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Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises
The coupled 1:2 resonance in a symmetric case and parametric amplification model
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran |
This paper deals with the coupled Hamiltonian $ 1 $:$ 2 $ resonance, i.e. the Hamiltonian $ 1 $:$ 2 $:$ 1 $:$ 2 $ resonance. This resonance is of the first order. We isolate several integrable cases. Our main focus is on two models. In the first part of the paper, we present a discrete symmetric normal form truncated to order three and we compute the relative equilibria for its corresponding system. In the second part, the paper is devoted to the study of the Hamiltonian describing the four-wave mixing (FWM) model. In addition to the Hamiltonian, the corresponding system possesses three more independent integrals. We use these integrals to obtain estimates for the phase space and total energy. Further, we compute the relative equilibria of the FWM system for the $ 1 $:$ 2 $:$ 1 $:$ 2 $ resonance. Finally, we carry out some numerical experiments for the detuned system.
References:
[1] |
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, in Dynamical Systems III, Encyc. Math. Sciences, Springer-Verlag, Berlin, 2006. |
[2] |
H. W. Broer, G. A. Lunter and G. Vegter,
Equivariant singularity theory with distinguished parameters: Two case studies of resonant Hamiltonian systems, Phys. D, 112 (1998), 64-80.
doi: 10.1016/S0167-2789(97)00202-9. |
[3] |
H. Broer and F. Takens, Dynamical Systems and Chaos, Appl. Math. Sciences, Vol. 172, Springer, New York, 2011.
doi: 10.1007/978-1-4419-6870-8. |
[4] |
R. Bruggeman and F. Verhulst,
The inhomogeneous Fermi-Pasta-Ulam chain. A case study of the $1:2:3$ Resonance, Acta Appl. Math., 152 (2017), 111-145.
doi: 10.1007/s10440-017-0115-4. |
[5] |
G. Cappellini and S. Trillo, Third-order three-wave mixxing in single-mode fibers: Exact solutions and spatial instability effects, J. Opt. Soc. Am. B., 8 (1991), 824-838. Google Scholar |
[6] |
O. Christov,
Non-integrability of first order resonances of Hamiltonian systems in three degrees of freedom, Celestial Mech. Dynam. Astronom., 112 (2012), 147-167.
doi: 10.1007/s10569-011-9389-4. |
[7] |
C. De Angelis, M. Santagiustina and S. Trillo,
Four-photon homoclinic instabilities in nonlinear highly birefringent media, Phys. Rev. A., 51 (1995), 774-791.
doi: 10.1103/PhysRevA.51.774. |
[8] |
J. J. Duistermaat,
Non-integrability of the $1$ : $2$ : $1$-resonance, Ergodic Theory Dynam. Systems, 4 (1984), 553-568.
doi: 10.1017/S0143385700002649. |
[9] |
J. Egea, S. Ferrer and J. C. van der Meer,
Bifurcations of the Hamiltonian fourfold $1$ : $1$ resonance with toroidal symmetry, J. Nonlinear Sci., 21 (2011), 835-874.
doi: 10.1007/s00332-011-9102-5. |
[10] |
D. D. Holm and P. Lynch,
Stepwise precession of the resonant swinging spring, SIAM J. Appl. Dyn. Syst., 1 (2002), 44-64.
doi: 10.1137/S1111111101388571. |
[11] |
G. Haller and S. Wiggins,
Geometry and chaos near resonant equilibria of 3-DOF Hamiltonian systems, Physica D, 90 (1996), 319-365.
doi: 10.1016/0167-2789(95)00247-2. |
[12] |
H. Hanßmann, Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems. Results and Examples, Lecture Notes Math., 1893, Springer-Verlag, Berlin, Heidelberg, 2007. Google Scholar |
[13] |
H. Hanßmann, R. Mazrooei-Sebdani, F. Verhulst, The $1: 2: 4$ resonance in a particle chain, preprint, 2020, arXiv: 2002.01263. Google Scholar |
[14] |
G. Y. Kryuchkyan and K. V. Kheruntsyan,
Four-wave mixing with non-degenerate pumps: Steady states and squeezing in the presence of phase modulation, Quantum Semiclass. Opt., 7 (1995), 529-539.
doi: 10.1088/1355-5111/7/4/010. |
[15] |
M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices, Cambridge University, Cambridge, 2008.
doi: 10.1017/CBO9780511600265. |
[16] |
S. Medvedev and B. Bednyakova, Hamiltonian approach for optimization of phase-sensitive double-pumped parametric amplifiers, Opt. Express., 26 (2018), 15503.
doi: 10.1364/OE.26.015503. |
[17] |
H. Pourbeyram and A. Mafi,
Four-wave mixing of a laser and its frequency-doubled version in a multimode optical fiber, Photonics, 2 (2015), 906-915.
doi: 10.3390/photonics2030906. |
[18] |
J. R. Ott, H. Steffensen, K. Rottwitt and C. J. Mckinstrie, Geometric interpreation of four-wave mixing, Phys. Rev. A., 88 (2013), 043805. Google Scholar |
[19] |
A. A. Redyuk, A. E. Bednyakova, S. B. Medvedev, M. P. Fedoruk and S. K. Turitsyn, Simple Geometric interpreation of signal evolution in phase-sensitive fibre optic parametric amplifier, Opt. Express., 25 (2017), 223-231. Google Scholar |
[20] |
D. A. Sadovski and B. I. Zhilinski,
Hamiltonian systems with detuned $1$:$1$:$2$ resonance: Manifestation of bidromy, Ann. Physics, 322 (2007), 164-200.
doi: 10.1016/j.aop.2006.09.011. |
[21] |
J. A. Sanders, F. Verhulst and J. Murdock, Averaging methods in nonlinear dynamical systems. Second Edition., Applied Mathematical Sciences, , Vol. 59, Springer, New York, 2007. |
[22] |
S. Trillo and S. Wabnitz,
Dynamics of the nonlinear modulational instability in optical fibers, Opt. Lett., 16 (1991), 986-988.
doi: 10.1364/OL.16.000986. |
[23] |
E. van der Aa,
First order resonances in three-degrees-of-freedom systems, Celestial Mech., 31 (1983), 163-191.
doi: 10.1007/BF01686817. |
[24] |
E. van der Aa and J. A. Sanders, The $1$: $2$: $1$-resonance, its periodic orbits and integrals, in Asymptotic Analysis: From Theory to Application, Lecture Notes Math., Vol. 711, Springer, 1979,187–208. Google Scholar |
[25] |
E. van der Aa and F. Verhulst,
Asymptotic integrability and periodic solutions of a Hamiltonian system in $1$ : $2$ : $2$-resonance, SIAM J. Math. Anal., 15 (1984), 890-911.
doi: 10.1137/0515067. |
[26] |
F. Verhulst,
Integrability and non-integrability of Hamiltonian normal forms, Acta Appl. Math., 137 (2015), 253-272.
doi: 10.1007/s10440-014-9998-5. |
[27] | L. Vivien and L. Pavesi, Handbook of Silicon Photonics. First Edition, CRC Press, Taylor & Francis Group, 2013. Google Scholar |
[28] |
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. Second Edition, in Texts in Appl. Math., Springer-Verlag, New York, 2003. |
show all references
References:
[1] |
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, in Dynamical Systems III, Encyc. Math. Sciences, Springer-Verlag, Berlin, 2006. |
[2] |
H. W. Broer, G. A. Lunter and G. Vegter,
Equivariant singularity theory with distinguished parameters: Two case studies of resonant Hamiltonian systems, Phys. D, 112 (1998), 64-80.
doi: 10.1016/S0167-2789(97)00202-9. |
[3] |
H. Broer and F. Takens, Dynamical Systems and Chaos, Appl. Math. Sciences, Vol. 172, Springer, New York, 2011.
doi: 10.1007/978-1-4419-6870-8. |
[4] |
R. Bruggeman and F. Verhulst,
The inhomogeneous Fermi-Pasta-Ulam chain. A case study of the $1:2:3$ Resonance, Acta Appl. Math., 152 (2017), 111-145.
doi: 10.1007/s10440-017-0115-4. |
[5] |
G. Cappellini and S. Trillo, Third-order three-wave mixxing in single-mode fibers: Exact solutions and spatial instability effects, J. Opt. Soc. Am. B., 8 (1991), 824-838. Google Scholar |
[6] |
O. Christov,
Non-integrability of first order resonances of Hamiltonian systems in three degrees of freedom, Celestial Mech. Dynam. Astronom., 112 (2012), 147-167.
doi: 10.1007/s10569-011-9389-4. |
[7] |
C. De Angelis, M. Santagiustina and S. Trillo,
Four-photon homoclinic instabilities in nonlinear highly birefringent media, Phys. Rev. A., 51 (1995), 774-791.
doi: 10.1103/PhysRevA.51.774. |
[8] |
J. J. Duistermaat,
Non-integrability of the $1$ : $2$ : $1$-resonance, Ergodic Theory Dynam. Systems, 4 (1984), 553-568.
doi: 10.1017/S0143385700002649. |
[9] |
J. Egea, S. Ferrer and J. C. van der Meer,
Bifurcations of the Hamiltonian fourfold $1$ : $1$ resonance with toroidal symmetry, J. Nonlinear Sci., 21 (2011), 835-874.
doi: 10.1007/s00332-011-9102-5. |
[10] |
D. D. Holm and P. Lynch,
Stepwise precession of the resonant swinging spring, SIAM J. Appl. Dyn. Syst., 1 (2002), 44-64.
doi: 10.1137/S1111111101388571. |
[11] |
G. Haller and S. Wiggins,
Geometry and chaos near resonant equilibria of 3-DOF Hamiltonian systems, Physica D, 90 (1996), 319-365.
doi: 10.1016/0167-2789(95)00247-2. |
[12] |
H. Hanßmann, Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems. Results and Examples, Lecture Notes Math., 1893, Springer-Verlag, Berlin, Heidelberg, 2007. Google Scholar |
[13] |
H. Hanßmann, R. Mazrooei-Sebdani, F. Verhulst, The $1: 2: 4$ resonance in a particle chain, preprint, 2020, arXiv: 2002.01263. Google Scholar |
[14] |
G. Y. Kryuchkyan and K. V. Kheruntsyan,
Four-wave mixing with non-degenerate pumps: Steady states and squeezing in the presence of phase modulation, Quantum Semiclass. Opt., 7 (1995), 529-539.
doi: 10.1088/1355-5111/7/4/010. |
[15] |
M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices, Cambridge University, Cambridge, 2008.
doi: 10.1017/CBO9780511600265. |
[16] |
S. Medvedev and B. Bednyakova, Hamiltonian approach for optimization of phase-sensitive double-pumped parametric amplifiers, Opt. Express., 26 (2018), 15503.
doi: 10.1364/OE.26.015503. |
[17] |
H. Pourbeyram and A. Mafi,
Four-wave mixing of a laser and its frequency-doubled version in a multimode optical fiber, Photonics, 2 (2015), 906-915.
doi: 10.3390/photonics2030906. |
[18] |
J. R. Ott, H. Steffensen, K. Rottwitt and C. J. Mckinstrie, Geometric interpreation of four-wave mixing, Phys. Rev. A., 88 (2013), 043805. Google Scholar |
[19] |
A. A. Redyuk, A. E. Bednyakova, S. B. Medvedev, M. P. Fedoruk and S. K. Turitsyn, Simple Geometric interpreation of signal evolution in phase-sensitive fibre optic parametric amplifier, Opt. Express., 25 (2017), 223-231. Google Scholar |
[20] |
D. A. Sadovski and B. I. Zhilinski,
Hamiltonian systems with detuned $1$:$1$:$2$ resonance: Manifestation of bidromy, Ann. Physics, 322 (2007), 164-200.
doi: 10.1016/j.aop.2006.09.011. |
[21] |
J. A. Sanders, F. Verhulst and J. Murdock, Averaging methods in nonlinear dynamical systems. Second Edition., Applied Mathematical Sciences, , Vol. 59, Springer, New York, 2007. |
[22] |
S. Trillo and S. Wabnitz,
Dynamics of the nonlinear modulational instability in optical fibers, Opt. Lett., 16 (1991), 986-988.
doi: 10.1364/OL.16.000986. |
[23] |
E. van der Aa,
First order resonances in three-degrees-of-freedom systems, Celestial Mech., 31 (1983), 163-191.
doi: 10.1007/BF01686817. |
[24] |
E. van der Aa and J. A. Sanders, The $1$: $2$: $1$-resonance, its periodic orbits and integrals, in Asymptotic Analysis: From Theory to Application, Lecture Notes Math., Vol. 711, Springer, 1979,187–208. Google Scholar |
[25] |
E. van der Aa and F. Verhulst,
Asymptotic integrability and periodic solutions of a Hamiltonian system in $1$ : $2$ : $2$-resonance, SIAM J. Math. Anal., 15 (1984), 890-911.
doi: 10.1137/0515067. |
[26] |
F. Verhulst,
Integrability and non-integrability of Hamiltonian normal forms, Acta Appl. Math., 137 (2015), 253-272.
doi: 10.1007/s10440-014-9998-5. |
[27] | L. Vivien and L. Pavesi, Handbook of Silicon Photonics. First Edition, CRC Press, Taylor & Francis Group, 2013. Google Scholar |
[28] |
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. Second Edition, in Texts in Appl. Math., Springer-Verlag, New York, 2003. |







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