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In this paper we investigate the conditions under which periodic solutions of the nonlinear oscillator ẍ + x = 0 persist when the differential equation is perturbed so as to become ẍ + x + εx cos t+ γẋ = 0. For any frequency ω, there exists a threshold for the damping coefficient γ, above which there is no periodic orbit with period 2π/ω. We conjecture that… (More)

We consider a class of differential equations, ẍ + γẋ + g(x) = f(ωt), with ω ∈ R, describing one-dimensional dissipative systems subject to a periodic or quasi-periodic (Diophantine) forcing. We study existence and properties of trajectories with the same quasi-periodicity as the forcing. For g(x) = x, p ∈ N, we show that, when the dissipation coefficient… (More)

- Michele Bartuccelli, J. H. B. Deane, Guido Gentile, Stephen A. Gourley
- Applied Mathematics and Computation
- 2004

We consider a parametrically-driven nonlinear ODE, which encompasses a simple model of an electronic circuit known as a parametric amplifier, whose linearisation has a zero eigenvalue. By adopting two different approaches, we obtain conditions for the origin to be a global attractor which is approached (a) non-monotonically and (b) monotonically. In case… (More)

- Yuliya N. Kyrychko, Stephen A. Gourley, Michele Bartuccelli
- SIAM J. Math. Analysis
- 2006

In this paper we derive a stage-structured model for a single species on a finite onedimensional lattice. There is no migration into or from the lattice. The resulting system of equations, to be solved for the total adult population on each patch, is a system of delay equations involving the maturation delay for the species, and the delay term is nonlocal… (More)

We consider a model for the injection-locked frequency divider, and study analytically the locking onto rational multiples of the driving frequency. We provide explicit formulae for the width of the plateaux appearing in the devil’s staircase structure of the lockings, and in particular show that the largest plateaux correspond to even integer values for… (More)

We consider dissipative one-dimensional systems subject to a periodic force and study numerically how a time-varying friction affects the dynamics. As a model system, particularly suited for numerical analysis, we investigate the driven cubic oscillator in the presence of friction. We find that, if the damping coefficient increases in time up to a final… (More)

We consider a class of second order ordinary differential equations describing one-dimensional systems with a quasiperiodic analytic forcing term and in the presence of damping. As a physical application one can think of a resistor– inductor–varactor circuit with a periodic sor quasiperiodicd forcing function, even if the range of applicability of the… (More)

The differential equation ẍ+γẋ+x = f(t) with f(t) positive, periodic and continuous is studied. After describing some physical applications of this equation, we construct a variety of invariant sets for it, thereby partitioning the phase plane into regions in which solutions grow without bound and also those in which bounded periodic solutions exist.

We consider a class of ordinary differential equations describing one-dimensional analytic systems with a quasiperiodic forcing term and in the presence of damping. In the limit of large damping, under some generic nondegeneracy condition on the force, there are quasiperiodic solutions which have the same frequency vector as the forcing term. We prove that… (More)

We present some integrable time-dependent systems of classical dynamics, and we apply the results to the equation ẍ + f(t)x = 0, with f a positive non-decreasing differentiable function; some of the results are extended to the nonlinear case. Moreover we investigate the conditions for the solutions to be bounded and we study their asymptotic behaviour.