpha (-5 - p + 8 \alpha )), \\ \gamma _{14}= & {} 4 \omega ^4 + 2 b^2 (2 \omega ^2 + 22 r^{*2} + \varepsilon ^2 \alpha ( p-1 + \alpha ) + \\&\varepsilon (5 {A_1^{(1)}}^2 + 7 {A_1^{(2)}}^2) ( p-1 + 2 \alpha )) + 2 b (8 r (\omega ^2 + 3 r^2) \\&- \varepsilon r ({A_1^{(2)}}^2 (15 + 7 p - 52 \alpha ) + {A_1^{(1)}}^2 (7 + 15 p - 36 \alpha )) \\&+ \omega ^2 (1 + p - 4 \alpha )) + \varepsilon ^3 ( \alpha -1) \alpha ^2 + \varepsilon ^2 \alpha (-{A_1^{(2)}}^2 \\&(13 + p - 20 \alpha ) + {A_1^{(1)}}^2 ( 12 \alpha -7 - 3 p ))) + \omega ^2 \\&(44 r^2 + \varepsilon ^2 \alpha (-3 + p + 4 \alpha ) + 2 \varepsilon ({A_1^{(1)}}^2 ( p + 14 \alpha \\&-5 ) + {A_1^{(2)}}^2 ( 3 p + 18 \alpha -7))) + \varepsilon (\alpha (24 r^3 + 2 \varepsilon ^2 \\&({A_1^{(1)}}^2 + 3 {A_1^{(2)}}^2) ( \alpha -1) \alpha + \varepsilon (7 {A_1^{(1)}}^4 + 22 {A_1^{(1)}}^2 \\&{A_1^{(2)}}^2 + 15 {A_1^{(2)}}^4) ( 2 \alpha -1)) - p r (24 {A_1^{(1)}}^4 +\\&3 {A_1^{(2)}}^2 (8 {A_1^{(2)}}^2 + 5 \varepsilon \alpha ) + {A_1^{(1)}}^2 (48 {A_1^{(2)}}^2 + 7 \varepsilon \alpha ))), \\ \gamma _{15}= & {} 8r^* (\omega ^2 + 3 r^{*2}) + 2\varepsilon ^2 ({A_1^{(1)}}^2 + 3 {A_1^{(2)}}^2) \alpha ( p-1 + \alpha )\\&+ \varepsilon (\omega ^2 + 7 {A_1^{(1)}}^4 + 22 {A_1^{(1)}}^2 {A_1^{(2)}}^2 + 15 {A_1^{(2)}}^4) ( p-1 + 2 \alpha ), \\ \gamma _{16}= & {} 2 \omega ^4 + 18 r^{*4} - 6 \varepsilon r^{*2} ({A_1^{(2)}}^2 (3 + p - 10 \alpha ) + {A_1^{(1)}}^2 \\ \end{aligned}$$
$$\begin{aligned}&(1 + 3 p - 6 \alpha )) + 2 \varepsilon ^3 ({A_1^{(1)}}^2 + 3 {A_1^{(2)}}^2) (\alpha -1) \alpha ^2 - \\&\omega ^2 (-12 r^{*2} + 2 \varepsilon ({A_1^{(2)}}^2 (3 + p - 10 \alpha ) + {A_1^{(1)}}^2 (1 +\\&3 p - 6 \alpha )) + \varepsilon ^2 (1 + p - 2 \alpha ) \alpha ) + \varepsilon ^2 \alpha (-2 {A_1^{(1)}}^2 \\&{A_1^{(2)}}^2 (17 + 5 p - 28\alpha ) + {A_1^{(1)}}^4 ( 16 \alpha -9 - 5 p) + 3 {A_1^{(2)}}^4\\&( p + 16 \alpha -11)), \\ \gamma _{17}= & {} 8 \omega ^2 r^* (\omega ^2 + 3 r^{*2}) + \varepsilon ^3 ({A_1^{(1)}}^2 + 3 {A_1^{(2)}}^2)^2 (\alpha -1)\alpha ^2 \\&+ \varepsilon ^2 ({A_1^{(1)}}^2 + 3 {A_1^{(2)}}^2) \alpha (-3 r^{*2} (1 + p - 2 \alpha ) + \omega ^2\\&( p + 4\alpha -3)) + \varepsilon (-9 r^{*4} (p - \varepsilon ) + \omega ^4 ( 3 \alpha -1) \\&+ \omega ^2 r^* (3 {A_1^{(2)}}^2 (3 p + 12\alpha -5) + {A_1^{(1)}}^2 ( p + 20 \alpha -7))),\\ \gamma _{21}= & {} (\varepsilon (1 + 3 p - 6 \alpha )-8 b - 8r^* ), \\ \gamma _{22}= & {} 22r^*+6 b (8 r^* + \varepsilon (-1 + 3 \alpha )) +(34 \varepsilon \alpha -5 \varepsilon - \\&19 \varepsilon p ) {A_1^{(1)}}^2 +(38 \varepsilon \alpha - 7 \varepsilon - 17 \varepsilon p ){A_1^{(2)}}^2 +6 b^2 + 8 w^2 \\&- 4 \varepsilon ^2 \alpha - 2 \varepsilon ^2 p \alpha + 7 \varepsilon ^2 \alpha ^2, \\ \gamma _{23}= & {} 8 r^* (4 \omega ^2 + 3 r^{*2}) + 2 \varepsilon ^3 (-1 + \alpha ) \alpha ^2 + 3 b^2 \\&(8 r^* + \varepsilon (-1 + p + 2 \alpha )) + k (-r^* ({A_1^{(2)}}^2 (15 + \\&29 p - 74 \alpha ) + {A_1^{(1)}}^2 (7 + 37 p - 58 \alpha )) + 4 \omega ^2 \\&(-1 + 3 \alpha )) + 2 \varepsilon ^2 \alpha ({A_1^{(1)}}^2(-6 - 4 p + 11 \alpha ) + \\&{A_1^{(2)}}^2 (-10 - 4 p + 17\alpha )) + b (8 \omega ^2 + 88 r^2 - 4 \varepsilon \\&{A_1^{(1)}}^2 (5 + p - 16 \alpha ) + \varepsilon ^2 \alpha (-7 + p + 10\alpha ) + 4 \\&\varepsilon {A_1^{(2)}}^2 (-7 + p + 20 \alpha )), \\ \gamma _{24}= & {} (2 \omega ^2 + 22 r^{*2} + \varepsilon ^2 \alpha (-1 + p + \alpha ) + \varepsilon (5 {A_1^{(1)}}^2 \\&+ 7 {A_1^{(2)}}^2) (-1 + p + 2 \alpha )), \\ \gamma _{25}= & {} 16 r^* (\omega ^2 + 3 r^{*2}) + \varepsilon ^3 (-1 + \alpha ) \alpha ^2 + 2 \varepsilon (-r^* \\&({A_1^{(2)}}^2 (15 - 4 p - 41 \alpha ) + {A_1^{(2)}}^2 (7 + 4 p - 25\alpha )) +\\&\omega ^2 (-1 + 3 \alpha )) + \varepsilon ^2 \alpha (-{A_1^{(1)}}^2 (9 + p - 14 \alpha ) + \\&{A_1^{(2)}}^2 (-19 + 5 p + 26 \alpha )), \\ \gamma _{26}= & {} 3 \omega ^4 + 9 r^{*4} - 3 \varepsilon r^{*2} ({A_1^{(2)}}^2 (3 + 5 p - 14 \alpha ) + {A_1^{(1)}}^2 \\&(1 + 7 p - 10 \alpha )) + 2 \varepsilon ^3 ({A_1^{(1)}}^2 + 3 {A_1^{(2)}}^2) (-1 + \alpha )\\&\alpha ^2 + \varepsilon ^2 \alpha (-2 {A_1^{(1)}}^2 {A_1^{(2)}}^2 (14 + 8 p - 25 \alpha ) - 3 {A_1^{(2)}}^4 \\&(8 + 2 p - 13\alpha ) + {A_1^{(1)}}^4 (-8 - 6 p + 15 \alpha )) + \\&\omega ^2 (28 r^{*2} + \varepsilon ^2 \alpha (-2 + 3 \alpha ) - 2 k ({A_1^{(1)}}^2 (3 + p\\&- 10 \alpha ) - {A_1^{(2)}}^2 (-5 + p + 14 \alpha ))), \end{aligned}$$
$$\begin{aligned} \gamma _{31}= & {} (4 b + 8 r^* - \varepsilon (1 + p - 4 \alpha )), \\ \gamma _{32}= & {} -a (4 b^2 + 8\omega ^2 - 10 \varepsilon {A_1^{(1)}}^2 - 14 \varepsilon p {A_1^{(1)}}^2 22r^{*2} - 14 k {A_1^{(2)}}^2 \\&- 10 \varepsilon p {A_1^{(2)}}^2 + 48 b r^* - 5 \varepsilon ^2 \alpha - \varepsilon ^2 p \alpha + 44 \varepsilon {A_1^{(1)}}^2 \alpha + \\&52 \varepsilon {A_1^{(2)}}^2 \alpha + 8 \varepsilon ^2 \alpha ^2 + 2 b \varepsilon (-3 + p + 8 \alpha )), \\ \gamma _{33}= & {} 8 r^* + \varepsilon (p-1 + 2\alpha ), \\ \gamma _{34}= & {} (4 \omega ^2 + 44 r^{*2} + \varepsilon ^2 \alpha (-3 + p + 4 \alpha ) + 2 \varepsilon ({A_1^{(1)}}^2 (-5 \\&+ p + 14 \alpha ) + {A_1^{(2)}}^2 (3 p-7 + 18 \alpha ))), \\ \gamma _{35}= & {} (16 \omega ^2 - 22 \varepsilon {A_1^{(1)}}^2 - 22 \varepsilon p {A_1^{(1)}}^2 + 72 {A_1^{(1)}}^4 - 13 \varepsilon ^2 \alpha - \\&\varepsilon ^2 p \alpha + 88 \varepsilon {A_1^{(1)}}^2 \alpha + 20 \varepsilon ^2 \alpha ^2) {A_1^{(1)}}^2 + (16 \omega ^2 - 22 \varepsilon {A_1^{(1)}}^2\\&- 22 \varepsilon p {A_1^{(1)}}^2 + 72 {A_1^{(1)}}^4 - 13 \varepsilon ^2 \alpha - \varepsilon ^2 p \alpha + 88 \varepsilon {A_1^{(1)}}^2\alpha +\\&20 \varepsilon ^2 \alpha ^2) {A_1^{(2)}}^2 -2 \varepsilon \omega ^2 + 6 \varepsilon \omega ^2 \alpha - \varepsilon ^3 \alpha ^2 + \varepsilon ^3 \alpha ^3,\\ \gamma _{41}= & {} \varepsilon (3 + p - 10 \alpha )-8 b - 24 r^*, \\ \gamma _{42}= & {} 22r^{*2} +16br^*+(18\alpha - 3 p-5 )\varepsilon {A_1^{(1)}}^2 + (22\alpha -p \\&-7)\varepsilon {A_1^{(2)}}^2+6 a^2 + b^2 + 3 w^2 - 2 k^2 \alpha + 3 \varepsilon ^2 \alpha ^2 + \\&b \varepsilon (p-2 + 5 \alpha );~ r^*=(A_1^{(1)*}+A_2^{(1)*}). \end{aligned}$$

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Gowthaman, I., Sathiyadevi, K., Chandrasekar, V.K. et al. Symmetry breaking-induced state-dependent aging and chimera-like death state. Nonlinear Dyn 101, 53–64 (2020). https://doi.org/10.1007/s11071-020-05766-5

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