RENDICONTI DEL SEMINARIO MATEMATICO

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116 D. Papini - F. Zanolincase in which Ŵ 0 is the image of a continuous curve θ 0 : [0, 1[ → R 2 , the set Ŵ ′ 0 cansometimes be described as the image of a continuous curve too, by means of the mapθ 1 : [0, 1[ ∋ s ↦→ ζ(t 1 ; t 0 ,θ 0 (s)) (this, of course, is not always guaranteed, in factwe are not assuming in this example that all the solutions of the Cauchy problems canbe defined on [t 0 , t 1 ] and therefore, without further assumptions on f, it could happenthat θ 1 may be not defined on the whole interval [0, 1[ ). See also [9, 10, 13, 23] fordifferent topological approaches where the uniqueness of the solutions for the Cauchyproblem is not assumed.In [55], dealing with some boundary value problems associated to the nonlinearscalar ODE(2) u ′′ + q(t)g(u) = 0and assuming that q : [t 0 , t 1 ] → R is a continuous and piece-wise monotone functionand g : R → R, withg(s)s > 0, for s ̸= 0,a locally Lipschitz continuous mapping satisfying a condition of superlinear growth atinfinity, we considered the case in which there is τ ∈]t 0 , t 1 [ such thatq(t) > 0 for t ∈]t 0 ,τ[ and q(t) < 0 for t ∈]τ, t 1 [and then we found in the phase-plane two conical shells W(+) := W(r, R) and W(−):= −W(r, R), with(3) W(r, R) := {(x, y) ∈ R 2 : x ≥ 0, y ≥ 0, r 2 ≤ x 2 + y 2 ≤ R 2 },such that the following path-stretching property holds:(H ± ) for every path σ(±) contained in W(±) and meeting the inner and the outercircumferences at the boundary of W(±) there are sub-paths γ i (±) (i = 1, 2)contained in the domain D ϕ of the map ϕ : R 2 ∋ z 0 ↦→ ζ(t 1 ; t 0 , z 0 ) ∈ R 2 andsuch that ϕ(γ 1 (±)) is contained in W(±) and meets both the inner and the outercircumferences, as well as ϕ(γ 2 (±)) is contained in W(∓) and meets both theinner and the outer circumferences. ∗The points z 0 = (x 0 , y 0 ) belonging to each of the sub-paths γ i (±) (i = 2, 1) are initialpoints of system(4) x ′ = y, y ′ = −q(t)g(x),for t = t 0 for which the corresponding solution ζ(·;t 0 , z 0 ) = (u(·), u ′ (·)) is such thatu(t) has a sufficiently large (but fixed in advance) number of zeros in the interval ]t 0 ,τ[and then, either exactly one zero or no zeros at all (and a zero for its derivative) in theinterval ]τ, t 1 [ .∗ In [55] we used a definition of path which is slightly different from the one considered here in Section1.5; however, this does not effect the validity of property (H ± ). Actually, for the moment, as long as we arein the introductory part of this work and also for the sake of simplicity in the exposition, we prefer to be alittle vague about the precise concept of path we are going to use.
Periodic points and chaotic dynamics 117Figure 1: An illustration of the situation which occurs in [55]. Any path σ = σ(+)contained in the first quadrant and meeting the inner and the outer circumferences ofthe conical shell W(+) (like the one drawn with a thinner line) contains (at least) twosub-paths γ 1 = γ 1 (+) and γ 2 = γ 2 (+) (like those drawn with a thicker line) satisfyingthe following property: through the map ϕ, one of the two sub-paths (namely, γ 1 ) istransformed to a path (drawn with a thicker line) contained in the first quadrant andcrossing the set W(+), while the other sub-path (that is, γ 2 ) is transformed to a path(drawn with a thicker line) contained in the third quadrant and crossing the set W(−).The same happens with respect to any path σ(−) contained in W(−) and intersectingthe inner and the outer circumferences at the boundary of W(−).
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Periodic solutions of difference eq
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36 J. Fan - S. JiangWe shall consid
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38 J. Fan - S. Jiangshock fronts fo
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40 J. Fan - S. Jiangwhich are indep
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42 J. Fan - S. Jiang3. Proof of The
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44 J. Fan - S. JiangHere and in wha
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46 J. Fan - S. JiangBy Lemma 5 and
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48 J. Fan - S. JiangNext, we show t
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50 J. Fan - S. Jiangwhich implies u
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52 J. Fan - S. Jiang[42] ROUSSET F.
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54 M. Francawhere F(u,|x|) = ∫ u0
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56 M. FrancaWe start from the speci
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58 M. FrancaY-POXSPẋ=0Figure 1: A
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60 M. Franca0 ≤ u ≤ U and r ≥
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62 M. Francaspeaking, when f is pos
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64 M. Franca0, then, from an elemen
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Page 74 and 75: 70 M. Francar > 0. Now we can state
Page 76 and 77: 72 M. Francadynamical system of the
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Page 84 and 85: 80 M. Francawith slow decay.Moreove
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RENDICONTI DEL SEMINARIO MATEMATICO

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