Lorenz Attractors With Arbitrary Expanding Dimension

A Rigorous ODE Solver and Smale’s 14th Problem

“…From this measure it is possible to construct an SRB measure µ R with supp(µ R ) = for the return map R, and also an SRB measure µ ϕ for the flow, see [1] or [24]. We will now prove that f is locally eventually onto: the map is singular at the origin since itself is a stable leaf, and projects to {0}.…”

“…Note that the parameters k 1 , k 2 , and k 3 , and the eigenvalues λ 1 , λ 2 , and λ 3 only depend on the parameters σ, β, and appearing in (1). By inserting the classical parameter values, we get the following approximate system: From now on, we will always refer to (2) as the Lorenz equations.…”

“…The initial rectangle N i is flowed to the first plane (1) by using an Euler method with rigorous error estimates. In the plane (1) , we take the rectangular hull of the largest image of N i , giving us a new starting rectangle R (1) .…”

“…In the plane (1) , we take the rectangular hull of the largest image of N i , giving us a new starting rectangle R (1) . This rectangle is then flowed to (2) and so on.…”

“…There are, however, rare occasions when this is not the case: sometimes a mixed derivative ( i ) x j (i = j) may vanish (the fact that the nonmixed derivatives are always positive is confirmed by the computer program). If, say, 0 ∈ ( 1 ) x 2 (R (k) ), we must consider the whole line segment connecting 4 p (4) and p (1) in order to estimate the upper bound of [R…”