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By Tarantello G.

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The last row corresponds to the pseudo-arclength equation, which is not included in the nq = 1 count. J. B. P. Kern´evez, Numerical analysis and control of bifurcation problems (II): Bifurcation in infinite dimensions, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 1(4) (1991) 745–772 c 1991 World Scientific Publishing; reproduced with permission. the discrete boundary conditions are bi (u0 , uN , µ, λ) = 0, i = 1, . . , nb , and the integrals constraints can be discretized as N m ωj,i qk (uj− mi , µ, λ) = 0, k = 1, .

Let   x u = y, z and write the Lorenz equations as u′ (t) = f (u(t)). 82, and corresponding normalized eigenvectors v1 , v2 , and v3 , respectively. We want to compute the stable manifold of the origin. We compute an initial orbit u(t), for t from 0 to T (where T < 0), with u(0) close to 0 in the eigenspace spanned by v1 and v2 , that is, 44 Eusebius J Doedel Y 15. 10. 5. 0. -5. -10. -15. -15. 5. 0. -5. -10. 15. 10. X Fig. 29. 24). u(0) = 0 + ε sin(θ) cos(θ) v2 , v1 − |µ2 | |µ1 | for, say, θ = 0.

14). u′k (t) = T f (uk (t), λk ), uk (0) = uk (1), 1 0 1 0 uk (t)∗ u′k−1 (t) dt = 0, (uk (t) − uk−1 (t))∗ u˙ k−1 (t) dt + (Tk − Tk−1 ) T˙k−1 + (λk − λk−1 ) λ˙ k−1 = ∆s, where u(·), f (·) ∈ Rn , λ, T ∈ R. 4 A Vertical Family of Periodic Orbits Consider the system of equations u′1 = λu1 − u2 , u′2 = u1 (1 − u1 ). 14) Note that u(t) ≡ 0 is a stationary solution for all λ. Another stationary 1 solution is u(t) ≡ . −λ The bifurcation diagram for u(t) ≡ 0 is shown in Fig. 17, but we can also analyze the behavior analytically.

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A quantization property for blow up solutions of singular Liouville-type equations by Tarantello G.


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