RENDICONTI DEL SEMINARIO MATEMATICO

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6 K.T. Alligood - E. Sander - J.A. YorkeURULy p i jFigure 3: The tangency point y is contained in U L ∩ U R , and is thus not an explosionpoint. However, the preimages i and j of y are explosion points. U R and U L aredepicted again with a solid and dashed line respectively.(a)(b)F(S)00 11pS00 11pSF(S)Figure 4: Homoclinic tangency points in the plane may or may not be explosion points.A set S and a large iterate f K (S) = F(S). In (a), it is never possible for the homoclinictangency point to be an explosion point. In part (b), it depends on the eigenvalues.
Explosions in dimensions one through three 72.2. Tangencies in two dimensionsIn two dimensions, a heteroclinic intersection is a hallmark for chaotic behavior in theform of a Smale horseshoe. Thus a homoclinic tangency is a bifurcation point. It is notalways an explosion point. Figure 4(a) shows a configuration of stable and unstablemanifolds at a homoclinic tangency which can never be an explosion, since there isa horseshoe forming locally prior to tangency. In Figure 4(a), the situation is not soclear, since a large iterate of a rectangular set S may or may not map across itselfunder a large iterate. In fact, whether this is a possible explosion point depends on therelative strength of contraction and expansion at the fixed point p. Palis and Takens [25]classified homoclinic explosion bifurcations, as shown in Figure 4 by describing whichplanar homoclinic tangencies can be explosion points as a function of the placementof the tangency point, sign of the eigenvalues, and area contraction or expansion of themap.An explosion can also occur as a result of tangencies between stable and unstablemanifolds of different fixed or periodic points: heteroclinic tangencies. In this case,in order for a heteroclinic tangency to result in any sort of recurrence, there needs tobe a means of return. A natural way is another heteroclinic intersection, resulting in aheteroclinic cycle. These ideas are given in the following definitions.DEFINITION 3 (n-connection). An n-connection is a sequence of points {ρ 1 , t 1 ,ρ 2 , t 2 ,...,ρ n } such that for all i, ρ i is a fixed point, and for each i < n, t i is aheteroclinic point such that t i ∈ W u (ρ i ) ∩ W s (ρ i+1 ).DEFINITION 4 (Cycle). A cycle is an n-connection {ρ 1 , t 1 ,ρ 2 , t 2 ,...,ρ n } suchthat ρ n = ρ 1 .Generically, we can assume a unique tangency in a cycle, with all other heteroclinicintersections being transverse. Thus any heteroclinic cycle can be reduced to aheteroclinic cycle containing only two periodic or fixed points. In a previous paper [1],we gave necessary and sufficient conditions for planar heteroclinic bifurcations to resultin explosions. The classification does not involve eigenvalue conditions and areacontraction/expansion conditions. Rather, the classification is a set of necessary andsufficient conditions for explosions at heteroclinic tangency based on the configurationof the manifolds, in addition to another “chain” condition on the behavior. Figure 5shows two heteroclinic cycles with a tangency. In (a), any curve which is arbitrarilyclose to the heteroclinic cycle necessarily intersects both the unstable manifold of pand the stable manifold of q, whereas in (b), this is not the case. The tangency point in(a) is an explosion point and in (b) it is not; just as in the one-dimensional case, the keyto an explosion is that different points of the cycle are on opposite sides of a dividingmanifold. Therefore, any ǫ-chain must cross the cycle. Such heteroclinic tangencyexplosion bifurcations are known as crossing bifurcations.
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Page 56 and 57: 52 J. Fan - S. Jiang[42] ROUSSET F.
Page 58 and 59: 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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62 M. Francaspeaking, when f is pos
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64 M. Franca0, then, from an elemen
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66 M. FrancaAnalogously assume that
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68 M. FrancaCorollary 3 we easily g
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70 M. Francar > 0. Now we can state
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72 M. Francadynamical system of the
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74 M. FrancaW u (τ)ON8XN 9ẋ=0E(
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78 M. FrancaNote that with this app
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80 M. Francawith slow decay.Moreove
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82 M. Franca∑ Mi=N+1A i > 0. Then
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86 M. Francah. We denote by f (v, s
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88 M. Franca[32] KABEYA Y., YANAGID
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