Low-dimensional non-linear dynamical systems and generalized entropy

Abstract

Low-dimensional non-linear maps are prototype models to study the emergence of complex behavior in nature. They may exhibit power-law sensitivity to initial conditions at the edge of chaos which can be naturally formulated within the generalized Tsallis statistics prescription which i s c haracterized by the entropic index q. General scaling arguments provide a direct relation between the entropic index q and the scaling exponents associated with the extremal sets of the multifractal critical attractor. The above result comes in favor of recent conjectures that Tsallis statistics is the natural frame for studying systems with a fractal-like structure in the phase-space. Power-law sensitivity i n high-dimensional dissipative and Hamiltonian systems are also discussed within the present picture. I IntroductionLow-dimensional non-linear maps are the prototype models to study the emergence of complex behavior in dynamical systems. Their typical behavior include the occurrence of bifurcation instabilities, long-range correlated sequences, fractal structures and chaos, which are commonly observed in a great variety of systems ranging from uids, magnetism, biology, social sciences and many others 1 .The study of the sensitivity to initial conditions of non-linear systems is one of the most important tools used to investigate the nature of the phase-space attractor. It is usually characterized by the Liapunov exponent , de ned for the simple case of a one-dimensional dynamical variable x as xt x0e t x0 ! 0; t ! 1 1 where x0 is the distance between two initially nearby orbits in an equivalent point of view, it is the uncertainty on the precise initial condition. If 0 the system is said to be strongly sensitive to the initial condition with the uncertainty on the dynamical variable growing exponentially in time and this characterizes a chaotic motion in the phase-space. On the other hand, if 0 the system becomes strongly insensitive to the initial condition which is expected for any state whose dynamical attractor is an orbit with a nite period. The problem of the sensitivity to initial conditions can be reformulated in an entropic language as a process of information loss in the case of chaotic behavior or recovery for periodic attractors. Within this context, it is useful to introduce the Kolmogorov-Sinai entropy K. It is basically the rate of variation of the Boltzmann- where S0 and SN are the entropies of the system evaluated at times t = 0 and t = N for maps = 1. With the simplifying assumption that at time t there are Wt occupied cells with the same occupation number, we h a ve from equation 2 that Wt = W 0e Kt ;3 which is equivalent to equation 1 for the sensitivity to initial condition and provides the well-known Pesin

Damage Spreading in the Bak–sneppen Model: Sensitivity to the Initial Conditions and Equilibration Dynamics

“…At this point it is worth to explore the similarities between the damage spreading analysis of the dynamical properties of extended critical systems and recent results concerning the sensitivity to initial conditions of lowdimensional dissipative maps poised at criticality [14][15][16][17][18][19][20][21][22]. More precisely, the numerical analysis of such systems (i.e., logistic map [14], logistic-like maps [15], circular maps [18], asymmetric logistic maps [19], single-site map [21], Henon map [22]) has shown that, at critical points such as the chaos threshold, tangent bifurcations etc, the standard type of the sensitivity to the initial conditions given by the sensitivity function…”

“…The same dynamical regimes have been identified in the onset of chaos of low-dimensional non-linear dynamical maps. The distance between two nearby orbits was shown to diverge following a power-law whose exponent is directly related to geometric exponents characterizing the extremal sets of the critical dynamical attractor [14][15][16][17][18][19][20][21][22]. On the other hand, the long-time relaxation towards the dynamical attractor is governed by a new exponent, which seems to be related to the fractal dimension of the support of the dynamical attractor in phase-space [23].…”

“…By measuring how the difference between two initially close configurations evolves in time under the same noise, it was shown that an initial power-law divergence sets up before the distance saturates in a finite size-dependent plateau. This slow short-time power-law dynamics is characteristics of systems poised at critical states such as at the critical point of second-order phase transitions [13] and the onset of chaos in non-linear dynamical systems [14][15][16][17][18][19][20][21][22].…”

Convergence to the critical attractor of dissipative maps: Log-periodic oscillations, fractality, and nonextensivity

“…The above scaling relation has been shown to hold for the families of generalized logistic and circle maps [3,[10][11][12]. However, the temporal evolution of critical dynamical systems can be strongly dependent on the particular initial ensemble.…”

Damage spreading in the Bak–Sneppen and ballistic deposition models: critical dynamics and nonextensivity

“…Among others, low-dimensional dissipative dynamical systems constitute one class of such examples. Indeed, nonextensive thermostatistics has been shown to be connected to the sensitivity properties of low-dimensional dissipative maps [3][4][5][6][7][8][9][10][11]. More precisely, the numerical analysis of such systems (i.e., logistic map [3], logistic-like maps [4], circular maps [7], asymmetric logistic maps [8], single-site map [9], Henon map [10]), as well as a very recent analytical treatment [11], have shown that, at the critical points such as the chaos threshold, tangent bifurcations, etc., the standard type of sensitivity to the initial conditions, given by the sensitivity function…”