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

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

with a trace that has for mean value

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

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

which has for Jacobian matrix

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.