RENDICONTI DEL SEMINARIO MATEMATICO

118 D. Papini - F. ZanolinFigure 2: An illustration of the possible behavior of a solution u(t) of (2) for t ∈[t 0 , t 1 ], with (u(t 0 ), u ′ (t 0 )) belonging to γ 1 (+). The solution oscillates a certain (large)number of times in ]t 0 ,τ[ . At the time τ of switching from q > 0 to q < 0 the point(u(τ), u ′ (τ)) lies in the interior of the fourth quadrant of the phase-plane. Then, in theinterval [τ, t 1 ] we have u(t) > 0 with u ′ (t) vanishing exactly once.Figure 3: An illustration of the possible behavior of a solution u(t) of (2) for t ∈[t 0 , t 1 ], with (u(t 0 ), u ′ (t 0 )) belonging to γ 2 (+). The solution oscillates a certain (large)number of times in ]t 0 ,τ[ . At the time τ of switching from q > 0 to q < 0 the point(u(τ), u ′ (τ)) lies in the interior of the fourth quadrant of the phase-plane. Then, in theinterval [τ, t 1 ] we have u ′ (t) < 0 with u(t) vanishing exactly once.