Chaotic saddles in nonlinear modulational interactions in a plasma

Abstract

A nonlinear model of modulational processes in the subsonic regime involving a linearly unstable wave and two linearly damped waves with different damping rates in a plasma is studied numerically. We compute the maximum Lyapunov exponent as a function of the damping rates in a two-parameter space, and identify shrimp-shaped self-similar structures in the parameter space. By varying the damping rate of the low-frequency wave, we construct bifurcation diagrams and focus on a saddle-node bifurcation and an interior crisis associated with a periodic window. We detect chaotic saddles and their stable and unstable manifolds, and demonstrate how the connection between two chaotic saddles via coupling unstable periodic orbits can result in a crisis-induced intermittency. The relevance of this work for the understanding of modulational processes observed in plasmas and fluids is discussed.

Zig-zag networks of self-excited periodic oscillations in a tunnel diode and a fiber-ring laser

“…For many decades, the great interest has been to investigate mainly the structure of the phase space of flows, with particular emphasis on the possible transitions from order to chaos and a plethora of instabilities associated with these transitions [1][2][3]. More recently, extensive numerical simulations have revealed unexpected regularities in a complementary setting, namely, in the control parameter space of systems as diverse as electronic circuits, laser systems, and modulational interactions in a plasma, in chemical and biophysical oscillators, and in many other paradigmatic flows covering a large spectrum of practical applications [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Such regularities emerged while attempting to classify systematically all collective oscillations supported by the aforementioned applications.…”

Alfvén waves in space and astrophysical dusty plasmas

“…Low-dimensional chaos of DNLS can also be studied by considering Alfvén modulational wave coupling processes (Sánchez-Arriaga et al, 2009;Miranda et al, 2012). Sánchez-Arriaga et al (2009) studied a truncation model of DNLS with three resonant traveling waves.…”

“…The aforementioned studies on Alfvén chaos elucidate how Alfvén intermittency appears in space and astrophysical plasmas, such as the magnetic intermittent turbulence observed in the solar wind (Koga et al, 2007;Sahraoui, 2008;Chian and Miranda, 2009;Chian and Muñoz, 2011), due to nonlinear dynamical phenomena such as the type-I intermittency induced by a saddle-node bifurcation, the crisisinduced intermittency induced by an interior crisis, and the extrinsic intermittency induced by a noise (Hada et al, 1990;Chian et al, 1998Chian et al, , 2006Chian et al, , 2007Rempel et al, 2006Rempel et al, , 2008Sánchez-Arriaga et al, 2009;Miranda et al, 2012).…”

“…In the presence of dissipative and growth terms, the system exhibits very complex dynamics including multi-attractors and period-doubling bifurcation to chaos. Miranda et al (2012) studied the truncated three-wave model of DNLS involving a linearly unstable Alfvén wave and two linearly damped Alfvén waves with different damping rates. The maximum Lyapunov exponent was computed as a function of the damping rates in a two-parameter phase space, and shrimp-shaped self-similar structures in the parameter phase space were identified.…”

Edge of chaos and genesis of turbulence

“…Moreover, we compute the Lyapunov spectrum [23] solving the variational equation for the flux Jacobian matrix from the STCS trajectories (see Miranda et al [24] for further details). The increase of the degree of spatial disorder with increasing driver amplitude is accompanied by an increase of temporal chaos.…”