On Poincaré's perturbation theorem and a theorem of W. S. Loud

Leach, Douglas Edward

doi:10.1016/0022-0396(70)90122-1

Cited by 62 publications

(32 citation statements)

References 4 publications

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“…Our result, in a sense, completes a series of investigations originated by W. S. Loud [4]. Also see [1], [2], [3], and [5]. Our techniques are different from those of the authors cited above.…”

supporting

confidence: 71%

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Periodically perturbed conservative systems

Ahmad¹

1974

Bull. Amer. Math. Soc.

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supporting

confidence: 71%

“…Using this lemma we prove that our theorem follows from a generalization of Poincaré's perturbation theorem (see [3]). The proof of Lemma 1 is too long to give here.…”

Section: )mentioning

confidence: 94%

Periodically perturbed conservative systems

Ahmad¹

1974

Bull. Amer. Math. Soc.

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“…To prove the lemma we use a continuation argument similar to one used in [12]. Let u(t,£,n,s) denote the solution of the initial value problem For each integer p > 0, let I denote the set of numbers 5 € [0,1 ] such that there are exactly p points satisfying (2.21).…”

Section: Jo Jo Jomentioning

confidence: 99%

Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities

Lazer¹,

McKenna²

1989

Trans. Amer. Math. Soc.

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Abstract.We give conditions for the existence, uniqueness, and asymptotic stability of periodic solutions of a second-order differential equation with piecewise linear restoring and 2fl-periodic forcing where the range of the derivative of the restoring term possibly contains the square of an integer. With suitable restrictions on the restoring and forcing in the undamped case, we give a necessary and sufficient condition.

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“…In this case, if there exists an integer TV > 0 such that 4n2N2/T2 < a < b < 4n2(N + \)2/T2 and (2) holds, then there exists a unique F-periodic solution of ( 1 ). This follows from work of Loud [6], under the additional assumption that a certain symmetry condition holds, and from work of Leach [5] in the general case. The Loud-Leach result can also be obtained from a theorem concerning Hammerstein integral equations due to Dolph in [4].…”

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confidence: 95%

On the Existence of Stable Periodic Solutions of Differential Equations of Duffing Type

Lazer¹,

McKenna²

1990

Proceedings of the American Mathematical Society

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Abstract.We consider a second-order differential equation periodic in t with period 7" > 0 and with linear damping. Bounds are given for the derivative of the restoring force wnich will guarantee the existence and uniqueness of a /"-periodic solution such that the unique 7"-periodic solution is asymptotically stable. These conditions also rule out the existence of additional periodic solutions which are subharmonics of order 2 .In this note we consider periodic solutions of the differential equation (1) u +ku + g(t, u) = 0 where zc > 0 is a constant, g and its partial derivative with respect to the second variable, denoted by D2g, are continuous, and g is F-periodic in t for some T > 0 . Our goal is to find conditions of the form (2) a < D2g(t, Í) < b for (t, ¿;) e R which will guarantee the existence and uniqueness of a Tperiodic solution u0 which is locally, exponentially, asymptotically stable, i.e. such that there exist constants C > 0 and a > 0 such that if u is another solution with |m(0) -m0(0)| and \u (0) -u'0(0)\ sufficiently small, then \u(t) -u0(t)\ < Cde~at, \u'(t) -u'0(t)\ < Cde~nt for all t > 0, where d = |«(0) -The existence and uniqueness part of this problem has been considered in several papers when zc = 0. In this case, if there exists an integer TV > 0 such that 4n2N2/T2 < a < b < 4n2(N + \)2/T2 and (2) holds, then there exists a unique F-periodic solution of ( 1 ). This follows from work of Loud [6], under the additional assumption that a certain symmetry condition holds, and

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On Poincaré's perturbation theorem and a theorem of W. S. Loud

Cited by 62 publications

References 4 publications

Periodically perturbed conservative systems

Periodically perturbed conservative systems

Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities

On the Existence of Stable Periodic Solutions of Differential Equations of Duffing Type

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