New Formulation of the Chaotic Intermittency

Theoretical Evaluation of the Reinjection Probability Density Function in Chaotic Intermittency

“…It is characterized by almost ordered motion in state space disrupted by randomly distributed chaotic bursts. Regular or laminar phases represent pseudo-equilibrium and pseudo-periodic solutions, while the bursts are related with chaotic behavior [2][3][4][5][6][7][8][9].…”

“…Type II intermittency occurs by a Hopf bifurcation, and two complex conjugate eigenvalues leave the unit circle. Type III intermittency appears by a subcritical period-doubling bifurcation, and then an eigenvalue leaves the unit circle across −1 [2,3,6,7]. For intermittency types I, II, and III, the characteristic local maps are [6] x n+1 = ε + x n + a x p n type I…”

“…The reinjection probability density function (RPD function) is utilized to describe the reinjection mechanism. This function determines the probability that trajectories are returned or reinjected close to the unstable or vanished fixed point and inside the laminar interval [2][3][4]6,7]. The RPD function is defined by the chaotic dynamics of the system and strongly influences the correct description of the chaotic intermittency.…”

“…A novel theory has been formulated, enabling the assessment of the RPD function through either an analytical representation of a map or by utilizing a set of numerical or experimental data [38]. In this theory, the RPD function is described by a power law with exponent α, and it works correctly for types I, II, III, and V intermittency with and without noise [6,7] φ…”

“…The theory establishes that the reinjection process depends on the map derivative at pre-reinjection points, i.e., points x n that verify x n+i = F i (x n ) where x n+i ∈ L, n, and i are integer numbers [6,39]. Therefore, the pre-reinjection points drive the reinjection process.…”