Application to Dynamical Systems. An Example with Discontinuous RPD: The Derivative Nonlinear Schrodinger Equation

Abstract

In this chapter type-I intermittency considering discontinuous RPD function in one-dimensional maps is studied. We employ and extend the M(x) function methodology, developed in Chap. 5, to study type-I intermittency with discontinuous RPD functions in one-dimensional maps with quadratic local form. The theoretical methodology implements a more general function M(x), called here global M(x) function. The discontinuous RPD functions are produced by the existence of at least two different reinjection mechanisms. One of them is generated by trajectories passing close to the zero-derivative point of the quadratic local map (local minimum of the map); these trajectories produce a high concentration of reinjection points inside of a subinterval close to the lower limit of the laminar interval. Therefore, the RPD function is discontinuous and it presents a huge density close to of the laminar interval lower limit. On the other hand, the characteristic relation, $\bar{l} \propto \varepsilon ^{-1/2}$, was found. Then, the characteristic relation for classical type-I intermittency holds although the RPD is not uniform. Also, the elevated density close to the lower limit of the laminar interval increases the average laminar length. This result can be understood because the maximum laminar length verifies the relation l(−c, c) ∝ ɛ ^−1∕2, and the average laminar length, due to the high local concentration, is a fraction of the maximum laminar length. Finally, the extended methodology to evaluate the function M(x) has been implemented to deal with type-I intermittency in the three wave truncation model for the derivative nonlinear Schrodinger equation. The numerical results and the analytical predictions for the nonlinear functions M(x) and for the discontinuous RPD functions present very good accuracy.