By Thomas Erneux

This available publication deals a common creation to hold up differential equations. it's a revised model of a sequence of lectures given on the Université Libre de Bruxelles after which on the Université Joseph Fourier (Grenoble) through the spring of 2003.

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**Sample text**

The plant organ tip, as seen from a distant viewpoint, describes an ellipse and, in three dimensions, the tip traces a helix. Such “circumnutations” of sunﬂower seedlings were modelled in 1967 by Israelsson and Johnsson [113]. According to the model, the movement was completely dependent on gravity. However, a Spacelab experiment in 1983 showed that under microgravity conditions oscillations were still occurring. They are, however, less regular [37, 116]. Here, y denotes the angle with the plumb line.

10 starting from a square wave initial proﬁle. 10 shows sustained oscillations with a period close to the delay that disappear on the long time-scale. By tuning parameters, it is possible to observe the oscillations for longer periods. The phenomenon has been called metastability. 80) where ε = τ −1 , f (x) = −1 if x < 0, f (x) = 1 if x > 0, and f (0) = 0 [3]. 11. 10: Top: slowly varying oscillations followed by a sudden jump to the steady-state x = −2. 45 (−τ < t < −2τ /3 and -τ /3 < t < 0) and x = 3 (−2τ /3 < t < −τ /3).

41) The linear stability boundaries are found by introducing θ = exp(iσs) into Eq. 41). 42) 2μσ − k sin(στ ) = 0. 43) The solution for k = k(τ ) can be determined analytically. If we wish to avoid the inverse trigonometric functions, we may obtain the solution in parametric form using x ≡ στ /2 ≥ 0 as parameter. Eliminating k in Eqs. 43), and inserting σ = 2x/τ , we obtain a quadratic equation for τ . It always admits a positive real root given by τ = 2 μx tan(x) + |x| μ2 tan2 (x) + 1 . 43) with σ = 2xτ −1 : k= 4μx .