By H.W Broer, B Krauskopf, Gert Vegter

Contributed via shut colleagues, buddies, and previous scholars of Floris Takens, international research of Dynamical structures is a liber amicorum devoted to Takens for his sixtieth birthday. the 1st bankruptcy is a replica of Takens's 1974 paper "Forced oscillators and bifurcations" that was once formerly to be had merely as a preprint of the college of Utrecht. between different vital effects, it includes the unfolding of what's referred to now because the Bogdanov-Takens bifurcation. the remainder chapters hide themes as different as bifurcation conception, Hamiltonian mechanics, homoclinic bifurcations, routes to chaos, ergodic thought, renormalization idea, and time sequence research. In its entirety, the booklet bears witness to the impression of Takens at the smooth thought of dynamical structures and its purposes. This ebook is a must-read for somebody drawn to this energetic and interesting box.

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**Example text**

The dashed curves are also contained in p(Z, τ | Y0 , t0 ) but not in p(Y, t; Z, τ ). Roughly speaking, they are ﬁltered out due to conditional probability p(Y, t; | Z, τ ; Y0 , t0 ) in the expression p(Y, t; Z, τ ; Y0 , t0 ) = p(Y, t; | Z, τ ; Y0 , t0 )p(Z, τ | Y0 , t0 )p(Y0 , t0 ). On the other hand, the product p(Y, t; | Z, τ )p(Z, τ | Y0 , t0 )p(Y0 , t0 ) considers also the dotted lines, which contribute particularly to p(Y, t | Z, τ ). Thus, we have to expect p(Y, t; | Z, τ ; Y0 , t0 ) = p(Y, t; | Z, τ ).

The third term and the ﬁfth term can be collected in one expression. 74) and considering ε → 0, (3) + (5) = −H dZdXf (X) W (Z | X; t) p (X, t | Y, t ) . 81) Notice that we use the symbol H to indicate the principal value integral. We ﬁnally get for the fourth term (4) = H dXdZf (Z) W (Z | X; t) p (X, t | Y, t ) . 83) and after integrating by parts, we get ∂ dXf (X) p (X, t | Y, t ) = dXf (X) ∂t ∂ ∂2 1 × − Aα (X, t) + Bαβ (X, t) p (X, t | Y, t ) ∂Xα 2 ∂Xα ∂Xβ α αβ +H dZdX [f (X) − f (Z)] W (X | Z; t) p (Z, t | Y, t ) .

162) − [a(x, t)p (x, t | y, t )] . 162). This result can be generalized for the case of a system of N stochastic diﬀerential equations with R Wiener processes. 139). 164) with µ Bαβ (Y, t) = bα,k (Y, t)bβ,k (Y, t) and Aα (Y, t) = aα (Y, t).