Abstract
A new perturbation method for a weakly nonlinear two-dimensional discrete-time dynamical system is presented. The proposed technique generalizes the asymptotic perturbation method that is valid for continuous-time systems and detects periodic or almost-periodic orbits and their stability. Two equations for the amplitude and the phase of solutions are derived and their fixed points correspond to limit cycles for the starting nonlinear map. The method is applied to various nonlinear (autonomous or not) two-dimensional maps. For the autonomous maps we derive the conditions for the appearance of a supercritical Hopf bifurcation and predict the characteristics of the corresponding limit cycle. For the nonautonomous maps we show amplitude-response and frequency-response curves. Under appropriate conditions, we demonstrate the occurrence of saddle-node bifurcations of cycles and of jumps and hysteresis effects in the system response (cusp catastrophe). Modulated motion can be observed for very low values of the external excitation and an infinite-period bifurcation occurs if the external excitation increases. Analytic approximate solutions are in good agreement with numerically obtained solutions.
Similar content being viewed by others
References
Rasband, S. N., Chaotic Dynamics of Nonlinear Systems, Wiley, New York, 1990.
Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, (corrected third printing), Springer-Verlag, New York, 1990.
Subramanian, R. and Krishnan, A., 'Non-linear discrete time systems analysis by multiple time perturbation techniques', Journal of Sound and Vibration 63, 1979, 325–335.
Atadan, S., 'Periodic behavior in nonlinear, discrete-time systems and its application to a digital control system', Dynamics and Stability of Systems 7, 1992, 151–161.
Luongo, A., 'Perturbation methods for nonlinear autonomous discrete-time dynamical systems', Nonlinear Dynamics 10, 1996, 317–331.
Maccari, A., 'Dissipative bidimensional systems and resonant excitations', International Journal of Non-Linear Mechanics 33, 1998, 713–726.
Maccari, A., 'Approximate solution of a class of nonlinear oscillators in resonance with a periodic excitation', Nonlinear Dynamics 15, 1998, 329–343.
Maccari, A., 'The dissipative nonlocal oscillator', Nonlinear Dynamics 16, 1998, 307–320.
Nayfeh, A. H., Perturbation Methods, Wiley, New York, 1973.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Maccari, A. A Perturbation Method for Nonlinear Two-Dimensional Maps. Nonlinear Dynamics 19, 295–312 (1999). https://doi.org/10.1023/A:1008354207308
Issue Date:
DOI: https://doi.org/10.1023/A:1008354207308