RENDICONTI DEL SEMINARIO MATEMATICO

Explosions in dimensions one through three 13The density of the new part of the attractor depends on this area, in that this is the onlyway for points in A old to enter A new . The square root in this formula is due to thequadratic tangency. Let L n = f n (L). Then A(L m ) = M A(L), where M depends onthe eigenvalues of q. These are the key ingredients giving rise to the scaling law.4.2. A new scaling law in three dimensionsFor the three-dimensional non-tangency bifurcation described in Section 3, we havedemonstrated numerically that there is a linear relationship between the logarithm ofthe mean transient length K and the logarithm of the distance from the bifurcationparameter (cf. [3]).Denote the eigenvalues of q by |λ 1 | > |λ 2 | > 1 > |µ|, and those of p by|β 1 | < |β 2 | < 1 < |α|. The set of points in the attractor (and thus on the twodimensionalunstable manifold U(q)) which exit A old must do so by coming very closeto the unstable manifold of p. Starting near p, the area of U(q) within ǫ of U(p) isapproximated using α and β 2 . We also need to know the fraction of points whichexit A old near U(p) which re-enter the attractor. This is done using the two unstableeigenvalues for the linearization at q. The estimate leads to the new scaling law, whichgive good agreement with numerical calculation. It states that the mean transient lengthis K(η) = η γ , whereγ = 1 + log |λ 1| log |α|+log |λ 2 | log |β 2 | .4.3. Unstable dimension variabilityThe low density of the new part of the attractor has interesting numerical implicationsin terms of testing for unstable dimension variability. Precisely, near the parameter atwhich a crisis occurs, the standard test for UDV is not applicable.In an attractor with a dense orbit which exhibits UDV, we know that the denseorbit comes arbitrarily close to the stable manifold of each fixed or periodic saddlepoint. Thus it is possible to find a sequence within the dense orbit which stays close toa fixed or periodic point for any prescribed number of iterates after any finite transientis removed. For simplicity, assume p and q are fixed points with different numbersof unstable directions in an attractor which exhibits UDV. Since the middle Lyapunovexponent of the fixed points have opposite signs, the orbit must have arbitrarily longfinite time sequences with the middle Lyapunov exponent being negative, and arbitrarilylong finite time sequences with the middle Lyapunov exponent being positive. Thestandard test for UDV uses this fluctuation of Lyapunov exponents around zero [10].In the case of UDV after a crossing bifurcation, although it is theoretically correctthat the Lyapunov exponents fluctuate around zero, the density of the attractor isquite low near the newly added fixed point. Therefore, it is computationally infeasibleto use the Lyapunov exponent test for UDV. See [3] for detailed numerical calculationsillustrated using the example depicted in Figure 8.