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July  2022, 9(3): 451-464. doi: 10.3934/jcd.2022010

Computation of nonreciprocal dynamics in nonlinear materials

Department of Mechanical, Industrial and Aerospace Engineering, Concordia University, Montreal, QC, H3G 1M8, Canada

Received  October 2021 Revised  November 2021 Published  July 2022 Early access  April 2022

The reciprocity theorem in elastic materials states that the response of a linear, time-invariant system to an external load remains invariant with respect to interchanging the locations of the input and output. In the presence of nonlinear forces within a material, circumventing the reciprocity invariance requires breaking the mirror symmetry of the medium, thus allowing different wave propagation characteristics in opposite directions along the same transmission path. This work highlights the application of numerical continuation methods for exploring the steady-state nonreciprocal dynamics of nonlinear periodic materials in response to external harmonic drive. Using the archetypal example of coupled oscillators, we apply continuation methods to analyze the influence of nonlinearity and symmetry on the reciprocity invariance. We present symmetry-breaking bifurcations for systems with and without mirror symmetry, and discuss their influence on the nonreciprocal dynamics. Direct computation of the reciprocity bias allows the identification of response regimes in which nonreciprocity manifests itself as a phase shift in the output displacements. Various operating regimes, bifurcations and manifestations of nonreciprocity are identified and discussed throughout the work.

Citation: Behrooz Yousefzadeh. Computation of nonreciprocal dynamics in nonlinear materials. Journal of Computational Dynamics, 2022, 9 (3) : 451-464. doi: 10.3934/jcd.2022010
References:
[1]

J. D. Achenbach, Reciprocity and related topics in elastodynamic, Applied Mechanics Reviews, 59 (2006), 13-32.  doi: 10.1115/1.2110262.

[2]

H. A. Ardeh and M. S. Allen, Investigating cases of jump phenomenon in a nonlinear oscillatory system, Topics in Nonlinear Dynamics, 1 (2013), 299-318.  doi: 10.1007/978-1-4614-6570-6_28.

[3]

W.-J. BeynA. ChampneysE. J. DoedelW. GovaertsY. A. Kuznetsov and B. Sandstede, Numerical continuation, and computation of normal forms, Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 149-219.  doi: 10.1016/S1874-575X(02)80025-X.

[4]

A. BlanchardT. P. Sapsis and A. F. Vakakis, Non-reciprocity in nonlinear elastodynamics, Journal of Sound and Vibration, 412 (2017), 326-335.  doi: 10.1016/j.jsv.2017.09.039.

[5]

N. BoechlerG. Theocharis and C. Daraio, Bifurcation-based acoustic switching and rectification, Nature Materials, 10 (2011), 665-668.  doi: 10.1038/nmat3072.

[6]

J. J. Bramburger and B. Sandstede, Spatially localized structures in lattice dynamical systems, Journal of Nonlinear Science, 30 (2020), 603-644.  doi: 10.1007/s00332-019-09584-x.

[7]

M. BrandenbourgerX. LocsinE. Lerner and C. Coulais, Non-reciprocal robotic metamaterials, Nature Communications, 10 (2019), 4608.  doi: 10.1038/s41467-019-12599-3.

[8]

C. CalozA. AlùS. TretyakovD. SounasK. Achouri and Z. Deck-Léger, Electromagnetic nonreciprocity, Physical Review Applied, 10 (2018), 047001.  doi: 10.1103/PhysRevApplied.10.047001.

[9]

M. Cenedese and G. Haller, How do conservative backbone curves perturb into forced responses? A Melnikov function analysis, Proceedings of the Royal Society A, 476 (2020), 20190494, 26 pp. doi: 10.1098/rspa.2019.0494.

[10]

C. ChongM. A. PorterP. G. Kevrekidis and C. Daraio, Nonlinear coherent structures in granular crystals, Journal of Physics: Condensed Matter, 20 (2017), 413003.  doi: 10.1088/1361-648X/aa7672.

[11]

F. Dercole and Y. A. Kuznetsov, SlideCont: An Auto97 driver for bifurcation analysis of Filippov systems, ACM Transactions on Mathematical Software, 31 (2005), 95-119.  doi: 10.1145/1055531.1055536.

[12]

T. DetrouxL. RensonL. Masset and G. Kerschen, The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems, Computer Methods in Applied Mechanics and Engineering, 296 (2015), 18-38.  doi: 10.1016/j.cma.2015.07.017.

[13]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems. Theory and Applications, Applied Mathematical Sciences, 163. Springer-Verlag London, Ltd., London, 2008. doi: 10.1007/978-1-84628-708-4.

[14]

E. J. Doedel, Lecture notes on numerical analysis of nonlinear equations, Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems. Understanding Complex Systems. Springer, Dordrecht, (2007), 1–49. doi: 10.1007/978-1-4020-6356-5_1.

[15]

E. J. Doedel and B. E. Oldeman, AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations, Concordia University, Montreal, Canada, 2012.

[16]

F. J. Fahy, Some applications of the reciprocity principle in experimental vibroacoustics, Acoustical Physics, 49 (2003), 217-229.  doi: 10.1134/1.1560385.

[17]

L. FangA. DarabiA. MojahedA. F. Vakakis and M. J. Leamy, Broadband non-reciprocity with robust signal integrity in a triangle-shaped nonlinear 1D metamaterial, Nonlinear Dynamics, 100 (2020), 1-13.  doi: 10.1007/s11071-020-05520-x.

[18]

I. GrinbergA. F. Vakakis and O. V. Gendelman, Acoustic diode: Wave non-reciprocity in nonlinearly coupled waveguides, Wave Motion, 83 (2018), 49-66.  doi: 10.1016/j.wavemoti.2018.08.005.

[19]

M. E. Henderson, Multiple parameter continuation: Computing implicitly defined $k$-manifolds, International Journal of Bifurcation and Chaos, 12 (2002), 451-476.  doi: 10.1142/S0218127402004498.

[20]

M. JohanssonG. KopidakisS. Lepri and S. Aubry, Transmission thresholds in time-periodically driven nonlinear disordered systems, Europhysics Letters, 86 (2009), 10009.  doi: 10.1209/0295-5075/86/10009.

[21]

J. KozlowskiU. Parlitz and W. Lauterborn, Bifurcation analysis of two coupled periodically driven Duffing oscillators, Physical Review E, 51 (1995), 1861-1867.  doi: 10.1103/PhysRevE.51.1861.

[22]

Y. A. KuznetsovS. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, International Journal of Bifurcation and Chaos, 13 (2003), 2157-2188.  doi: 10.1142/S0218127403007874.

[23]

H. Lamb, On reciprocal theorems in dynamics, Proceedings of the London Mathematical Society, 19 (1887/88), 144-151.  doi: 10.1112/plms/s1-19.1.144.

[24]

S. Lepri and G. Casati, Asymmetric wave propagation in nonlinear systems, Physical Review Letters, 106 (2011), 164101.  doi: 10.1103/PhysRevLett.106.164101.

[25]

S. Lepri and A. Pikovsky, Nonreciprocal wave scattering on nonlinear string-coupled oscillators, Chaos, 24 (2014), 043119, 9 pp. doi: 10.1063/1.4899205.

[26]

B. LiangX. S. GuoJ. TuD. Zhang and J. C. Cheng, An acoustic rectifier, Nature Materials, 9 (2010), 989-992.  doi: 10.1038/nmat2881.

[27]

Z. Lu and A. N. Norris, Unilateral and nonreciprocal transmission through bilinear spring systems, Extreme Mechanics Letters, 42 (2021), 101087.  doi: 10.1016/j.eml.2020.101087.

[28]

P. ManiadisG. Kopidakis and S. Aubry, Energy dissipation threshold and self-induced transparency in systems with discrete breathers, Physica D, 216 (2006), 121-135.  doi: 10.1016/j.physd.2006.01.023.

[29]

J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Dover Publications, Inc., New York, 1994.

[30]

H. Masoud and H. A. Stone, The reciprocal theorem in fluid dynamics and transport phenomena, Journal of Fluid Mechanics, 879 (2019), P1, 78 pp. doi: 10.1017/jfm.2019.553.

[31]

K. J. MooreJ. BunyanS. TawfickO. V. GendelmanS. LiM. J. Leamy and A. F. Vakakis, Nonreciprocity in the dynamics of coupled oscillators with nonlinearity, asymmetry, and scale hierarchy, Physical Review E, 97 (2018), 012219.  doi: 10.1103/PhysRevE.97.012219.

[32]

F. J. Muñoz-AlmarazE. FreireJ. GalánE. J. Doedel and A. Vanderbauwhede, Continuation of periodic orbits in conservative and Hamiltonian systems, Physica D, 181 (2003), 1-38.  doi: 10.1016/S0167-2789(03)00097-6.

[33]

H. NassarB. YousefzadehR. FleuryM. RuzzeneA. AlùC. DaraioA. N. NorrisG. Huang and M. R. Haberman, Nonreciprocity in acoustic and elastic materials, Nature Reviews Materials, 5 (2020), 667-685.  doi: 10.1038/s41578-020-0206-0.

[34]

S. Novak and R. G. Frehlich, Transition to chaos in the Duffing oscillator, Physical Review A, 26 (1982), 3660-3663.  doi: 10.1103/PhysRevA.26.3660.

[35]

U. Parlitz, Common dynamical features of periodically driven strictly dissipative oscillators, International Journal of Bifurcation and Chaos, 3 (1993), 703-715.  doi: 10.1142/S0218127493000611.

[36]

N. ReiskarimianA. NaguluT. Dinc and H. Krishnaswamy, Nonreciprocal electronic devices: A hypothesis turned into reality, IEEE Microwave Magazine, 20 (2019), 94-111.  doi: 10.1109/MMM.2019.2891380.

[37]

J.-A. Sepulchre and R. S. MacKay, Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators, Nonlinearity, 879 (10), 679-713.  doi: 10.1088/0951-7715/10/3/006.

[38]

J. W. Strutt, Some General Theorems relating to Vibrations, Proceedings of the London Mathematical Society, 4 (1871/73), 357-368.  doi: 10.1112/plms/s1-4.1.357.

[39]

T. Ten Wolde, Reciprocity measurements in acoustical and mechano-acoustical systems. Review of theory and applications, Acta Acustica united with Acustica, 96 (2010), 1-13.  doi: 10.3813/AAA.918250.

[40]

G. TheocharisM. KavousanakisP. G. KevrekidisC. DaraioM. A. Porter and I. G. Kevrekidis, Localized breathing modes in granular crystals with defects, Physical Review E, 80 (2009), 066601.  doi: 10.1103/PhysRevE.80.066601.

[41]

P. Thota and H. Dankowicz, TC-HAT ($\widehat {TC}$): A novel toolbox for the continuation of periodic trajectories in hybrid dynamical systems, SIAM Journal on Applied Dynamical Systems, 7 (2008), 1283-1322.  doi: 10.1137/070703028.

[42]

B. Yousefzadeh and C. Daraio, Complete delocalization in a defective periodic structure, Physical Review E, 96 (2017), 042219.  doi: 10.1103/PhysRevE.96.042219.

[43]

B. Yousefzadeh and A. S. Phani, Supratransmission in a disordered nonlinear periodic structures, Journal of Sound and Vibration, 380 (2016), 242-266.  doi: 10.1016/j.jsv.2016.06.001.

[44]

A. V. Yulin and A. R. Champneys, Discrete snaking: Multiple cavity solitons in saturable media, SIAM Journal on Applied Dynamical Systems, 9 (2010), 391-431.  doi: 10.1137/080734297.

show all references

References:
[1]

J. D. Achenbach, Reciprocity and related topics in elastodynamic, Applied Mechanics Reviews, 59 (2006), 13-32.  doi: 10.1115/1.2110262.

[2]

H. A. Ardeh and M. S. Allen, Investigating cases of jump phenomenon in a nonlinear oscillatory system, Topics in Nonlinear Dynamics, 1 (2013), 299-318.  doi: 10.1007/978-1-4614-6570-6_28.

[3]

W.-J. BeynA. ChampneysE. J. DoedelW. GovaertsY. A. Kuznetsov and B. Sandstede, Numerical continuation, and computation of normal forms, Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 149-219.  doi: 10.1016/S1874-575X(02)80025-X.

[4]

A. BlanchardT. P. Sapsis and A. F. Vakakis, Non-reciprocity in nonlinear elastodynamics, Journal of Sound and Vibration, 412 (2017), 326-335.  doi: 10.1016/j.jsv.2017.09.039.

[5]

N. BoechlerG. Theocharis and C. Daraio, Bifurcation-based acoustic switching and rectification, Nature Materials, 10 (2011), 665-668.  doi: 10.1038/nmat3072.

[6]

J. J. Bramburger and B. Sandstede, Spatially localized structures in lattice dynamical systems, Journal of Nonlinear Science, 30 (2020), 603-644.  doi: 10.1007/s00332-019-09584-x.

[7]

M. BrandenbourgerX. LocsinE. Lerner and C. Coulais, Non-reciprocal robotic metamaterials, Nature Communications, 10 (2019), 4608.  doi: 10.1038/s41467-019-12599-3.

[8]

C. CalozA. AlùS. TretyakovD. SounasK. Achouri and Z. Deck-Léger, Electromagnetic nonreciprocity, Physical Review Applied, 10 (2018), 047001.  doi: 10.1103/PhysRevApplied.10.047001.

[9]

M. Cenedese and G. Haller, How do conservative backbone curves perturb into forced responses? A Melnikov function analysis, Proceedings of the Royal Society A, 476 (2020), 20190494, 26 pp. doi: 10.1098/rspa.2019.0494.

[10]

C. ChongM. A. PorterP. G. Kevrekidis and C. Daraio, Nonlinear coherent structures in granular crystals, Journal of Physics: Condensed Matter, 20 (2017), 413003.  doi: 10.1088/1361-648X/aa7672.

[11]

F. Dercole and Y. A. Kuznetsov, SlideCont: An Auto97 driver for bifurcation analysis of Filippov systems, ACM Transactions on Mathematical Software, 31 (2005), 95-119.  doi: 10.1145/1055531.1055536.

[12]

T. DetrouxL. RensonL. Masset and G. Kerschen, The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems, Computer Methods in Applied Mechanics and Engineering, 296 (2015), 18-38.  doi: 10.1016/j.cma.2015.07.017.

[13]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems. Theory and Applications, Applied Mathematical Sciences, 163. Springer-Verlag London, Ltd., London, 2008. doi: 10.1007/978-1-84628-708-4.

[14]

E. J. Doedel, Lecture notes on numerical analysis of nonlinear equations, Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems. Understanding Complex Systems. Springer, Dordrecht, (2007), 1–49. doi: 10.1007/978-1-4020-6356-5_1.

[15]

E. J. Doedel and B. E. Oldeman, AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations, Concordia University, Montreal, Canada, 2012.

[16]

F. J. Fahy, Some applications of the reciprocity principle in experimental vibroacoustics, Acoustical Physics, 49 (2003), 217-229.  doi: 10.1134/1.1560385.

[17]

L. FangA. DarabiA. MojahedA. F. Vakakis and M. J. Leamy, Broadband non-reciprocity with robust signal integrity in a triangle-shaped nonlinear 1D metamaterial, Nonlinear Dynamics, 100 (2020), 1-13.  doi: 10.1007/s11071-020-05520-x.

[18]

I. GrinbergA. F. Vakakis and O. V. Gendelman, Acoustic diode: Wave non-reciprocity in nonlinearly coupled waveguides, Wave Motion, 83 (2018), 49-66.  doi: 10.1016/j.wavemoti.2018.08.005.

[19]

M. E. Henderson, Multiple parameter continuation: Computing implicitly defined $k$-manifolds, International Journal of Bifurcation and Chaos, 12 (2002), 451-476.  doi: 10.1142/S0218127402004498.

[20]

M. JohanssonG. KopidakisS. Lepri and S. Aubry, Transmission thresholds in time-periodically driven nonlinear disordered systems, Europhysics Letters, 86 (2009), 10009.  doi: 10.1209/0295-5075/86/10009.

[21]

J. KozlowskiU. Parlitz and W. Lauterborn, Bifurcation analysis of two coupled periodically driven Duffing oscillators, Physical Review E, 51 (1995), 1861-1867.  doi: 10.1103/PhysRevE.51.1861.

[22]

Y. A. KuznetsovS. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, International Journal of Bifurcation and Chaos, 13 (2003), 2157-2188.  doi: 10.1142/S0218127403007874.

[23]

H. Lamb, On reciprocal theorems in dynamics, Proceedings of the London Mathematical Society, 19 (1887/88), 144-151.  doi: 10.1112/plms/s1-19.1.144.

[24]

S. Lepri and G. Casati, Asymmetric wave propagation in nonlinear systems, Physical Review Letters, 106 (2011), 164101.  doi: 10.1103/PhysRevLett.106.164101.

[25]

S. Lepri and A. Pikovsky, Nonreciprocal wave scattering on nonlinear string-coupled oscillators, Chaos, 24 (2014), 043119, 9 pp. doi: 10.1063/1.4899205.

[26]

B. LiangX. S. GuoJ. TuD. Zhang and J. C. Cheng, An acoustic rectifier, Nature Materials, 9 (2010), 989-992.  doi: 10.1038/nmat2881.

[27]

Z. Lu and A. N. Norris, Unilateral and nonreciprocal transmission through bilinear spring systems, Extreme Mechanics Letters, 42 (2021), 101087.  doi: 10.1016/j.eml.2020.101087.

[28]

P. ManiadisG. Kopidakis and S. Aubry, Energy dissipation threshold and self-induced transparency in systems with discrete breathers, Physica D, 216 (2006), 121-135.  doi: 10.1016/j.physd.2006.01.023.

[29]

J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Dover Publications, Inc., New York, 1994.

[30]

H. Masoud and H. A. Stone, The reciprocal theorem in fluid dynamics and transport phenomena, Journal of Fluid Mechanics, 879 (2019), P1, 78 pp. doi: 10.1017/jfm.2019.553.

[31]

K. J. MooreJ. BunyanS. TawfickO. V. GendelmanS. LiM. J. Leamy and A. F. Vakakis, Nonreciprocity in the dynamics of coupled oscillators with nonlinearity, asymmetry, and scale hierarchy, Physical Review E, 97 (2018), 012219.  doi: 10.1103/PhysRevE.97.012219.

[32]

F. J. Muñoz-AlmarazE. FreireJ. GalánE. J. Doedel and A. Vanderbauwhede, Continuation of periodic orbits in conservative and Hamiltonian systems, Physica D, 181 (2003), 1-38.  doi: 10.1016/S0167-2789(03)00097-6.

[33]

H. NassarB. YousefzadehR. FleuryM. RuzzeneA. AlùC. DaraioA. N. NorrisG. Huang and M. R. Haberman, Nonreciprocity in acoustic and elastic materials, Nature Reviews Materials, 5 (2020), 667-685.  doi: 10.1038/s41578-020-0206-0.

[34]

S. Novak and R. G. Frehlich, Transition to chaos in the Duffing oscillator, Physical Review A, 26 (1982), 3660-3663.  doi: 10.1103/PhysRevA.26.3660.

[35]

U. Parlitz, Common dynamical features of periodically driven strictly dissipative oscillators, International Journal of Bifurcation and Chaos, 3 (1993), 703-715.  doi: 10.1142/S0218127493000611.

[36]

N. ReiskarimianA. NaguluT. Dinc and H. Krishnaswamy, Nonreciprocal electronic devices: A hypothesis turned into reality, IEEE Microwave Magazine, 20 (2019), 94-111.  doi: 10.1109/MMM.2019.2891380.

[37]

J.-A. Sepulchre and R. S. MacKay, Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators, Nonlinearity, 879 (10), 679-713.  doi: 10.1088/0951-7715/10/3/006.

[38]

J. W. Strutt, Some General Theorems relating to Vibrations, Proceedings of the London Mathematical Society, 4 (1871/73), 357-368.  doi: 10.1112/plms/s1-4.1.357.

[39]

T. Ten Wolde, Reciprocity measurements in acoustical and mechano-acoustical systems. Review of theory and applications, Acta Acustica united with Acustica, 96 (2010), 1-13.  doi: 10.3813/AAA.918250.

[40]

G. TheocharisM. KavousanakisP. G. KevrekidisC. DaraioM. A. Porter and I. G. Kevrekidis, Localized breathing modes in granular crystals with defects, Physical Review E, 80 (2009), 066601.  doi: 10.1103/PhysRevE.80.066601.

[41]

P. Thota and H. Dankowicz, TC-HAT ($\widehat {TC}$): A novel toolbox for the continuation of periodic trajectories in hybrid dynamical systems, SIAM Journal on Applied Dynamical Systems, 7 (2008), 1283-1322.  doi: 10.1137/070703028.

[42]

B. Yousefzadeh and C. Daraio, Complete delocalization in a defective periodic structure, Physical Review E, 96 (2017), 042219.  doi: 10.1103/PhysRevE.96.042219.

[43]

B. Yousefzadeh and A. S. Phani, Supratransmission in a disordered nonlinear periodic structures, Journal of Sound and Vibration, 380 (2016), 242-266.  doi: 10.1016/j.jsv.2016.06.001.

[44]

A. V. Yulin and A. R. Champneys, Discrete snaking: Multiple cavity solitons in saturable media, SIAM Journal on Applied Dynamical Systems, 9 (2010), 391-431.  doi: 10.1137/080734297.

Figure 1.  Output norms (5) at $ P = 0.5 $ for system (1) for (a) the system with mirror symmetry ($ \mu = 1 $), and (b) the system with broken symmetry ($ \mu = 2 $). Reciprocity invariance only holds for the system with mirror symmetry. The vertical lines indicate the frequencies at which the response is shown in the time domain in Figure 2. The diamond markers denote the saddle-node bifurcations occurring at the turning points
Figure 2.  Nonreciprocal dynamics of (1) for $ P = 0.5 $ and $ \mu = 2 $. Panel (a) shows the reciprocity bias, $ R $. The diamonds indicate saddle-node bifurcation points. Panels (b)-(e) show the periodic orbits at various values of forcing amplitude, $ \omega_f $, for the forward and backward configurations: (b) $ \omega_f = 1.036 $, (c) $ \omega_f = 1.500 $, (d) $ \omega_f = 2.948 $, (e) $ \omega_f = 3.079 $. These four frequencies are indicated in panel (a) with cross markers. Panel (d) corresponds to phase nonreciprocity, discussed in Section 5
Figure 3.  Reciprocal dynamics of the symmetric system ($ \mu = 1 $) at $ P = 2 $. (a) Output norm as a function of forcing frequency. The diamond, square, and pentagram markers indicate, respectively, saddle-node, Neimark-Sacker and symmetry-breaking bifurcation points. The inset shows the extra solution branches caused by symmetry-breaking bifurcations near $ \omega_f = 2.11 $. The vertical lines indicate the frequencies at which the response is shown in the time domain. (b) The two symmetry-broken branches of solutions. (c) The periodic orbit at $ \omega_f = 2.11 $. The markers correspond to solution branches in panels (a)-(b). (d) The periodic orbit at $ \omega_f = 0.45 $, which contains significant contributions at $ 3\omega_f $. (e) The anharmonic response at $ \omega_f = 4.0 $ between the two Neimark-Sacker bifurcation points (not computed using continuation)
Figure 4.  Nonreciprocal dynamics of the system with broken symmetry ($ \mu = 2 $) at $ P = 2 $. (a) Output norm as a function of forcing frequency. The diamond and pentagram markers indicate saddle-node and symmetry-breaking bifurcation points, respectively. (b)-(c) The symmetry-broken branches of solutions for the forward and backward configurations. The vertical lines indicate $ \omega_f = 0.3733 $. The diamond markers indicate Neimark-Sacker bifurcation points. (d) The periodic orbits at $ \omega_f = 1.975 $. (e) Output norms near nonlinear resonances. (f) The amplitudes of the third ($ A_3 $, thick curves) and fifth ($ A_5 $, thin curves) harmonics in the Fourier series expansion of the response. (g) The periodic orbit at $ \omega_f = 0.3733 $, indicated by vertical lines in panels (e)-(f)
Figure 5.  Locus of a symmetry-breaking bifurcation as a function of $ \mu $ at (a) $ P = 2 $ and (b) $ P = 3 $. The vertical line denotes the mirror-symmetric system
Figure 6.  Phase nonreciprocity for $ \mu = 2 $ as a function of forcing amplitude, $ P $. (a) The locus of phase nonreciprocity in the $ (P, \omega_f) $ plane. (b) The black curve shows the locus of phase nonreciprocity in the $ (\omega_f, R) $ plane. The gray curve is identical to the one in Figure 2(a). (c) The amplitude of the first harmonic in the Fourier series expansion of the periodic orbit, $ a^F $ and $ a^B $. (d) The phase difference between the forward and backward output displacements, $ \Delta \phi = \phi^F-\phi^B $
Figure 7.  Phase nonreciprocity at $ P = 0.5 $ as function of the symmetry-breaking parameter $ \mu $. (a) The normalized reciprocity bias. (b) The phase difference between the forward and backward output displacements, $ \Delta \phi = \phi^F-\phi^B $
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