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 & Continuous Dynamical Systems, 2021, 41 (12) : 5943-5978. doi: 10.3934/dcds.2021102
References:
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V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer, Berlin, 2006.  Google Scholar

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D. G. AronsonD. G. Ermentrout and N. Kopell, Amplitude response of coupled oscillators, Physica D, 41 (1990), 403-449.  doi: 10.1016/0167-2789(90)90007-C.  Google Scholar

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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.  Google Scholar

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[37]

O. A. Sultanov, Stability and bifurcation phenomena in asymptotically Hamiltonian systems, arXiv: 2006.12957. Google Scholar

[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.  Google Scholar

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H. Thieme, Asymptotically autonomous differential equations in the plane, Rocky Mountain J. Math., 24 (1994), 351-380.  doi: 10.1216/rmjm/1181072470.  Google Scholar

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C. I. UmK. 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.  Google Scholar

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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.  Google Scholar

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A. Wintner, The adiabatic linear oscillator, Amer. J. Math., 68 (1946), 385-397.  doi: 10.2307/2371822.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. Adler, A study of locking phenomena in oscillators, Proc. I.R.E., 34 (1946), 351-357.   Google Scholar

[2]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer, Berlin, 2006.  Google Scholar

[3]

D. G. AronsonD. G. Ermentrout and N. Kopell, Amplitude response of coupled oscillators, Physica D, 41 (1990), 403-449.  doi: 10.1016/0167-2789(90)90007-C.  Google Scholar

[4]

F. V. Atkinson, The asymptotic solution of second-order differential equations, Ann. Mat. Pura Appl., 37 (1954), 347-378.  doi: 10.1007/BF02415105.  Google Scholar

[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.  Google Scholar

[6]

N. N. Bogolubov and Yu. A. Mitropolsky, Asymptotic Methods in Theory of Nonlinear Oscillations, Gordon and Breach, New York, 1961.  Google Scholar

[7]

J. BrüningS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

V. Burd and P. Nesterov, Parametric resonance in adiabatic oscillators, Results. Math., 58 (2010), 1-15.  doi: 10.1007/s00025-010-0043-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

L. Friedland, Autoresonance in nonlinear systems, Scholarpedia, 4 (2009), 5473. Google Scholar

[17]

S. G. Glebov, O. M. Kiselev and N. Tarkhanov, Nonlinear Equations with Small Parameter, v. 1. Oscillations and resonances, De Gruyter, Berlin, 2017.  Google Scholar

[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.  Google Scholar
[19]

R. C. Grimmer, Asymptotically almost periodic solutions of differential equations, SIAM J. Appl. Math., 17 (1968), 109-115.  doi: 10.1137/0117012.  Google Scholar

[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.  Google Scholar

[21]

H. Hanßmann, Local and Semi-local Bifurcations in Hamiltonian Systems - Results and Examples, Springer, Berlin, 2007. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

L. A. Kalyakin, Asymptotic analysis of autoresonance models, Russian Math. Surveys., 63 (2008), 791-857.  doi: 10.1070/RM2008v063n05ABEH004560.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.  Google Scholar
[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.  Google Scholar

[35]

M. Rasmussen, Bifurcations of asymptotically autonomous differential equations, Set-Valued Anal., 16 (2008), 821-849.  doi: 10.1007/s11228-008-0089-5.  Google Scholar

[36]

B. Simon, On positive eigenvalues of one-body Schrödinger operators, Commun. Pure Appl. Math., 22 (1969), 531-538.  doi: 10.1002/cpa.3160220405.  Google Scholar

[37]

O. A. Sultanov, Stability and bifurcation phenomena in asymptotically Hamiltonian systems, arXiv: 2006.12957. Google Scholar

[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.  Google Scholar

[39]

H. Thieme, Asymptotically autonomous differential equations in the plane, Rocky Mountain J. Math., 24 (1994), 351-380.  doi: 10.1216/rmjm/1181072470.  Google Scholar

[40]

C. I. UmK. 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.  Google Scholar

[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.  Google Scholar

[42]

A. Wintner, The adiabatic linear oscillator, Amer. J. Math., 68 (1946), 385-397.  doi: 10.2307/2371822.  Google Scholar

[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.  Google Scholar

Figure 1.  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 $s_1 = 1$
Figure 2.  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 $H_0(x,y)$
Figure 3.  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 $\psi_\ast\approx -1.322$
Figure 4.  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 $\psi_\ast\approx -1.322$
Figure 5.  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 $H_0(x,y)$
Figure 6.  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 $\psi_\ast\approx -1.4289$
Figure 7.  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 $H_0(x,y)$
Figure 8.  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 $H_0(x,y)$
Figure 9.  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 $r = 2 R_\ast t^{-\nu-l/q}$
Figure 10.  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 $\psi_\ast\approx -0.615$
Figure 11.  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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