Autres Articles de :
3D flows

# 1986 The Nosé-Hoover system (Sprott A system)

Christophe LETELLIER
19/07/2017

Shūichi Nosé [1] and by William Hoover [2], [3] investigated a system of N particles (d degrees of freedom) in a given volume V and interacting (heat transfer) with an external system in such a way that the energy E is conserved. The equations governing the coordinates Q, the momentum P and the effective mass s, after a coordinate transformation, were

$\left\{&space;\begin{array}{l}&space;\dot{x}&space;=&space;ay&space;\\[0.1cm]&space;\dot{y}&space;=&space;-x&space;+&space;yz&space;\\[0.1cm]&space;\dot{z}&space;=&space;d&space;-&space;y^2&space;\end{array}&space;\right.$

which were rediscovered by Julian Clinton Sprott [4] as the Sprott A system. This system is a conservative system as shown by its Jacobian matrix

${\cal&space;J}_{\rm&space;Sprott&space;A}&space;=&space;\left[&space;\begin{array}{ccc}&space;0&space;&&space;a&space;&&space;0&space;\\[0.1cm]&space;-1&space;&&space;z&space;&&space;y&space;\\[0.1cm]&space;0&space;&&space;-2y&space;&&space;0&space;\end{array}&space;\right]$

with a trace that has for mean value

$\frac{1}{T}&space;\int_{t=0}^T&space;z&space;\,&space;{\rm&space;d}z&space;=&space;0$

Depending on initial conditions, the solution can be a chaotic sea, a quasi-periodic motion or a periodic orbit as evidenced by the Poincaré section

$\left\{&space;(x_n,&space;y_n)&space;\in&space;\mathbb{R}^2&space;~|~&space;z_n&space;=&space;0&space;\right\}$

for a = 0.2 (Fig. 1).

Fig. 1. Poincaré section of the Sprott A system.

The chaotic sea is mainly structured around the four islands corresponding to a period-4 orbit (Fig. 2).

Fig. 2. Chaotic sea structured around a period-4 orbit.

This system was recently modified by Cang and coworkers [5] as follows

$\left\{&space;\begin{array}{l}&space;\dot{x}=-cx&space;+&space;ay&space;\\[0.1cm]&space;\dot{y}&space;=&space;-ax&space;+&space;yz&space;\\[0.1cm]&space;\dot{z}&space;=&space;d&space;-bz&space;-y^2&space;\end{array}&space;\right.$

which has for Jacobian matrix

${\cal&space;J}_{\rm&space;Cang}&space;=&space;\left[&space;\begin{array}{ccc}&space;-c&space;&&space;a&space;&&space;0&space;\\[0.1cm]&space;-a&space;&&space;z&space;&&space;y&space;\\[0.1cm]&space;0&space;&&space;-2y&space;&&space;-b&space;\end{array}&space;\right]$

with a mean trace equal to -(b+c). With a=1, b=0, and d=1, a bifurcation diagram is computed versus parameter c, from c=0 associated with a conservative dynamics to c=1 corresponding to a strongly dissipative case (Fig. 3). The conservative case corresponds to the Sprott A system and the strongly dissipative case is just a "Lorenz-like" dynamics with a chaotic attractor topologically equivalent to the Lorenz attractor. As seen in the bifurcation diagram, there is not a continuous transition between the chaotic sea (c=0) and the Lorenz-like attractor (c=1). There is a huge window with period-1 limit cycle and a crisis at c=0.6. Along this line of the parameter space, there is no chaotic attractor with a weakly dissipation rate.

Fig. 3. Bifurcation diagram from a conservative to a dissipative dynamics.

[1] S. Nosé, A molecular dynamics method for simulations in the canonical ensemble, Molecular Physics, 52 (2), 255-268, 1984.

[2] W. G. Hoover, Nonlinear conductivity and entropy in the two-body Boltzmann gas, Journal of Statistical Physics, 42 (3/4), 587-600, 1986.

[3] H. A. Posch, W.oover & F. J. Vesely, Canonical dynamics of the Nosé oscillator : Stability, order, and chaos, Physical Review A, 33, 4253-5265, 1986.

[4] J. C. Sprott, Some simple chaotic flows, Physical Review E, 50 (2), 647-650, 1994.

[5] S. Cang, A. Wu, Z. Wang, Z. Chen, On a 3-D generalized Hamiltonian model with conservative and dissipative chaotic flows, Chaos, Solitons & Fractals, 99, 45-51, 2017.

ATOMOSYD © 2007-2018 |   |   |   |  MàJ . 02/09/2018