# American Institute of Mathematical Sciences

December  2021, 41(12): 5943-5978. doi: 10.3934/dcds.2021102

## Bifurcations in asymptotically autonomous Hamiltonian systems under oscillatory perturbations

 Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O., 29, Saint Petersburg 199178, Russia

Received  October 2020 Revised  January 2021 Published  December 2021 Early access  June 2021

Fund Project: Research is supported by the Russian Science Foundation grant 19-71-30002

The effect of decaying oscillatory perturbations on autonomous Hamiltonian systems in the plane with a stable equilibrium is investigated. It is assumed that perturbations preserve the equilibrium and satisfy a resonance condition. The behaviour of the perturbed trajectories in the vicinity of the equilibrium is investigated. Depending on the structure of the perturbations, various asymptotic regimes at infinity in time are possible. In particular, a phase locking and a phase drifting can occur in the systems. The paper investigates the bifurcations associated with a change of Lyapunov stability of the equilibrium in both regimes. The proposed stability analysis is based on a combination of the averaging method and the construction of Lyapunov functions.

Citation: Oskar A. Sultanov. Bifurcations in asymptotically autonomous Hamiltonian systems under oscillatory perturbations. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5943-5978. doi: 10.3934/dcds.2021102
##### References:
 [1] R. Adler, A study of locking phenomena in oscillators, Proc. I.R.E., 34 (1946), 351-357. [2] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer, Berlin, 2006. [3] D. G. Aronson, D. G. Ermentrout and N. Kopell, Amplitude response of coupled oscillators, Physica D, 41 (1990), 403-449.  doi: 10.1016/0167-2789(90)90007-C. [4] F. V. Atkinson, The asymptotic solution of second-order differential equations, Ann. Mat. Pura Appl., 37 (1954), 347-378.  doi: 10.1007/BF02415105. [5] M. Ben-Artzi and A. Devinatz, Spectral and scattering theory for the adiabatic oscillator and related potentials, J. Math. Phys., 20 (1979), 594-607.  doi: 10.1063/1.524128. [6] N. N. Bogolubov and Yu. A. Mitropolsky, Asymptotic Methods in Theory of Nonlinear Oscillations, Gordon and Breach, New York, 1961. [7] J. Brüning, S. Yu. Dobrokhotov and M. A. Poteryakhin, Averaging for Hamiltonian systems with one fast phase and small amplitudes, Math. Notes., 70 (2001), 599-607.  doi: 10.1023/A:1012918708490. [8] A. D. Bruno, Asymptotic behaviour and expansions of solutions of an ordinary differential equation, Russian Math. Surveys, 59 (2004), 429-480.  doi: 10.1070/RM2004v059n03ABEH000736. [9] A. D. Bruno and I. V. Goryuchkina, Boutroux asymptotic forms of solutions to Painlevé equations and power geometry, Doklady Mathematics, 78 (2008), 681-685.  doi: 10.1134/S1064562408050104. [10] V. Burd, Method of Averaging for Differential Equations on an Infinite Interval: Theory and Applications, Chapman & Hall/CRC, Boca Raton, 2007. doi: 10.1201/9781584888758. [11] V. Burd and P. Nesterov, Parametric resonance in adiabatic oscillators, Results. Math., 58 (2010), 1-15.  doi: 10.1007/s00025-010-0043-3. [12] T. Chakraborty and R. Rand, The transition from phase locking to drift in a system of two weakly coupled van der Pol oscillators, Int. J. Nonlin. Mech., 23 (1988), 369-376.  doi: 10.1016/0020-7462(88)90034-0. [13] S. Yu. Dobrokhotov and D. S. Minenkov, On various averaging methods for a nonlinear oscillator with slow time-dependent potential and a nonconservative perturbation, Regul. Chaot. Dyn., 15 (2010), 285-299.  doi: 10.1134/S1560354710020152. [14] J. D. Dollard and C. N. Friedman, Existence of the Møller wave operators for $V(r) = \gamma \sin(\mu r^\alpha)r^\beta$, Annals of Physics, 111 (1978), 251-266.  doi: 10.1016/0003-4916(78)90230-0. [15] A. S. Fokas, A. R. Its, A. A. Kapaev and V. Yu. Novokshenov, Painlevé Transcendents. The Riemann-Hilbert Approach, Amer. Math. Soc., Providence, 2006. doi: 10.1090/surv/128. [16] L. Friedland, Autoresonance in nonlinear systems, Scholarpedia, 4 (2009), 5473. [17] S. G. Glebov, O. M. Kiselev and N. Tarkhanov, Nonlinear Equations with Small Parameter, v. 1. Oscillations and resonances, De Gruyter, Berlin, 2017. [18] P. A. Glendinning, Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9780511626296. [19] R. C. Grimmer, Asymptotically almost periodic solutions of differential equations, SIAM J. Appl. Math., 17 (1968), 109-115.  doi: 10.1137/0117012. [20] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1983. doi: 10.1007/978-1-4612-1140-2. [21] H. Hanßmann, Local and Semi-local Bifurcations in Hamiltonian Systems - Results and Examples, Springer, Berlin, 2007. [22] M. M. Hapaev, Averaging in Stability Theory: A Study of Resonance Multi-frequency Systems, Kluwer Academic Publishers, Dordrecht, Boston, 1993. doi: 10.1007/978-94-011-2644-1. [23] W. A. Harris and D. A. Lutz, Asymptotic integration of adiabatic oscillators, J. Math. Anal. Appl., 51 (1975), 76-93.  doi: 10.1016/0022-247X(75)90142-0. [24] L. A. Kalyakin, Synchronization in a nonisochronous nonautonomous system, Theoret. and Math. Phys., 181 (2014), 1339-1348.  doi: 10.1007/s11232-014-0216-4. [25] L. A. Kalyakin, Asymptotic analysis of autoresonance models, Russian Math. Surveys., 63 (2008), 791-857.  doi: 10.1070/RM2008v063n05ABEH004560. [26] A. Kiselev, Absolutely continuous spectrum of one-dimensional Schrödinger operators and Jacobi matrices with slowly decreasing potentials, Commun. Math. Phys., 179 (1996), 377-400.  doi: 10.1007/BF02102594. [27] V. V. Kozlov and S. D. Furta, Asymptotic Solutions of Strongly Nonlinear Systems of Differential Equations, Springer, New York, 2013. doi: 10.1007/978-3-642-33817-5. [28] L. K. B. Li and M. P. Juniper, Phase trapping and slipping in a forced hydrodynamically self-excited jet, J. Fluid Mech., 735 (2013), R5. doi: 10.1017/jfm.2013.533. [29] M. Lukic, A class of Schrödinger operators with decaying oscillatory potentials, Commun. Math. Phys., 326 (2014), 441-458.  doi: 10.1007/s00220-013-1851-6. [30] L. Markus, Aymptotically autonomous differential systems, in Contributions to the Theory of Nonlinear Oscillations III, Ann. Math. Stud., vol. 36 (ed. S. Lefschetz), Princeton University Press, (1956), 17–29. [31] A. I. Neishtadt, The separation of motions in systems with rapidly rotating phase, J. Appl. Math. Mech., 48 (1984), 133-139.  doi: 10.1016/0021-8928(84)90078-9. [32] P. N. Nesterov, Construction of the asymptotics of the solutions of the one-dimensional Schrödinger equation with rapidly oscillating potential, Math. Notes, 80 (2006), 233-243.  doi: 10.1007/s11006-006-0132-5. [33] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743. [34] C. Pötzsche, Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach, Discrete Contin. Dynam. Systems - B, 14 (2010), 739-776.  doi: 10.3934/dcdsb.2010.14.739. [35] M. Rasmussen, Bifurcations of asymptotically autonomous differential equations, Set-Valued Anal., 16 (2008), 821-849.  doi: 10.1007/s11228-008-0089-5. [36] B. Simon, On positive eigenvalues of one-body Schrödinger operators, Commun. Pure Appl. Math., 22 (1969), 531-538.  doi: 10.1002/cpa.3160220405. [37] O. A. Sultanov, Stability and bifurcation phenomena in asymptotically Hamiltonian systems, arXiv: 2006.12957. [38] H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267. [39] H. Thieme, Asymptotically autonomous differential equations in the plane, Rocky Mountain J. Math., 24 (1994), 351-380.  doi: 10.1216/rmjm/1181072470. [40] C. I. Um, K. H. Yeon and T. F. George, The quantum damped harmonic oscillator, Phys. Rep., 362 (2002), 63-192.  doi: 10.1016/S0370-1573(01)00077-1. [41] F. Verhulst, Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics, Springer, New York, 2005. doi: 10.1007/0-387-28313-7. [42] A. Wintner, The adiabatic linear oscillator, Amer. J. Math., 68 (1946), 385-397.  doi: 10.2307/2371822. [43] J. S. W. Wong and T. A. Burton, Some properties of solutions of $u''(t)+a(t)f(u)g(u')=0$. II, Monatsh. Math., 69 (1965), 368-374.  doi: 10.1007/BF01297623.

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##### References:
 [1] R. Adler, A study of locking phenomena in oscillators, Proc. I.R.E., 34 (1946), 351-357. [2] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer, Berlin, 2006. [3] D. G. Aronson, D. G. Ermentrout and N. Kopell, Amplitude response of coupled oscillators, Physica D, 41 (1990), 403-449.  doi: 10.1016/0167-2789(90)90007-C. [4] F. V. Atkinson, The asymptotic solution of second-order differential equations, Ann. Mat. Pura Appl., 37 (1954), 347-378.  doi: 10.1007/BF02415105. [5] M. Ben-Artzi and A. Devinatz, Spectral and scattering theory for the adiabatic oscillator and related potentials, J. Math. Phys., 20 (1979), 594-607.  doi: 10.1063/1.524128. [6] N. N. Bogolubov and Yu. A. Mitropolsky, Asymptotic Methods in Theory of Nonlinear Oscillations, Gordon and Breach, New York, 1961. [7] J. Brüning, S. Yu. Dobrokhotov and M. A. Poteryakhin, Averaging for Hamiltonian systems with one fast phase and small amplitudes, Math. Notes., 70 (2001), 599-607.  doi: 10.1023/A:1012918708490. [8] A. D. Bruno, Asymptotic behaviour and expansions of solutions of an ordinary differential equation, Russian Math. Surveys, 59 (2004), 429-480.  doi: 10.1070/RM2004v059n03ABEH000736. [9] A. D. Bruno and I. V. Goryuchkina, Boutroux asymptotic forms of solutions to Painlevé equations and power geometry, Doklady Mathematics, 78 (2008), 681-685.  doi: 10.1134/S1064562408050104. [10] V. Burd, Method of Averaging for Differential Equations on an Infinite Interval: Theory and Applications, Chapman & Hall/CRC, Boca Raton, 2007. doi: 10.1201/9781584888758. [11] V. Burd and P. Nesterov, Parametric resonance in adiabatic oscillators, Results. Math., 58 (2010), 1-15.  doi: 10.1007/s00025-010-0043-3. [12] T. Chakraborty and R. Rand, The transition from phase locking to drift in a system of two weakly coupled van der Pol oscillators, Int. J. Nonlin. Mech., 23 (1988), 369-376.  doi: 10.1016/0020-7462(88)90034-0. [13] S. Yu. Dobrokhotov and D. S. Minenkov, On various averaging methods for a nonlinear oscillator with slow time-dependent potential and a nonconservative perturbation, Regul. Chaot. Dyn., 15 (2010), 285-299.  doi: 10.1134/S1560354710020152. [14] J. D. Dollard and C. N. Friedman, Existence of the Møller wave operators for $V(r) = \gamma \sin(\mu r^\alpha)r^\beta$, Annals of Physics, 111 (1978), 251-266.  doi: 10.1016/0003-4916(78)90230-0. [15] A. S. Fokas, A. R. Its, A. A. Kapaev and V. Yu. Novokshenov, Painlevé Transcendents. The Riemann-Hilbert Approach, Amer. Math. Soc., Providence, 2006. doi: 10.1090/surv/128. [16] L. Friedland, Autoresonance in nonlinear systems, Scholarpedia, 4 (2009), 5473. [17] S. G. Glebov, O. M. Kiselev and N. Tarkhanov, Nonlinear Equations with Small Parameter, v. 1. Oscillations and resonances, De Gruyter, Berlin, 2017. [18] P. A. Glendinning, Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9780511626296. [19] R. C. Grimmer, Asymptotically almost periodic solutions of differential equations, SIAM J. Appl. Math., 17 (1968), 109-115.  doi: 10.1137/0117012. [20] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1983. doi: 10.1007/978-1-4612-1140-2. [21] H. Hanßmann, Local and Semi-local Bifurcations in Hamiltonian Systems - Results and Examples, Springer, Berlin, 2007. [22] M. M. Hapaev, Averaging in Stability Theory: A Study of Resonance Multi-frequency Systems, Kluwer Academic Publishers, Dordrecht, Boston, 1993. doi: 10.1007/978-94-011-2644-1. [23] W. A. Harris and D. A. Lutz, Asymptotic integration of adiabatic oscillators, J. Math. Anal. Appl., 51 (1975), 76-93.  doi: 10.1016/0022-247X(75)90142-0. [24] L. A. Kalyakin, Synchronization in a nonisochronous nonautonomous system, Theoret. and Math. Phys., 181 (2014), 1339-1348.  doi: 10.1007/s11232-014-0216-4. [25] L. A. Kalyakin, Asymptotic analysis of autoresonance models, Russian Math. Surveys., 63 (2008), 791-857.  doi: 10.1070/RM2008v063n05ABEH004560. [26] A. Kiselev, Absolutely continuous spectrum of one-dimensional Schrödinger operators and Jacobi matrices with slowly decreasing potentials, Commun. Math. Phys., 179 (1996), 377-400.  doi: 10.1007/BF02102594. [27] V. V. Kozlov and S. D. Furta, Asymptotic Solutions of Strongly Nonlinear Systems of Differential Equations, Springer, New York, 2013. doi: 10.1007/978-3-642-33817-5. [28] L. K. B. Li and M. P. Juniper, Phase trapping and slipping in a forced hydrodynamically self-excited jet, J. Fluid Mech., 735 (2013), R5. doi: 10.1017/jfm.2013.533. [29] M. Lukic, A class of Schrödinger operators with decaying oscillatory potentials, Commun. Math. Phys., 326 (2014), 441-458.  doi: 10.1007/s00220-013-1851-6. [30] L. Markus, Aymptotically autonomous differential systems, in Contributions to the Theory of Nonlinear Oscillations III, Ann. Math. Stud., vol. 36 (ed. S. Lefschetz), Princeton University Press, (1956), 17–29. [31] A. I. Neishtadt, The separation of motions in systems with rapidly rotating phase, J. Appl. Math. Mech., 48 (1984), 133-139.  doi: 10.1016/0021-8928(84)90078-9. [32] P. N. Nesterov, Construction of the asymptotics of the solutions of the one-dimensional Schrödinger equation with rapidly oscillating potential, Math. Notes, 80 (2006), 233-243.  doi: 10.1007/s11006-006-0132-5. [33] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743. [34] C. Pötzsche, Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach, Discrete Contin. Dynam. Systems - B, 14 (2010), 739-776.  doi: 10.3934/dcdsb.2010.14.739. [35] M. Rasmussen, Bifurcations of asymptotically autonomous differential equations, Set-Valued Anal., 16 (2008), 821-849.  doi: 10.1007/s11228-008-0089-5. [36] B. Simon, On positive eigenvalues of one-body Schrödinger operators, Commun. Pure Appl. Math., 22 (1969), 531-538.  doi: 10.1002/cpa.3160220405. [37] O. A. Sultanov, Stability and bifurcation phenomena in asymptotically Hamiltonian systems, arXiv: 2006.12957. [38] H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267. [39] H. Thieme, Asymptotically autonomous differential equations in the plane, Rocky Mountain J. Math., 24 (1994), 351-380.  doi: 10.1216/rmjm/1181072470. [40] C. I. Um, K. H. Yeon and T. F. George, The quantum damped harmonic oscillator, Phys. Rep., 362 (2002), 63-192.  doi: 10.1016/S0370-1573(01)00077-1. [41] F. Verhulst, Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics, Springer, New York, 2005. doi: 10.1007/0-387-28313-7. [42] A. Wintner, The adiabatic linear oscillator, Amer. J. Math., 68 (1946), 385-397.  doi: 10.2307/2371822. [43] J. S. W. Wong and T. A. Burton, Some properties of solutions of $u''(t)+a(t)f(u)g(u')=0$. II, Monatsh. Math., 69 (1965), 368-374.  doi: 10.1007/BF01297623.
The evolution of $r(t) = \sqrt{x^2(t)+y^2(t)}$ for solutions of (7) with $x(1) = 0.4$, $y(1) = 0$, $a = 4$, and $s_1 = 1$
The evolution of $(x(t),y(t))$, $r(t)$, $|\theta(t)|$ for solutions of (47) with $h = 1/6$, $a_0 = a_1 = b_1 = s_2 = 0$, $s_1 = 1$, where $x(t) = r(t)\cos(\theta(t)+S(t))$, $y(t) = -r(t)\sin(\theta(t)+S(t))$. The blue points correspond to initial data $(x(1),y(1))$. Gray solid curves correspond to level lines of $H_0(x,y)$
The evolution of $(x(t),y(t))$, $r(t)$, $\theta(t)$ for solutions of (47) with $h = 1/6$, $a_0 = a_1 = 0.8$, $b_1 = 0.6$, $s_1 = -0.8$, $s_2 = -1/6$, where $x(t) = r(t)\cos(\theta(t)+S(t))$, $y(t) = -r(t)\sin(\theta(t)+S(t))$. The blue points correspond to initial data $(x(1),y(1))$. The gray solid curves correspond to level lines of $H_0(x,y)$. The gray dashed curve corresponds to $\theta = \psi_\ast$, where $\psi_\ast\approx -1.322$
The evolution of $(x(t),y(t))$, $r(t)$, $\theta(t)$ for solutions of (47) with $h = 0$, $a_0 = 1$, $a_1 = 0.8$, $b_0 = 0$, $b_1 = 0.6$, $s_1 = -1$, $s_2 = -1/6$, where $x(t) = r(t)\cos(\theta(t)+S(t))$, $y(t) = -r(t)\sin(\theta(t)+S(t))$. The blue point corresponds to initial data $(x(1),y(1))$. The gray solid curve corresponds to $r = r(1) t^{{|\vartheta_4|}/{2}}$. The gray dashed curve corresponds to $\theta = \psi_\ast$, where $\psi_\ast\approx -1.322$
The evolution of $(x(t),y(t))$, $r(t)$, $|\theta(t)|$ for solutions of (47) with $h = 1/6$, $a_0 = a_1 = 0.8$, $b_1 = 0.6$, $s_1 = -0.8$, $s_2 = 1$, where $x(t) = r(t)\cos(\theta(t)+S(t))$, $y(t) = -r(t)\sin(\theta(t)+S(t))$. The blue points correspond to initial data $(x(1),y(1))$. The gray solid curves correspond to level lines of $H_0(x,y)$
The evolution of $(x(t),y(t))$, $R(t)$, $\theta(t)$ for solutions of (50) with $h = 1/6$, $a_0 = a_1 = 0.8$, $b_1 = 0.6$, $s_1 = -1$, $s_2 = 0$, where $x(t) = r(t)\cos(\theta(t)+S(t)/2)$, $y(t) = -r(t)\sin(\theta(t)+S(t)/2)$. The blue points correspond to initial data $(x(1),y(1))$. The gray solid curves correspond to level lines of $H_0(x,y)$. The gray dashed curves correspond to $\theta = \psi_\ast$, where $\psi_\ast\approx -1.4289$
The evolution of $(x(t),y(t))$, $R(t)$, $|\theta(t)|$ for solutions of (50) with $h = 1/6$, $a_0 = a_1 = 0.8$, $b_1 = 0.6$, $s_1 = s_2 = 0$, where $x(t) = r(t)\cos(\theta(t)+S(t)/2)$, $y(t) = -r(t)\sin(\theta(t)+S(t)/2)$. The blue points correspond to initial data $(x(1),y(1))$. The gray solid curves correspond to level lines of $H_0(x,y)$
The evolution of $(x(t),y(t))$, $r(t)$, $|\theta(t)|$ for solutions of (51) with $h = 1/6$, $b_0 = -1/4$, $z_0 = 0.6$, $s_2 = 1$, $a_0 = a_1 = b_1 = z_1 = s_4 = 0$, where $x(t) = r(t)\cos(\theta(t)+S(t))$, $y(t) = -r(t)\sin(\theta(t)+S(t))$. The blue point corresponds to initial data $(x(1),y(1))$. The gray solid curves correspond to level lines of $H_0(x,y)$
The evolution of $(x(t),y(t))$, $r(t)$, $|\theta(t)|$ for solutions of (51) with $h = 1/6$, $s_2 = 1$, $a_0 = a_1 = b_1 = z_1 = s_4 = 0$, where $x(t) = r(t)\cos(\theta(t)+S(t))$, $y(t) = -r(t)\sin(\theta(t)+S(t))$. The blue points correspond to initial data $(x(1),y(1))$. The gray solid curves correspond to level lines of $H_0(x,y)$. The gray dashed curve corresponds to $r = 2 R_\ast t^{-\nu-l/q}$
The evolution of $(x(t),y(t))$, $r(t)$, $\theta(t)$ for solutions of (51) with $h = 0$, $a_0 = -1$, $a_1 = 1$, $s_2 = 1$, $s_4 = -1/4$, $b_1 = 0$, $\vartheta_8\approx -0.235$, where $x(t) = r(t)\cos(\theta(t)+S(t))$, $y(t) = -r(t)\sin(\theta(t)+S(t))$. The blue points correspond to initial data $(x(1),y(1))$. The gray dashed curves correspond to $\theta = \psi_\ast$, where $\psi_\ast\approx -0.615$
The evolution of $(x(t),y(t))$, $r(t)$, $|\theta(t)|$ for solutions of (51) with $h = 0$, $a_0 = -1$, $a_1 = 1$, $s_2 = 1$, $s_4 = 0$, where $x(t) = r(t)\cos(\theta(t)+S(t))$, $y(t) = -r(t)\sin(\theta(t)+S(t))$. The blue points correspond to initial data $(x(1),y(1))$
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