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. 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.

[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. 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.

[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. PikovskyM. 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. 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.

[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.

show all references

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. 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.

[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. 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.

[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. PikovskyM. 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. 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.

[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.

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 $(x(1),y(1))$
[1]

Jinlong Bai, Xuewei Ju, Desheng Li, Xiulian Wang. On the eventual stability of asymptotically autonomous systems with constraints. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4457-4473. doi: 10.3934/dcdsb.2019127

[2]

Sigurdur Hafstein, Skuli Gudmundsson, Peter Giesl, Enrico Scalas. Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 939-956. doi: 10.3934/dcdsb.2018049

[3]

Shubo Zhao, Ping Liu, Mingchao Jiang. Stability and bifurcation analysis in a chemotaxis bistable growth system. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1165-1174. doi: 10.3934/dcdss.2017063

[4]

Xiaoyue Li, Xuerong Mao. Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 523-545. doi: 10.3934/dcds.2009.24.523

[5]

Johan Matheus Tuwankotta, Eric Harjanto. Strange attractors in a predator–prey system with non-monotonic response function and periodic perturbation. Journal of Computational Dynamics, 2019, 6 (2) : 469-483. doi: 10.3934/jcd.2019024

[6]

Jacson Simsen, Mariza Stefanello Simsen. On asymptotically autonomous dynamics for multivalued evolution problems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3557-3567. doi: 10.3934/dcdsb.2018278

[7]

Jaume Llibre, Claudio A. Buzzi, Paulo R. da Silva. 3-dimensional Hopf bifurcation via averaging theory. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 529-540. doi: 10.3934/dcds.2007.17.529

[8]

Shanshan Liu, Maoan Han. Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3115-3124. doi: 10.3934/dcdss.2020133

[9]

Jaume Llibre, Amar Makhlouf, Sabrina Badi. $3$ - dimensional Hopf bifurcation via averaging theory of second order. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1287-1295. doi: 10.3934/dcds.2009.25.1287

[10]

Hongyong Cui. Convergences of asymptotically autonomous pullback attractors towards semigroup attractors. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3525-3535. doi: 10.3934/dcdsb.2018276

[11]

Wen-Chiao Cheng, Yun Zhao, Yongluo Cao. Pressures for asymptotically sub-additive potentials under a mistake function. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 487-497. doi: 10.3934/dcds.2012.32.487

[12]

Thorsten Hüls. A model function for non-autonomous bifurcations of maps. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 351-363. doi: 10.3934/dcdsb.2007.7.351

[13]

Florian Dorsch, Hermann Schulz-Baldes. Random Möbius dynamics on the unit disc and perturbation theory for Lyapunov exponents. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 945-976. doi: 10.3934/dcdsb.2021076

[14]

Yipeng Chen, Yicheng Liu, Xiao Wang. Exponential stability for a multi-particle system with piecewise interaction function and stochastic disturbance. Evolution Equations and Control Theory, 2022, 11 (3) : 729-748. doi: 10.3934/eect.2021023

[15]

Carlos Arnoldo Morales, M. J. Pacifico. Lyapunov stability of $\omega$-limit sets. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 671-674. doi: 10.3934/dcds.2002.8.671

[16]

Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631

[17]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084

[18]

Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979

[19]

Soliman A. A. Hamdallah, Sanyi Tang. Stability and bifurcation analysis of Filippov food chain system with food chain control strategy. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1631-1647. doi: 10.3934/dcdsb.2019244

[20]

Oleg Makarenkov, Paolo Nistri. On the rate of convergence of periodic solutions in perturbed autonomous systems as the perturbation vanishes. Communications on Pure and Applied Analysis, 2008, 7 (1) : 49-61. doi: 10.3934/cpaa.2008.7.49

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (129)
  • HTML views (222)
  • Cited by (0)

Other articles
by authors

[Back to Top]