From integrability to chaotic motion

We point out a link between integrability and chaotic motion by showing that the algebraic structure of the integrals of motion of a system can be used to get information on the chaotic solutions of this system or to reveal the parameters values for which there is no chaotic motion in this system. We will illustrate this on 3-D systems :

$$\dot{\vec{x}}=\vec{F}(x,y,z) \mbox{ with } \vec{F}=(P(x,y,z),Q(x,y,z),R(x,y,z))^T$$ (1)

First we will track the path from integrability to chaotic motion using the Lorenz [4] system as an example.

We will first introduce our method on a classical system, the Lorenz system which comes from a truncation of the Boussinesq equations which model the movement of a fluid between two isothermal planes [4]  :

$$\vec{F}=(\sigma\,(y-x),x\,(r-z)-y,x\,y-b\,z)^T$$ (2)

We first notice that the divergence of the vector field is constant and negative :  $\vec{\nabla} \cdot \vec{F}=-(\sigma+b+1)<0$. This fact comes from the fluid system which is highly dissipative. System (2) has three positive parameters ( $r,\sigma,b$) and hence a parameter space included in ${\mathrm{I\!R\!}}\,^3$. Depending on the values of the parameters, the system may exhibit : 1 equilibrium point, 3 equilibrium points, 3 equilibrium points and a chaotic attractor [5] . For most of the parameters values, one cannot integrate equations (2), i.e. one cannot express the solutions $x(t),\,y(t),\,z(t)$ in an analytic way, hence one numerically calculates approximations to these solutions.

Complete integrability
Nevertheless, for some precise values of the parameters, we may integrate equations (2) : if b=1, $\sigma=\frac{1}{2}$, $\forall r$, the system fulfils the Painlevé test [1] and admits two integrals of motion :

$$I_1=(y^2+z^2)\,e^{2t} \quad , \quad I_2=(x^2-z)\,e^{t}\, ,$$ (3)

i.e. $\dot{I}_{1,2} \stackrel{\mbox{\rm\scriptsize def}}{=}\frac{dI_{1,2}}{dt} \equiv 0$. The trajectories evolve on the intersections of the two surfaces defined by the equations $I_1=\mbox{const}_1 \mbox{ and } I_2=\mbox{const}_2$. Moreover, we may write the solutions in term of elliptic functions [2,3] . In this particular case, where of course there is no chaos, we have a lot of information on the solutions of (2) but for only a small part of the parameters space (a line in a three dimensional parameter space).

Partial integrability
If we relax the integrability conditions to $b=2\sigma$, $\forall r$, the system no longer fulfils the Painlevé test and we cannot write down the solutions in terms of known functions. There is only one integral of motion left :

$$I_3=(x^2-2\,\sigma\,z)\,e^{2\sigma\,t}\, ,$$ (4)

  

Figure 1: When the system has an integral of motion, the phase space in full with semi-permeable surfaces. The invariant surface (4) contains the two stable equilibrium points which are in this case the global attractor of the system.

which ensures the absence of chaotic motion (via Bendixon-Poincaré theorem). Hence we have less information than before on the system, but on a larger part of the parameters space (a plane in the three dimensional parameter space).

Non integrability
If one relax again the integrability conditions to $b \neq 2\,\sigma$, $\forall r$, there is no integral of motion left. Moreover for $r>r_{ho}(\sigma,b)$a chaotic behaviour (asymptotic or transient) is possible. Now we may wonder how to continue to find information on the system or how to use the analytic results existing before, now that chaos has arisen.

We make the following remark : when the system admits I₃ as an integral of motion (for $b=2\,\sigma$), the trajectories evolve on the surfaces defined by $I_3=\mbox{const}$. Hence the surfaces :

$$I_4=a_1\,x^2+a_2\,z+a_3=0$$ (5)

should have interesting properties even when $b \neq 2\,\sigma$. It is clear so far that these new surfaces I₄ will not be integrals of motion, i.e. $\dot{I}_4$ will not be identically null, but after some calculations, we discover that under certain conditions, the following inequality :

$$\dot{I}_4 <0 \quad , \quad \forall (x,y,z) / I_4=0$$ (6)

holds ( $\dot{I}_4 >0$ can be found with symmetrical conditions). In this case we have a weaker relation than before : we have a function I(x,y,z) such that $\dot{I}$ is of constant sign and no longer identically null. We have even less information than before, but on a much (much) larger region of the parameter space than before : $b \neq 2\sigma$, $\forall r$ : two full half spaces that even contain values of parameters for which the system is chaotic !

Geometrically speaking
To understand the consequences in the phase portrait of the inequality (6), one has to consider that $\dot{I}_4$ is $\vec{\nabla} I_4 \cdot \vec{F}$where $\vec{\nabla} I_4$ is the normal vector of the surface I₄=0. So when (6) holds, the surface I₄=0is crossed by the trajectories always in the same direction with regard to its normal vector. Such a surface will be called semi-permeable. For the system (2), the surfaces $I_4=x^2-2\sigma\,z+a_3=0$ are semi-permeable when $(b-2\sigma)$has the same sign as a₃.

  

Figure: The family of parabolas $x^2-2 \sigma z+a_3=0$ with a₃<0in the case where $(\frac{8}{3}=)b<2 \sigma(=20)$. All the parabolas are semi-permeable. They are crossed by the flow upward. The chaotic attractor is stuck in the zone of phase space where there are no semi-permeable surfaces.

The shape of these surfaces enables us to give algebraic bounds to the spread of the chaotic attractor (see figure (2)). Moreover with the help of the other integrals of motion of system (2), we can bound sharply its chaotic attractor. In certain other cases the semi-permeable surfaces prevent the chaotic attractor from existing (these results are developed in [6] . Semi-permeable surfaces have also been used in the study of some others 3-D dissipative dynamical systems [7,8] .

References

• [1] W. Steeb, N. Euler, Nonlinear Evolution Equations and Painlevé Test (1988) Singapore : World Scientific.
• [2] T. C. Bountis, A. Ramani, B. Grammaticos and B. Dorizzi, Physica A 128 (1984) 268-288.
• [3] G. Levine and M. Tabor, Physica D 33 (1988) 189-210.
• [4] E. Lorenz, J. Atmosph. Sci. 20 (1963) 130.
• [5] C. Sparrow, The Lorenz Equations (1982) Springer-Verlag.
• [6] H. Giacomini, S. Neukirch, Physics Letters A 227 (1997) 309-318.
• [7] S. Neukirch, H. Giacomini, Physical Review E 61 #5 (2000)
• [8] T. McMillen, The NonLinear Journal 1 (1998) 1-10, available at
  http://www.math.arizona.edu/ $\sim$ goriely/nljournal/nljournal.html.