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Example text

15), then clearly Now, we can verify the basic assumptions. 12. 34), respectively. Moreover, conditions H1), H2) and H3) are satisfied. Proof. Condition H1) was already verified. 35) holds, also condition H2) is fulfilled. Now suppose that dim N(I − Pξ (x0 , 0, µ, α)) > 1. We recall that f+ (x0 , y0 ) ∈ N(I − Pξ (x0 , 0, µ, α)). Since N(I − Pξ (x0 , 0, µ, α)) is linear, there is a vector v¯ ∈ N(I − Pξ (x0 , 0, µ, α)) such that v¯ , f+ (x0 , y0 ) = 0. Then we can write v¯ = (0, v)∗ . 18) for Pξ we look for the image of v¯ under the mapping Pξ (x0 , 0, µ, α).

48). 4. e. x0 (V) is an immersed submanifold of codimension 1. Related problems have been studied in [28] for smooth dynamical systems. Then we can suppose that x0 (β) = (β1 , . . , βn−1 , 0) = (β, 0). Let us denote ξ¯ = (ξ, 0) for ξ ∈ V. We take Σ = 57 58 ´ Poincare-Andronov-Melnikov Analysis for Non-Smooth Systems Rn−1 × {0} ∩ Ω+ and a new Poincar´e mapping P : V × (−ε0 , ε0 ) × R p → Σ defined as P(ξ, ε, µ) = x+ (t2 (0, ξ¯, ε, µ), x− (t1 (0, ξ¯, ε, µ), x+ (0, ξ¯)(t1 (0, ξ¯, ε, µ), ε, µ)) (t2 (0, ξ¯, ε, µ), ε, µ))(t3 (0, ξ, ε, µ), ε, µ), where t3 (·, ·, ·, ·) is a solution close to T ξ of equation ¯ ε, µ), x− (t1 (0, ξ, ¯ ε, µ), x+ (0, ξ) ¯ x+ (t2 (0, ξ, ¯ ε, µ), ε, µ))(t2 (0, ξ, ¯ ε, µ), ε, µ))(t, ε, µ), en = 0 (t1 (0, ξ, with en = (0, .

Let us denote ξ¯ = (ξ, 0) for ξ ∈ V. We take Σ = 57 58 ´ Poincare-Andronov-Melnikov Analysis for Non-Smooth Systems Rn−1 × {0} ∩ Ω+ and a new Poincar´e mapping P : V × (−ε0 , ε0 ) × R p → Σ defined as P(ξ, ε, µ) = x+ (t2 (0, ξ¯, ε, µ), x− (t1 (0, ξ¯, ε, µ), x+ (0, ξ¯)(t1 (0, ξ¯, ε, µ), ε, µ)) (t2 (0, ξ¯, ε, µ), ε, µ))(t3 (0, ξ, ε, µ), ε, µ), where t3 (·, ·, ·, ·) is a solution close to T ξ of equation ¯ ε, µ), x− (t1 (0, ξ, ¯ ε, µ), x+ (0, ξ) ¯ x+ (t2 (0, ξ, ¯ ε, µ), ε, µ))(t2 (0, ξ, ¯ ε, µ), ε, µ))(t, ε, µ), en = 0 (t1 (0, ξ, with en = (0, .

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