Abstract

Secure communication systems based on the chaos in erbium-doped fiber lasers are proposed and studied with two schemes: message masking and chaos shift keying. The effect of the parameter mismatch between the transmitter and the receiver on synchronization of chaos is also studied. The synchronization is maintained quite well with mismatches of parameters of 5% or even more by use of a comatching method.

© 1998 Optical Society of America

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References

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  1. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64, 821–824 (1990).
    [CrossRef] [PubMed]
  2. L. M. Pecora and T. L. Carroll, “Driving systems with chaotic signals,” Phys. Rev. A 44, 2374–2383 (1991); T. L. Carroll and L. M. Pecora, “Synchronizing chaotic circuits,” IEEE Trans. Circuits Syst. 38, 453–456 (1991).
    [CrossRef] [PubMed]
  3. M. Cuomo and A. V. Oppenheim, “Circuit implementation of synchronized chaos with applications to communications,” Phys. Rev. Lett. 71, 65–68 (1993); K. M. Cuomo, A. V. Oppenheim, and S. H. Strogatz, “Synchronization of Lorenz-based chaotic circuits with applications to communications,” IEEE Trans. Circuits Syst. 40, 626–633 (1993).
    [CrossRef] [PubMed]
  4. L. Kocarev, K. S. Halle, K. Echert, L. O. Chua, and U. Parlitz, “Experimental demonstration of secure communications via chaotic synchronization,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 709–713 (1992).
    [CrossRef]
  5. H. Dedieu, M. P. Kennedy, and M. Hasler, “Chaos shift keying: modulation and demodulation of a chaotic carrier using self-synchronizing Chua’s circuits,” IEEE Trans. Circuits Syst. 40, 634–642 (1993).
    [CrossRef]
  6. U. Parlitz, L. O. Chua, Lj. Kocarev, K. S. Halle, and A. Shang, “Transmission of digital signals by chaotic synchronization,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 973–977 (1992).
    [CrossRef]
  7. R. Roy and K. S. Thornburg, “Experimental synchronization of chaotic lasers,” Phys. Rev. Lett. 72, 2009–2012 (1994).
    [CrossRef] [PubMed]
  8. T. Sugawara, M. Tachikawa, T. Tsukamoto, and T. Shimizu, “Observation of synchronization in laser chaos,” Phys. Rev. Lett. 72, 3502–3505 (1994).
    [CrossRef] [PubMed]
  9. P. Colet and R. Roy, “Digital communication with synchronized chaotic lasers,” Opt. Lett. 19, 2056–2058 (1994).
    [CrossRef] [PubMed]
  10. V. Annovazzi-Lodi, S. Donati, and A. Scire, “Synchronization of chaotic injected-laser systems and its application to optical cryptography,” IEEE J. Quantum Electron. 32, 953–959 (1996).
    [CrossRef]
  11. L. G. Luo and P. L. Chu, “Application of laser chaos to secure communication,” in Proceedings of 21st Australian Conference of Optical Fiber Technology (Institution of Radio and Electronics Engineers Australia, Sydney, 1996), pp. 245–248.
  12. P. Celka, “Chaotic synchronization and modulation of nonlinear time-delayed feedback optical systems,” IEEE Trans. Circuits Syst. 42, 455–463 (1995); P. Celka, “Synchronization of chaotic optical dynamical systems through 700 m of single mode fiber,” IEEE Trans. Circuits Syst. 43, 869–872 (1996).
    [CrossRef]
  13. L. Kocarev and U. Parlitz, “General approach for chaotic synchronization with applications to communication,” Phys. Rev. Lett. 74, 5028–5031 (1995); U. Parlitz, L. Kocarev, T. Stojanovski, and H. Preckel, “Encoding messages using chaotic synchronization,” Phys. Rev. E 53, 4351–4361 (1996).
    [CrossRef] [PubMed]
  14. G. D. Van Wiggeren and R. Roy, “Communications with chaotic lasers,” Science 279, 1198–1200 (1998).
    [CrossRef]
  15. J.-P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. 80, 2249–2252 (1998).
    [CrossRef]
  16. L. G. Luo and P. L. Chu, “Suppression of self-pulsing in an erbium-doped fiber laser,” Opt. Lett. 22, 1174–1176 (1997).
    [CrossRef] [PubMed]
  17. L. G. Luo, T. J. Tee, and P. L. Chu, “Bistability of erbium doped fiber laser,” Opt. Commun. 146, 151–157 (1998); L. G. Luo, T. J. Tee, and P. L. Chu, “Chaotic behavior in erbium doped fiber ring lasers,” J. Opt. Soc. Am. B 15, 972–978 (1998).
    [CrossRef]
  18. H. Haken, Light (Elsevier, New York, 1985), Vol. 2, p. 98.
  19. J. W. Arkwright, B. Wu, I. Skinner, and P. L. Chu, “Resonantly enhanced nonlinearity using stimulated down-pumping in neodymium-doped twin core fibre,” Electron. Lett. 30, 235–236 (1994).
    [CrossRef]

1998

G. D. Van Wiggeren and R. Roy, “Communications with chaotic lasers,” Science 279, 1198–1200 (1998).
[CrossRef]

J.-P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. 80, 2249–2252 (1998).
[CrossRef]

1997

1996

V. Annovazzi-Lodi, S. Donati, and A. Scire, “Synchronization of chaotic injected-laser systems and its application to optical cryptography,” IEEE J. Quantum Electron. 32, 953–959 (1996).
[CrossRef]

1994

R. Roy and K. S. Thornburg, “Experimental synchronization of chaotic lasers,” Phys. Rev. Lett. 72, 2009–2012 (1994).
[CrossRef] [PubMed]

T. Sugawara, M. Tachikawa, T. Tsukamoto, and T. Shimizu, “Observation of synchronization in laser chaos,” Phys. Rev. Lett. 72, 3502–3505 (1994).
[CrossRef] [PubMed]

P. Colet and R. Roy, “Digital communication with synchronized chaotic lasers,” Opt. Lett. 19, 2056–2058 (1994).
[CrossRef] [PubMed]

J. W. Arkwright, B. Wu, I. Skinner, and P. L. Chu, “Resonantly enhanced nonlinearity using stimulated down-pumping in neodymium-doped twin core fibre,” Electron. Lett. 30, 235–236 (1994).
[CrossRef]

1993

H. Dedieu, M. P. Kennedy, and M. Hasler, “Chaos shift keying: modulation and demodulation of a chaotic carrier using self-synchronizing Chua’s circuits,” IEEE Trans. Circuits Syst. 40, 634–642 (1993).
[CrossRef]

1992

U. Parlitz, L. O. Chua, Lj. Kocarev, K. S. Halle, and A. Shang, “Transmission of digital signals by chaotic synchronization,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 973–977 (1992).
[CrossRef]

L. Kocarev, K. S. Halle, K. Echert, L. O. Chua, and U. Parlitz, “Experimental demonstration of secure communications via chaotic synchronization,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 709–713 (1992).
[CrossRef]

1990

L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64, 821–824 (1990).
[CrossRef] [PubMed]

Annovazzi-Lodi, V.

V. Annovazzi-Lodi, S. Donati, and A. Scire, “Synchronization of chaotic injected-laser systems and its application to optical cryptography,” IEEE J. Quantum Electron. 32, 953–959 (1996).
[CrossRef]

Arkwright, J. W.

J. W. Arkwright, B. Wu, I. Skinner, and P. L. Chu, “Resonantly enhanced nonlinearity using stimulated down-pumping in neodymium-doped twin core fibre,” Electron. Lett. 30, 235–236 (1994).
[CrossRef]

Carroll, T. L.

L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64, 821–824 (1990).
[CrossRef] [PubMed]

Chu, P. L.

L. G. Luo and P. L. Chu, “Suppression of self-pulsing in an erbium-doped fiber laser,” Opt. Lett. 22, 1174–1176 (1997).
[CrossRef] [PubMed]

J. W. Arkwright, B. Wu, I. Skinner, and P. L. Chu, “Resonantly enhanced nonlinearity using stimulated down-pumping in neodymium-doped twin core fibre,” Electron. Lett. 30, 235–236 (1994).
[CrossRef]

Chua, L. O.

L. Kocarev, K. S. Halle, K. Echert, L. O. Chua, and U. Parlitz, “Experimental demonstration of secure communications via chaotic synchronization,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 709–713 (1992).
[CrossRef]

U. Parlitz, L. O. Chua, Lj. Kocarev, K. S. Halle, and A. Shang, “Transmission of digital signals by chaotic synchronization,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 973–977 (1992).
[CrossRef]

Colet, P.

Dedieu, H.

H. Dedieu, M. P. Kennedy, and M. Hasler, “Chaos shift keying: modulation and demodulation of a chaotic carrier using self-synchronizing Chua’s circuits,” IEEE Trans. Circuits Syst. 40, 634–642 (1993).
[CrossRef]

Donati, S.

V. Annovazzi-Lodi, S. Donati, and A. Scire, “Synchronization of chaotic injected-laser systems and its application to optical cryptography,” IEEE J. Quantum Electron. 32, 953–959 (1996).
[CrossRef]

Echert, K.

L. Kocarev, K. S. Halle, K. Echert, L. O. Chua, and U. Parlitz, “Experimental demonstration of secure communications via chaotic synchronization,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 709–713 (1992).
[CrossRef]

Goedgebuer, J.-P.

J.-P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. 80, 2249–2252 (1998).
[CrossRef]

Halle, K. S.

L. Kocarev, K. S. Halle, K. Echert, L. O. Chua, and U. Parlitz, “Experimental demonstration of secure communications via chaotic synchronization,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 709–713 (1992).
[CrossRef]

U. Parlitz, L. O. Chua, Lj. Kocarev, K. S. Halle, and A. Shang, “Transmission of digital signals by chaotic synchronization,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 973–977 (1992).
[CrossRef]

Hasler, M.

H. Dedieu, M. P. Kennedy, and M. Hasler, “Chaos shift keying: modulation and demodulation of a chaotic carrier using self-synchronizing Chua’s circuits,” IEEE Trans. Circuits Syst. 40, 634–642 (1993).
[CrossRef]

Kennedy, M. P.

H. Dedieu, M. P. Kennedy, and M. Hasler, “Chaos shift keying: modulation and demodulation of a chaotic carrier using self-synchronizing Chua’s circuits,” IEEE Trans. Circuits Syst. 40, 634–642 (1993).
[CrossRef]

Kocarev, L.

L. Kocarev, K. S. Halle, K. Echert, L. O. Chua, and U. Parlitz, “Experimental demonstration of secure communications via chaotic synchronization,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 709–713 (1992).
[CrossRef]

Kocarev, Lj.

U. Parlitz, L. O. Chua, Lj. Kocarev, K. S. Halle, and A. Shang, “Transmission of digital signals by chaotic synchronization,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 973–977 (1992).
[CrossRef]

Larger, L.

J.-P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. 80, 2249–2252 (1998).
[CrossRef]

Luo, L. G.

Parlitz, U.

U. Parlitz, L. O. Chua, Lj. Kocarev, K. S. Halle, and A. Shang, “Transmission of digital signals by chaotic synchronization,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 973–977 (1992).
[CrossRef]

L. Kocarev, K. S. Halle, K. Echert, L. O. Chua, and U. Parlitz, “Experimental demonstration of secure communications via chaotic synchronization,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 709–713 (1992).
[CrossRef]

Pecora, L. M.

L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64, 821–824 (1990).
[CrossRef] [PubMed]

Porte, H.

J.-P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. 80, 2249–2252 (1998).
[CrossRef]

Roy, R.

G. D. Van Wiggeren and R. Roy, “Communications with chaotic lasers,” Science 279, 1198–1200 (1998).
[CrossRef]

P. Colet and R. Roy, “Digital communication with synchronized chaotic lasers,” Opt. Lett. 19, 2056–2058 (1994).
[CrossRef] [PubMed]

R. Roy and K. S. Thornburg, “Experimental synchronization of chaotic lasers,” Phys. Rev. Lett. 72, 2009–2012 (1994).
[CrossRef] [PubMed]

Scire, A.

V. Annovazzi-Lodi, S. Donati, and A. Scire, “Synchronization of chaotic injected-laser systems and its application to optical cryptography,” IEEE J. Quantum Electron. 32, 953–959 (1996).
[CrossRef]

Shang, A.

U. Parlitz, L. O. Chua, Lj. Kocarev, K. S. Halle, and A. Shang, “Transmission of digital signals by chaotic synchronization,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 973–977 (1992).
[CrossRef]

Shimizu, T.

T. Sugawara, M. Tachikawa, T. Tsukamoto, and T. Shimizu, “Observation of synchronization in laser chaos,” Phys. Rev. Lett. 72, 3502–3505 (1994).
[CrossRef] [PubMed]

Skinner, I.

J. W. Arkwright, B. Wu, I. Skinner, and P. L. Chu, “Resonantly enhanced nonlinearity using stimulated down-pumping in neodymium-doped twin core fibre,” Electron. Lett. 30, 235–236 (1994).
[CrossRef]

Sugawara, T.

T. Sugawara, M. Tachikawa, T. Tsukamoto, and T. Shimizu, “Observation of synchronization in laser chaos,” Phys. Rev. Lett. 72, 3502–3505 (1994).
[CrossRef] [PubMed]

Tachikawa, M.

T. Sugawara, M. Tachikawa, T. Tsukamoto, and T. Shimizu, “Observation of synchronization in laser chaos,” Phys. Rev. Lett. 72, 3502–3505 (1994).
[CrossRef] [PubMed]

Thornburg, K. S.

R. Roy and K. S. Thornburg, “Experimental synchronization of chaotic lasers,” Phys. Rev. Lett. 72, 2009–2012 (1994).
[CrossRef] [PubMed]

Tsukamoto, T.

T. Sugawara, M. Tachikawa, T. Tsukamoto, and T. Shimizu, “Observation of synchronization in laser chaos,” Phys. Rev. Lett. 72, 3502–3505 (1994).
[CrossRef] [PubMed]

Van Wiggeren, G. D.

G. D. Van Wiggeren and R. Roy, “Communications with chaotic lasers,” Science 279, 1198–1200 (1998).
[CrossRef]

Wu, B.

J. W. Arkwright, B. Wu, I. Skinner, and P. L. Chu, “Resonantly enhanced nonlinearity using stimulated down-pumping in neodymium-doped twin core fibre,” Electron. Lett. 30, 235–236 (1994).
[CrossRef]

Electron. Lett.

J. W. Arkwright, B. Wu, I. Skinner, and P. L. Chu, “Resonantly enhanced nonlinearity using stimulated down-pumping in neodymium-doped twin core fibre,” Electron. Lett. 30, 235–236 (1994).
[CrossRef]

IEEE J. Quantum Electron.

V. Annovazzi-Lodi, S. Donati, and A. Scire, “Synchronization of chaotic injected-laser systems and its application to optical cryptography,” IEEE J. Quantum Electron. 32, 953–959 (1996).
[CrossRef]

IEEE Trans. Circuits Syst.

H. Dedieu, M. P. Kennedy, and M. Hasler, “Chaos shift keying: modulation and demodulation of a chaotic carrier using self-synchronizing Chua’s circuits,” IEEE Trans. Circuits Syst. 40, 634–642 (1993).
[CrossRef]

Int. J. Bifurcation Chaos Appl. Sci. Eng.

U. Parlitz, L. O. Chua, Lj. Kocarev, K. S. Halle, and A. Shang, “Transmission of digital signals by chaotic synchronization,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 973–977 (1992).
[CrossRef]

L. Kocarev, K. S. Halle, K. Echert, L. O. Chua, and U. Parlitz, “Experimental demonstration of secure communications via chaotic synchronization,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 709–713 (1992).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

J.-P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. 80, 2249–2252 (1998).
[CrossRef]

L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64, 821–824 (1990).
[CrossRef] [PubMed]

R. Roy and K. S. Thornburg, “Experimental synchronization of chaotic lasers,” Phys. Rev. Lett. 72, 2009–2012 (1994).
[CrossRef] [PubMed]

T. Sugawara, M. Tachikawa, T. Tsukamoto, and T. Shimizu, “Observation of synchronization in laser chaos,” Phys. Rev. Lett. 72, 3502–3505 (1994).
[CrossRef] [PubMed]

Science

G. D. Van Wiggeren and R. Roy, “Communications with chaotic lasers,” Science 279, 1198–1200 (1998).
[CrossRef]

Other

L. G. Luo, T. J. Tee, and P. L. Chu, “Bistability of erbium doped fiber laser,” Opt. Commun. 146, 151–157 (1998); L. G. Luo, T. J. Tee, and P. L. Chu, “Chaotic behavior in erbium doped fiber ring lasers,” J. Opt. Soc. Am. B 15, 972–978 (1998).
[CrossRef]

H. Haken, Light (Elsevier, New York, 1985), Vol. 2, p. 98.

L. G. Luo and P. L. Chu, “Application of laser chaos to secure communication,” in Proceedings of 21st Australian Conference of Optical Fiber Technology (Institution of Radio and Electronics Engineers Australia, Sydney, 1996), pp. 245–248.

P. Celka, “Chaotic synchronization and modulation of nonlinear time-delayed feedback optical systems,” IEEE Trans. Circuits Syst. 42, 455–463 (1995); P. Celka, “Synchronization of chaotic optical dynamical systems through 700 m of single mode fiber,” IEEE Trans. Circuits Syst. 43, 869–872 (1996).
[CrossRef]

L. Kocarev and U. Parlitz, “General approach for chaotic synchronization with applications to communication,” Phys. Rev. Lett. 74, 5028–5031 (1995); U. Parlitz, L. Kocarev, T. Stojanovski, and H. Preckel, “Encoding messages using chaotic synchronization,” Phys. Rev. E 53, 4351–4361 (1996).
[CrossRef] [PubMed]

L. M. Pecora and T. L. Carroll, “Driving systems with chaotic signals,” Phys. Rev. A 44, 2374–2383 (1991); T. L. Carroll and L. M. Pecora, “Synchronizing chaotic circuits,” IEEE Trans. Circuits Syst. 38, 453–456 (1991).
[CrossRef] [PubMed]

M. Cuomo and A. V. Oppenheim, “Circuit implementation of synchronized chaos with applications to communications,” Phys. Rev. Lett. 71, 65–68 (1993); K. M. Cuomo, A. V. Oppenheim, and S. H. Strogatz, “Synchronization of Lorenz-based chaotic circuits with applications to communications,” IEEE Trans. Circuits Syst. 40, 626–633 (1993).
[CrossRef] [PubMed]

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Figures (10)

Fig. 1
Fig. 1

Erbium-doped fiber laser system with application to secure communication with the message-masking scheme: EDF, erbium-doped fiber; OM, optical modulator; WDM, wavelength-division-multiplexing coupler.

Fig. 2
Fig. 2

(a) Strange attractor of chaos on the plane of the field ELA and the population inversion DA. (b) Corresponding power spectrum of the chaotic light. ka0=3.3×107 s-1, ga=6.7 ×107 s-1, ma=0.03; ωa=3.5×105 s-1; ωs=3.14 ×105 s-1; IPA=10; ca=0.02.

Fig. 3
Fig. 3

Simulation of the working process of secure communication with the message-masking scheme. The corresponding parameters are respectively identical except kb0=3.4×107 s-1 and cb=0.02: (a) |Sin|2, (b) |ELA+Sin|2, (c) |ELB|2, (d) log10|ELA-ELB|2, and (e) |Srec|2=|ELA+Sin-ELB|2.

Fig. 4
Fig. 4

Erbium-doped fiber laser system for secure communication with the chaotic-shift-keying scheme.

Fig. 5
Fig. 5

(a) Chaotic strange attractor for the chaotic-shift-keying scheme. (b) Corresponding power spectrum of the chaotic light. ka0=3.3×107 s-1; ga=6.7×107 s-1; ma=0.03; ωa=2.2×105 s-1; IPA=10.

Fig. 6
Fig. 6

Process of secure communication with chaotic shift keying. The corresponding parameters are respectively identical except kb=3.4×107 s-1 and cb=0.11: (a) ωs, (b) |ELA|2, (c) |ELB|2, (d) 10 log10|ELB-ELA|2; units in decibels.

Fig. 7
Fig. 7

Process of secure communication with chaotic shift keying. The decay time of the population on the lasing high level is 1 ms. The parameters are the same as those in Fig. 6 except for the modulation frequency and cb=0.14: (a) ωs, (b) |ELA|2, (c) |ELB|2, (d) 10 log10|ELB-ELA|2; units in decibels.

Fig. 8
Fig. 8

Synchronization scale versus variation of the coefficients of the field loss and the field-gain coefficient. A valley exists along the line of (δk=-0.1, δg=-0.09) to (δk=0.1, δg=0.09).

Fig. 9
Fig. 9

Synchronization scale versus variation of the field loss and the pump power. A valley is along the line of (δk=-0.067, δp=-0.1) to (δk=0.067, δp=0.1).

Fig. 10
Fig. 10

Binary codes transmitted with chaotic-shift-keying scheme with 5% mismatches of k. ka0=3.3×107 s-1; kb=3.135×107 s-1; IPA=10, IPB=9.26: (a) ωs, (b) |ELA|2, (c) |ELB|2, (d) 10 log10|ELB-ELA|2; units in decibels.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

E˙L=-kEL+gELD+ξ,
D˙=-1τ [(1+IP+EL2)D-IP+1],
k=k0(1+m sin ωt),
E˙LA=-ka(ELA-caSin)+gaELADA+ξLA,
D˙A=-1τ [(1+IPA+ELA2)DA-IPA+1],
ka=ka0(1+ma sin ωat),
E˙LB=-kb[ELB-cb(ELA+Sin)]+gbELBDB+ξLB,
D˙B=-1τ [(1+IPB+ELB2)DB-IPB+1],
kb=kb0(1+mb sin ωbt),
Sin=S0(1-sin ωst).
E˙LA=-kaELA+gaELADA+ξLA,
D˙A=-1τ [(1+IPA+ELA2)DA-IPA+1],
ka=ka0(1+ma sin ωat)
E˙LB=-kb(ELB-cbELA)+gbELBDB+ξLB,
D˙B=-1τ [(1+IPB+ELB2)DB-IPB+1],
kb=kb0(1+mb sin ωbt)
ωa=(2.2+0.5η)×105,
η=0t=(2n+1)T/21t=2nT/2,
gb=ga(1+δg),
kb0=ka01-cb (1+δk),
IPB=IPA(1+δP).

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