- Email: [email protected]
Chaotic signal-induced dynamics of degenerate optical parametric oscillator
Available online at www.sciencedirect.com
Chaos, Solitons and Fractals 36 (2008) 494–499 www.elsevier.com/locate/chaos
Chaotic signal-induced dynamics of degenerate optical parametric oscillator Ma Jun a b
a,b,*
, Jin Wu-Yin a, Li Yan-Long
a,b
School of science, Lanzhou University of Technology, Lanzhou 730050, China Institute of Theoretical Physics, Lanzhou University, Lanzhou 730050, China Accepted 29 June 2006
Abstract The degenerate optical parametric oscillator (DOPO) is investigated. We introduced a normal Lorenz chaotic signal to adjust the amplitude and period of the input electric field in order to influence the dynamics of the time-dependent system. Our numerical simulation results based on the phase figures and Lyapunov exponents spectrum confirm that the characters of the DOPO are determined by the amplitude of the input field, and the system could be controlled to reach n-periodical (n = 1, 2, 3, 4, 5, etc.) orbit, chaotic and/ or hyperchaotic and stable state by using a modified scheme based on the self-adaptive stratagem. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction Since the pioneering work of Ott et al. [1–3], peoples began to pay more attention to the topic of chaos and hyperchaos, including chaos control, anti-control and synchronization. Time-dependent and spatiotemporal chaotic systems [4–14] are investigated in many fields, such as chemical, engineering, economics, physics and so on. As an important application, chaos in Lasers attracts more attention. Generally, periodical signal is used to drive the chaotic system or adjust a certain parameter of the system in order to direct the system to reach the stable state or desired arbitrary orbit, and it proved its effectiveness in several models. Here, an anti-chaos scheme is used to investigate the dynamics of the degenerate optical parametric oscillator (DOPO) [15] by introducing a chaotic signal to adjust the input electric field of the DOPO. The DOPO is modified from the optical parametric oscillator (OPO), which is used to describe the broadly tunable source of highly coherent radiation. Bistability, self-pulsing, Hopf bifurcations and chaos of the DOPO were investigated for the first time by Pettiaux et al. [15], furthermore, DOPO is studied and testified by Oppo et al. [15–18]. In this letter, we are interesting in chaotic signal-induced dynamics of the DOPO by applying the anti-chaos control scheme, which introduces a Lorenz chaotic signal with different amplitudes into the DOPO in order to adjust the input filed. Our goal is to prove our scheme of anti-control chaos could control the DOPO to reach arbitrary desired state, stable state, n-periodical orbit, etc. The phase figures and Lyapunov exponents [12,13] of the controlled DOPO are plotted so that the evolution of the dynamics of the system could be approached. *
Corresponding author. E-mail address: [email protected] (M. Jun).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.06.072
M. Jun et al. / Chaos, Solitons and Fractals 36 (2008) 494–499
495
2. Schemes and numerical simulation results The DOPO system is defined as Eq. (1) [15]: ( A_ 1 ¼ ð1 þ iD1 ÞA1 þ A1 A0 A_ 0 ¼ ð1 þ icÞA0 þ EA A21
ð1Þ
Fig. 1. Evolution of the variables vs time (t = 0, 100 time units) at EA = 0.1x3 (a), EA = 0.3x3 (b), phase figure of x vs y (c), x vs v (d) at EA = 1.0x3, the lyapunov exponents of the DOPO when the input filed EA is adjusted by the controller EA = Kx3 from K = 0 to K = 10 (e).
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where A1 and A0 are the complex amplitude of the subharmonic mode and fundamental mode of the field, respectively. Furthermore, D1 and D0 are the detuning parameters, c is the reduced decay rate of the fundamental mode and EA is the input field amplitude, which is set as real and positive generally. Feng and Shen [18] discussed its characters with parameters c = D0 = 1.0 and D1 = 5, and confirmed its hyperchaotic state when the input field EA = 10.8. Compared to the results in [18], here, we select the same parameters c = D0 = 1.0 and D1 = 5 while the input filed EA will be adjusted by the Lorenz chaotic signal with different amplitude. To understand the scheme in ease, Eq. (1) is equivalent to the following Eq. (2) by introducing a direct transform as Eq. (3). 8 x_ ¼ x þ D1 y þ xu þ yv > > > < y_ ¼ D x y yu þ xv 1 ð2Þ > _ u ¼ cu þ D0 v x2 þ y 2 þ EA > > : v_ ¼ cv D0 u 2xy ð3Þ A1 ¼ x þ iy; A0 ¼ u þ iv where x, y, u and v are real variables and i is the imaginary number. The equivalent point is (0, 0, 0, 0) in absence of the input EA (EA = 0). The Lorenz chaotic system is defined in the following equation: x_ 1 ¼ 10ðx2 x1 Þ;
x_ 2 ¼ 28x1 x2 x1 x3 ;
x_ 3 ¼ x1 x2 2:667x3
ð4Þ
It is reliable and interesting to adjust the amplitude of input filed EA with different signal. Here, we will investigate the dynamics evolution of Eq. (1) or/and (2) by adjusting the amplitude EA. EA ¼ Kx3
ð5Þ
Here, K is a positive constant and x3 is the variable of Eq. (4). The numerical results according to the controller (5) is plotted in Fig. 1 based on the fourth-order R K integration scheme at step size h = 0.001, and the Lyapunov exponent spectrum of the controlled system are computated with the effective scheme in [12,13]. The numerical simulation results confirm that the first two variables x, y or the subharmonic mode A1 begin to reach zero when the amplitude K of the driving chaotic signal is small, furthermore, more numerical simulation results show the subharmonic mode will become chaotic when the amplitude increased, e.g. K = 1.0, the results are plotted in Fig. 1c and d, the chaotic attractors are observed in our numerical simulation. The relevant Lyapunov exponents are plotted when the amplitude of the driving chaotic signal is increased in Fig. 1e. We can conclude that weaker K will direct the whole system to become stable while stronger K occurs chaotic state or hyperchaotic state. Furthermore, the external electric field is adjusted by the first or/and second variable in Eq. (3) as EA = Kx2 and/or EA = Kx1, and all the variables in Eq. (1) or/and (2) are investigated by introducing the controller EA = 0.3 x2 into the system, and the Lyapunov exponents [12,13] subjected to the controller EA = 0.3 x2 are plotted in Fig. 2. The numerical results confirm that the first two variable x and y reached zero soon while the last two ones keep chaotic, which means that the subharmonic mode could be influenced greatly. More numerical results show that stronger K in controller EA = Kx2 and EA = Kx1 causes the four variables or the subharmonic mode and fundamental mode of the field to become chaotic, and the Lyapunov. Exponents are plotted vs K in Fig. 3.
Fig. 2. Evolution of the variables vs time (t = 0, 100 time units) at EA = 0.3x2 for x, u vs time (a), y, v vs time (b).
M. Jun et al. / Chaos, Solitons and Fractals 36 (2008) 494–499
497
Fig. 3. Lyapunov exponents vs K for controller EA = K x1 (a), EA = K x2 (b)
The numerical simulation results illustrate that the system will become chaotic as the intensity of input filed increased. Then we investigate the improved controller as EA = Kxx3, EA = Kyx3 and EA = Kvx3, the external input electric filed modulated by the controller EA = Kxx3, EA = Kyx3 and EA = Kvx3 will stop work when the variables x, y and
Fig. 4. n-periodical orbit at controller EA = 8 + 0.01x1, 2-P for t = 26 time units x vs y (a), x vs u (b); 3-P for t = 35 time units x vs y (c), x vs u (d); 4-P for t = 45 time units x vs y (e), x vs u (f); 5-P for t = 55 time units x vs y (g), x vs u (h).
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M. Jun et al. / Chaos, Solitons and Fractals 36 (2008) 494–499
Fig. 4 (continued)
v reach zero. Then we calculated the Lyapunov exponents of Eq. (1) and/or (2) subjected to the improved controller above, the numerical simulation results (all negative Lyapunov exponents) confirmed that all the variables in Eq. (1) or (2) becomes zero and stable. Compared to the periodical signal driving and/or periodical signal modulated scheme, here we would like to testify our scheme whether it is effective to control the system to n-periodical state. Here, we introduce the input filed as EA = 8 + 0.01 x1 into Eq. (1) or/and (2). The phase figures are plotted at different time in Fig. 4. The numerical simulation results show that the system can reach different periodical orbits at right time subjected to a chaotic input field EA = 8 + 0.01x1. More numerical simulation results confirm that any period orbit could be reached and observed in numerical simulation when appropriate chaotic input field is introduced into Eq. (1) and /or (2).
3. Conclusion Chaotic signal-induced dynamics of DOPO chaotic system are investigated. Lorenz chaotic signal is introduced into the DOPO system to influence its character by adjusting the amplitude of the input electric field. The scheme is different from the noise-induced and periodical signal-induced dynamics because there is much difference among a periodical signal, chaotic signal and noise. The DOPO can reach stable zero state, n-period (n = 1, 2, 3, 4, 5, etc.) orbit and chaotic and/or hyperchaotic state determined by the amplitude of the driving chaotic signal, it is another application and study of anti-control chaos.
M. Jun et al. / Chaos, Solitons and Fractals 36 (2008) 494–499
499
Acknowledgements We thanks to Professor Ying H.P. for useful discussion, and this work is supported partially by the Natural Natural Science foundation of China under Grant No. 10572056, 10405018 and the Natural Science foundation of Gansu Province under the Grant No. 3ZS042-B25-021.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys Rev Lett 1990;64:1196–9. Boccaletti S, Grebogi C, Lai YC, et al. The control of chaos: theory and applications. Phys Rep 2000;329:103–97. Boccaletti S, Kurths J, Osipov G, et al. The synchronization of chaotic systems. Phys Rep 2002;366:1–101. Xiao JH, Hu G, Yang JZ, et al. Controlling turbulence in the complex Ginzburg–Landau equation. Phys Rev Lett 1998;81:5552–5. Zhang H, Hu B, Hu G, et al. Turbulence control by developing a spiral wave with a periodic signal injection in the complex Ginzburg–Landau equation. Phys Rev E 2002;66:046303 (p. 5). Park JH, Kwon OM. A novel criterion for delayed feedback control of time-delay chaotic systems. Chaos, Solitons and Fractrals 2005;23:495–501. Zhang T,Wang J, Fei XY et al. Synchronization of coupled Fitzhugh-Nagumo systems via MIMO feedback linearization control. Chaos, Solitons and Fractrals, in press, doi:10.1016/j.chaos.2006.01.037. Liu SD, Liu SK, Fu ZT, et al. The Hopf bifurcation and spiral wave solution of the complex Ginzburg–Landau equation. Chaos, Solitons and Fractrals 2002;13:1377–81. Ahlstrom C, Johansson A, Hult P, et al. Chaotic dynamics of respiratory sounds. Chaos, Solitons and Fractrals 2006;29:1054–62. Hou ZH, Yang LF, Zuo XB, et al. Noise induced pattern transition and spatiotemporal stochastic resonance. Phys Rev Lett 1998;81:2854–7. Ma J, Pu ZS, Feng WJ, et al. Hyperchaos synchronization and control using intermittent feedback. Acta Phys Sin 2005;54:4602–9. He DH, Xu JX, Chen YH. Study on Lyapunov characteristic exponents of a nonlinear differential equation system. Acta Phys Sin 2000;49:0833–7. Lu J, Yang GL, Hyounkyun Oh, et al. Computing Lyapunov exponents of continuous dynamical systems: method of Lyapunov vectors. Chaos, Solitons and Fractrals 2005;23:1879–92. Ma J, Ying HP, Pu ZS. An anti-control scheme for spiral under Lorenz chaotic signals. Chinese Phys Lett 2005;22:1065–8; Ma J, Ying HP, Pan GW. Evolution of spiral waves under modulated electric fields. Chinese Phys Lett 2005;22:2176–9. Pettiaux NP, Li RD, Mandel P. Instabilities of the degenerate optical parametric oscillator. Opt Commun 1989;72:256–60. Oppo GL, Bramblla, Lugiato LA. Formation and evolution of roll patterns in optical parametric oscillators. Phys Rev A 1994;49:2028–32. Sialiunas K, Sa´nchez-Morcillo VJ. Spatial-localized structures in degenerate optical parametric oscillators. Phys Rev A 1998;57:1454–7. Feng XQ, Shen K. Phase synchronization and anti-phase synchronization of chaos for degenerate optical parametric oscillator. Chinese Phys 2005;14:1526–32.
Chaos, Solitons and Fractals 36 (2008) 494–499 www.elsevier.com/locate/chaos
Chaotic signal-induced dynamics of degenerate optical parametric oscillator Ma Jun a b
a,b,*
, Jin Wu-Yin a, Li Yan-Long
a,b
School of science, Lanzhou University of Technology, Lanzhou 730050, China Institute of Theoretical Physics, Lanzhou University, Lanzhou 730050, China Accepted 29 June 2006
Abstract The degenerate optical parametric oscillator (DOPO) is investigated. We introduced a normal Lorenz chaotic signal to adjust the amplitude and period of the input electric field in order to influence the dynamics of the time-dependent system. Our numerical simulation results based on the phase figures and Lyapunov exponents spectrum confirm that the characters of the DOPO are determined by the amplitude of the input field, and the system could be controlled to reach n-periodical (n = 1, 2, 3, 4, 5, etc.) orbit, chaotic and/ or hyperchaotic and stable state by using a modified scheme based on the self-adaptive stratagem. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction Since the pioneering work of Ott et al. [1–3], peoples began to pay more attention to the topic of chaos and hyperchaos, including chaos control, anti-control and synchronization. Time-dependent and spatiotemporal chaotic systems [4–14] are investigated in many fields, such as chemical, engineering, economics, physics and so on. As an important application, chaos in Lasers attracts more attention. Generally, periodical signal is used to drive the chaotic system or adjust a certain parameter of the system in order to direct the system to reach the stable state or desired arbitrary orbit, and it proved its effectiveness in several models. Here, an anti-chaos scheme is used to investigate the dynamics of the degenerate optical parametric oscillator (DOPO) [15] by introducing a chaotic signal to adjust the input electric field of the DOPO. The DOPO is modified from the optical parametric oscillator (OPO), which is used to describe the broadly tunable source of highly coherent radiation. Bistability, self-pulsing, Hopf bifurcations and chaos of the DOPO were investigated for the first time by Pettiaux et al. [15], furthermore, DOPO is studied and testified by Oppo et al. [15–18]. In this letter, we are interesting in chaotic signal-induced dynamics of the DOPO by applying the anti-chaos control scheme, which introduces a Lorenz chaotic signal with different amplitudes into the DOPO in order to adjust the input filed. Our goal is to prove our scheme of anti-control chaos could control the DOPO to reach arbitrary desired state, stable state, n-periodical orbit, etc. The phase figures and Lyapunov exponents [12,13] of the controlled DOPO are plotted so that the evolution of the dynamics of the system could be approached. *
Corresponding author. E-mail address: [email protected] (M. Jun).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.06.072
M. Jun et al. / Chaos, Solitons and Fractals 36 (2008) 494–499
495
2. Schemes and numerical simulation results The DOPO system is defined as Eq. (1) [15]: ( A_ 1 ¼ ð1 þ iD1 ÞA1 þ A1 A0 A_ 0 ¼ ð1 þ icÞA0 þ EA A21
ð1Þ
Fig. 1. Evolution of the variables vs time (t = 0, 100 time units) at EA = 0.1x3 (a), EA = 0.3x3 (b), phase figure of x vs y (c), x vs v (d) at EA = 1.0x3, the lyapunov exponents of the DOPO when the input filed EA is adjusted by the controller EA = Kx3 from K = 0 to K = 10 (e).
496
M. Jun et al. / Chaos, Solitons and Fractals 36 (2008) 494–499
where A1 and A0 are the complex amplitude of the subharmonic mode and fundamental mode of the field, respectively. Furthermore, D1 and D0 are the detuning parameters, c is the reduced decay rate of the fundamental mode and EA is the input field amplitude, which is set as real and positive generally. Feng and Shen [18] discussed its characters with parameters c = D0 = 1.0 and D1 = 5, and confirmed its hyperchaotic state when the input field EA = 10.8. Compared to the results in [18], here, we select the same parameters c = D0 = 1.0 and D1 = 5 while the input filed EA will be adjusted by the Lorenz chaotic signal with different amplitude. To understand the scheme in ease, Eq. (1) is equivalent to the following Eq. (2) by introducing a direct transform as Eq. (3). 8 x_ ¼ x þ D1 y þ xu þ yv > > > < y_ ¼ D x y yu þ xv 1 ð2Þ > _ u ¼ cu þ D0 v x2 þ y 2 þ EA > > : v_ ¼ cv D0 u 2xy ð3Þ A1 ¼ x þ iy; A0 ¼ u þ iv where x, y, u and v are real variables and i is the imaginary number. The equivalent point is (0, 0, 0, 0) in absence of the input EA (EA = 0). The Lorenz chaotic system is defined in the following equation: x_ 1 ¼ 10ðx2 x1 Þ;
x_ 2 ¼ 28x1 x2 x1 x3 ;
x_ 3 ¼ x1 x2 2:667x3
ð4Þ
It is reliable and interesting to adjust the amplitude of input filed EA with different signal. Here, we will investigate the dynamics evolution of Eq. (1) or/and (2) by adjusting the amplitude EA. EA ¼ Kx3
ð5Þ
Here, K is a positive constant and x3 is the variable of Eq. (4). The numerical results according to the controller (5) is plotted in Fig. 1 based on the fourth-order R K integration scheme at step size h = 0.001, and the Lyapunov exponent spectrum of the controlled system are computated with the effective scheme in [12,13]. The numerical simulation results confirm that the first two variables x, y or the subharmonic mode A1 begin to reach zero when the amplitude K of the driving chaotic signal is small, furthermore, more numerical simulation results show the subharmonic mode will become chaotic when the amplitude increased, e.g. K = 1.0, the results are plotted in Fig. 1c and d, the chaotic attractors are observed in our numerical simulation. The relevant Lyapunov exponents are plotted when the amplitude of the driving chaotic signal is increased in Fig. 1e. We can conclude that weaker K will direct the whole system to become stable while stronger K occurs chaotic state or hyperchaotic state. Furthermore, the external electric field is adjusted by the first or/and second variable in Eq. (3) as EA = Kx2 and/or EA = Kx1, and all the variables in Eq. (1) or/and (2) are investigated by introducing the controller EA = 0.3 x2 into the system, and the Lyapunov exponents [12,13] subjected to the controller EA = 0.3 x2 are plotted in Fig. 2. The numerical results confirm that the first two variable x and y reached zero soon while the last two ones keep chaotic, which means that the subharmonic mode could be influenced greatly. More numerical results show that stronger K in controller EA = Kx2 and EA = Kx1 causes the four variables or the subharmonic mode and fundamental mode of the field to become chaotic, and the Lyapunov. Exponents are plotted vs K in Fig. 3.
Fig. 2. Evolution of the variables vs time (t = 0, 100 time units) at EA = 0.3x2 for x, u vs time (a), y, v vs time (b).
M. Jun et al. / Chaos, Solitons and Fractals 36 (2008) 494–499
497
Fig. 3. Lyapunov exponents vs K for controller EA = K x1 (a), EA = K x2 (b)
The numerical simulation results illustrate that the system will become chaotic as the intensity of input filed increased. Then we investigate the improved controller as EA = Kxx3, EA = Kyx3 and EA = Kvx3, the external input electric filed modulated by the controller EA = Kxx3, EA = Kyx3 and EA = Kvx3 will stop work when the variables x, y and
Fig. 4. n-periodical orbit at controller EA = 8 + 0.01x1, 2-P for t = 26 time units x vs y (a), x vs u (b); 3-P for t = 35 time units x vs y (c), x vs u (d); 4-P for t = 45 time units x vs y (e), x vs u (f); 5-P for t = 55 time units x vs y (g), x vs u (h).
498
M. Jun et al. / Chaos, Solitons and Fractals 36 (2008) 494–499
Fig. 4 (continued)
v reach zero. Then we calculated the Lyapunov exponents of Eq. (1) and/or (2) subjected to the improved controller above, the numerical simulation results (all negative Lyapunov exponents) confirmed that all the variables in Eq. (1) or (2) becomes zero and stable. Compared to the periodical signal driving and/or periodical signal modulated scheme, here we would like to testify our scheme whether it is effective to control the system to n-periodical state. Here, we introduce the input filed as EA = 8 + 0.01 x1 into Eq. (1) or/and (2). The phase figures are plotted at different time in Fig. 4. The numerical simulation results show that the system can reach different periodical orbits at right time subjected to a chaotic input field EA = 8 + 0.01x1. More numerical simulation results confirm that any period orbit could be reached and observed in numerical simulation when appropriate chaotic input field is introduced into Eq. (1) and /or (2).
3. Conclusion Chaotic signal-induced dynamics of DOPO chaotic system are investigated. Lorenz chaotic signal is introduced into the DOPO system to influence its character by adjusting the amplitude of the input electric field. The scheme is different from the noise-induced and periodical signal-induced dynamics because there is much difference among a periodical signal, chaotic signal and noise. The DOPO can reach stable zero state, n-period (n = 1, 2, 3, 4, 5, etc.) orbit and chaotic and/or hyperchaotic state determined by the amplitude of the driving chaotic signal, it is another application and study of anti-control chaos.
M. Jun et al. / Chaos, Solitons and Fractals 36 (2008) 494–499
499
Acknowledgements We thanks to Professor Ying H.P. for useful discussion, and this work is supported partially by the Natural Natural Science foundation of China under Grant No. 10572056, 10405018 and the Natural Science foundation of Gansu Province under the Grant No. 3ZS042-B25-021.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys Rev Lett 1990;64:1196–9. Boccaletti S, Grebogi C, Lai YC, et al. The control of chaos: theory and applications. Phys Rep 2000;329:103–97. Boccaletti S, Kurths J, Osipov G, et al. The synchronization of chaotic systems. Phys Rep 2002;366:1–101. Xiao JH, Hu G, Yang JZ, et al. Controlling turbulence in the complex Ginzburg–Landau equation. Phys Rev Lett 1998;81:5552–5. Zhang H, Hu B, Hu G, et al. Turbulence control by developing a spiral wave with a periodic signal injection in the complex Ginzburg–Landau equation. Phys Rev E 2002;66:046303 (p. 5). Park JH, Kwon OM. A novel criterion for delayed feedback control of time-delay chaotic systems. Chaos, Solitons and Fractrals 2005;23:495–501. Zhang T,Wang J, Fei XY et al. Synchronization of coupled Fitzhugh-Nagumo systems via MIMO feedback linearization control. Chaos, Solitons and Fractrals, in press, doi:10.1016/j.chaos.2006.01.037. Liu SD, Liu SK, Fu ZT, et al. The Hopf bifurcation and spiral wave solution of the complex Ginzburg–Landau equation. Chaos, Solitons and Fractrals 2002;13:1377–81. Ahlstrom C, Johansson A, Hult P, et al. Chaotic dynamics of respiratory sounds. Chaos, Solitons and Fractrals 2006;29:1054–62. Hou ZH, Yang LF, Zuo XB, et al. Noise induced pattern transition and spatiotemporal stochastic resonance. Phys Rev Lett 1998;81:2854–7. Ma J, Pu ZS, Feng WJ, et al. Hyperchaos synchronization and control using intermittent feedback. Acta Phys Sin 2005;54:4602–9. He DH, Xu JX, Chen YH. Study on Lyapunov characteristic exponents of a nonlinear differential equation system. Acta Phys Sin 2000;49:0833–7. Lu J, Yang GL, Hyounkyun Oh, et al. Computing Lyapunov exponents of continuous dynamical systems: method of Lyapunov vectors. Chaos, Solitons and Fractrals 2005;23:1879–92. Ma J, Ying HP, Pu ZS. An anti-control scheme for spiral under Lorenz chaotic signals. Chinese Phys Lett 2005;22:1065–8; Ma J, Ying HP, Pan GW. Evolution of spiral waves under modulated electric fields. Chinese Phys Lett 2005;22:2176–9. Pettiaux NP, Li RD, Mandel P. Instabilities of the degenerate optical parametric oscillator. Opt Commun 1989;72:256–60. Oppo GL, Bramblla, Lugiato LA. Formation and evolution of roll patterns in optical parametric oscillators. Phys Rev A 1994;49:2028–32. Sialiunas K, Sa´nchez-Morcillo VJ. Spatial-localized structures in degenerate optical parametric oscillators. Phys Rev A 1998;57:1454–7. Feng XQ, Shen K. Phase synchronization and anti-phase synchronization of chaos for degenerate optical parametric oscillator. Chinese Phys 2005;14:1526–32.