Nonlinearity-generated resilience in large complex systems.

We consider a generic nonlinear extension of May's 1972 model by including all higher-order terms in the expansion around the chosen fixed point (placed at the origin) with random Gaussian coefficients. The ensuing analysis reveals that as long as the origin remains stable, it is surrounded by a "resilience gap": there are no other fixed points within a radius r_{*}>0 and the system is therefore expected to be resilient to a typical initial displacement small in comparison to r_{*}. The radius r_{*} is shown to vanish at the same threshold where the origin loses local stability, revealing a mechanism by which systems close to the tipping point become less resilient. We also find that beyond the resilience radius the number of fixed points in a ball surrounding the original point of equilibrium grows exponentially with N, making systems dynamics highly sensitive to far enough displacements from the origin.

Exact relations between multifractal exponents at the Anderson transition.

Two exact relations between mutlifractal exponents are shown to hold at the critical point of the Anderson localization transition. The first relation implies a symmetry of the multifractal spectrum linking the exponents with indices q<1/2 to those with q>1/2. The second relation connects the wave-function multifractality to that of Wigner delay times in a system with a lead attached.

Reducing nonideal to ideal coupling in random matrix description of chaotic scattering: application to the time-delay problem.

We write explicitly a transformation of the scattering phases reducing the problem of quantum chaotic scattering for systems with M statistically equivalent channels at nonideal coupling to that for ideal coupling. Unfolding the phases by their local density leads to universality of their local fluctuations for large M. A relation between the partial time delays and diagonal matrix elements of the Wigner-Smith matrix is revealed for ideal coupling. This helped us in deriving the joint probability distribution of partial time delays and the distribution of the Wigner time delay.