A peculiar periodic solution of a modified Duffing’s equation

1957 ◽  
Vol 44 (1) ◽  
pp. 23-33 ◽  
Author(s):  
H. L. Turrittin ◽  
W. J. A. Culmer
Keyword(s):  
Periodic Solution ◽  
Duffing's Equation ◽  
Duffing’S Equation

scholarly journals Boundedness and convergence of solutions of Duffing’s equation

1977 ◽  
Vol 66 ◽  
pp. 151-166 ◽  
Author(s):  
Kenichi Shiraiwa
Keyword(s):  
Periodic Solution ◽  
Asymptotic Stability ◽  
Periodic Function ◽  
Positive Constant ◽  
Boundedness Of Solutions ◽  
Duffing's Equation ◽  
Convergence Of Solutions ◽  
Duffing’S Equation

In this paper, we shall discuss boundedness of solutions of the equationunder suitable conditions. And we shall discuss asymptotic stability of a periodic solution and convergence of solutions for the equationfor a positive constant cand a periodic function e(t)under some restricted conditions.

Properties of cross-well chaos in an impacting system

1994 ◽  
Vol 347 (1683) ◽  
pp. 391-410 ◽  
Keyword(s):  
Phase Space ◽  
Poincaré Map ◽  
Poincare Map ◽  
Chaotic Motions ◽  
Higher Harmonics ◽  
Potential Wells ◽  
Duffing's Equation ◽  
Duffing’S Equation ◽  
Phase Space Transport ◽  
Basin Boundaries

In this paper we present results on chaotic motions in a periodically forced impacting system which is analogous to the version of Duffing’s equation with negative linear stiffness. Our focus is on the prediction and manipulation of the cross-well chaos in this system. First, we develop a general method for determining parameter conditions under which homoclinic tangles exist, which is a necessary condition for cross-well chaos to occur. We then show how one may manipulate higher harmonics of the excitation in order to affect the range of excitation amplitudes over which fractal basin boundaries between the two potential wells exist. We also experimentally investigate the threshold for cross-well chaos and compare the results with the theoretical results. Second, we consider the rate at which the system crosses from one potential well to the other during a chaotic motion and relate this to the rate of phase space flux in a Poincare map defined in terms of impact parameters. Results from simulations and experiments are compared with a simple theory based on phase space transport ideas, and a predictive scheme for estimating the rate of crossings under different parameter conditions is presented. The main conclusions of the paper are the following: (1) higher harmonics can be used with some effectiveness to extend the region of deterministic basin boundaries (in terms of the amplitude of excitation) but their effect on steady-state chaos is unreliable; (2) the rate at which the system executes cross-well excursions is related in a direct manner to the rate of phase space flux of the system as measured by the area of a turnstile lobe in the Poincare map. These results indicate some of the ways in which the chaotic properties of this system, and possibly others such as Duffing’s equation, are influenced by various system and input parameters. The main tools of analysis are a special version of Melnikov’s method, adapted for this piecewise-linear system, and ideas of phase space transport.

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