# Integrals of motion and semipermeable surfaces to bound the amplitude of a plasma instability

Sébastien Neukirch

► To cite this version:

Sébastien Neukirch. Integrals of motion and semipermeable surfaces to bound the amplitude of a plasma instability. Physical Review E, 2001, 63 (3), pp.036202. 10.1103/PhysRevE.63.036202. hal-05381163

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Integrals of motion and semi-permeable surfaces to bound
the amplitude of a plasma instability.

S. Neukirch*

Centre for Nonlinear Dynamics and its Applications, University College London
Gower Street, London WC1E 6BT, U.K.

November 17, 2000

Abstract

We study a dissipative dynamical system that models a parametric instability in a plasma. This instability is due to the interaction of a whistler with the ion acoustic wave and a plasma oscillation near the lower hybrid resonance. The amplitude of these three oscillations obey a 3D system of ordinary differential equations which exhibits chaos for certain parameter values. By using certain 'integrability informations' we have on the system, we get geometrical bounds for its chaotic attractor, leading to an upper bound for its Lyapunov dimension. On the other hand we also obtain ranges of values of the system's parameters for which there is no chaotic motion.

P.A.C.S. numbers : 05.45.ac / 02.30.Hq / 52.35.P

Keywords : plasma instability, integral of motion, transversal surface, homoclinic bifurcation,
Lyapunov exponent, ion acoustic wave, whistler.

# 1 Introduction

A *whistler* is a wave in a plasma which propagates parallel to the magnetic field. It is produced by currents outside the plasma at a frequency less than that of the electron cyclotron frequency. Also it is circulary polarized, rotating about the magnetic field in the same sense as the electrons in the plasma.

Interactions between these whistler waves and lower hybrid waves in a plasma are among
the important phenomena taking place in the ionosphere [1]. As it has been shown in [2], a
whistler can destabilize a magnetoactive plasma by exciting the lower hybrid wave together with

<footnote>*e-mail: s.neukirch@ucl.ac.uk</footnote>
the ion acoustic wave (the longitudinal compression wave in the ion density of a plasma). This parametric excitation, although restrained by the loss of energy which is given to the other non-resonant waves, may become chaotic for certain ranges of value of the pump amplitude. More specifically, the whistler at frequency $\omega_q$ excites a plasma wave at frequency $\omega_k$ and the ion acoustic wave at frequency $\Omega_\chi = \omega_q - \omega_k$. We call $a_k$ the normal amplitude of the wave at frequency $\omega_k$ and $b_\chi$ the normal amplitude of the ion acoustic wave. As a result of the decay of these excitations, at least a third synchronous wave is produced (of normal amplitude $a_{k_1}$) which is linearly damped and will act as a limiter for the instability. This elementary limiting process may nevertheless induce complicated oscillations of the three waves when the pump amplitude is increased.

The differential equations for the amplitudes of the three waves are obtained from the hydrodynamic equation for the radio-frequency oscillation of an electron gas and from the kinetic equation for the ion acoustic wave. The amplitudes are assumed to be constant in space. The evolution equations take the dimensionless form :

$$ 
\begin{aligned}
\dot{a}_k &= -b_\chi a_{k_1} - \nu_1 a_k + h b_\chi^* \\
\dot{b}_\chi &= a_k a_{k_1}^* - \nu_2 b_\chi + h a_k^* \\
a_{k_1} &= a_k b_\chi^* - a_{k_1}
\end{aligned}
\qquad (1) $$

where the amplitudes have been non-dimensionalized; $h$ is proportional to the amplitude of the electric field of the whistler and $\nu_1$ and $\nu_2$ are the damping decrements of the excited hybrid and acoustic waves normalized to the damping of the decay-induced (third) wave : $\nu_1 = \frac{\gamma_k}{\gamma_{k_1}}$, $\nu_2 = \frac{\gamma_\chi}{\gamma_{k_1}}$. Depending on the relative values of $h$ compared to $(\nu_1, \nu_2)$, the system can relax to trivial equilibrium (no oscillation) or stabilize on a steady oscillation or even present chaotic motion. By studying the dynamics of the phases of $a_k$, $b_\chi$ and $a_{k_1}$ it can be shown [2] that they correlate as $t \to +\infty$. Hence we shall study system (1) with real amplitudes.

## 2 The dynamics and route to chaos of the PRT system

We set $x = a_k$, $y = b_\chi$ and $z = a_{k_1}$ and $x, y, z \in \mathbb{R}^3$ and rewrite system (1) as :

$$ (\dot{x}, \dot{y}, \dot{z})^T = \vec{F}(x, y, z) = (h y - \nu_1 x - y z, h x - \nu_2 y + x z, xy - \nu_3 z)^T \qquad (2) $$

We will refer at this system as the PRT system as it has been introduced in [2]. The system is symmetrical about the transformation : $x \to -x, y \to -y$. The four parameters are assumed to
be positive. We will briefly recall its important features. The origin $O(0, 0, 0)$ is asymptotically stable for $h < h_{pk} = \sqrt{\nu_1 \nu_2}$. At $h = h_{pk}$, two stable equilibrium points :

$$M_{\pm} \left( \pm \sqrt{\frac{\nu_3}{\nu_1} z_0 (h - z_0)}, \pm \sqrt{\nu_1 \nu_3 \frac{z_0}{h - z_0}}, z_0 = \sqrt{h^2 - \nu_1 \nu_2} \right)$$

appear in a pitchfork bifurcation, as the origin loses its stability.

<figure><img src="image_2.png" /><figcaption>Figure 1: Double homoclinic trajectory for system (2) with $\nu_1 = 1$, $\nu_2 = 4$, $\nu_3 = 1$, $h \simeq 3.99$. The trajectory and 3 projections are drawn. For only slightly greater values of $h$, the system exhibits transient chaotic dynamics. Note that the homoclinic trajectory heads back toward the origin by positive $z$ (tangent to the $z$ axis).</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2: Schematic route to chaos for the system (2). With $\nu_1 = 1$, $\nu_2 = 4$, $\nu_3 = 1$, we have $h_{pk} = 2$, $h_{hf} \simeq 5$, $h_{he} \simeq 4.8$, $h_{ho} \simeq 3.99$. As we shall see, for some other values of $\nu_1$ and $\nu_2$, $h_{ho}$ does not exist.</figcaption></figure>

If one increases $h$ further, different bifurcations occur as the motion in phase space becomes more and more complicated (see figure 2) :

* At $h = h_{ho}$, an homoclinic bifurcation takes place : the 1-D unstable manifold of the origin (tangent to the $z = 0$ plane) becomes connected with its 2-D stable manifold (see figure 1). Note that in this figure, the homoclinic trajectory heads back toward the origin through
positive z and tangent to the z axis. Considering the orientation of the 2 stable eigenvectors and the respective values of the 2 real negative eigenvalues, the finishing part of the homoclinic orbit (if there is one) will lie in the z = 0 plane for $h < \sqrt{(\nu_3 - \nu_1)(\nu_3 - \nu_2)}$ & $\nu_3 > \nu_1$ & $\nu_3 > \nu_2$ and will be tangent to the z axis otherwise. In this latter case it could well be that the homoclinic orbit reaches back to the origin by negative z, but in numerical experiments, following the orbit while changing parameters, we only saw a configuration like in figure (1) (tangency to z axis, positive z) and we conjecture that it is always the case. We believe that this bifurcation plays an important role in the dynamics of the system and that the so-called homoclinic explosions, introduced in the study of the Lorenz system in [4], occur here also <fnref n="1" />. Moreover, we assume that the chaotic motion has its source in this homoclinic bifurcation since at this point, $h = h_{ho}$, a strange invariant set (not stable) is born.

* This set becomes stable at $h = h_{he}$ when the heteroclinic bifurcation takes place : the left (resp. right) part of the 1-D unstable manifold of the origin becomes connected with the 1-D stable manifold of the limit cycle (of saddle type) surrounding the equilibrium point $M_+$ (resp. $M_-$) (see figure 3). Note that up to now, the point $M \pm$ are still stable, so we have 3 competing attractors in the phase space.

* The $M \pm$ points lose their stability in a sub-critical Hopf bifurcation at $h = h_{hf}$.

The values of $h_{ho}$, $h_{he}$ and $h_{hf}$ depend on $\nu_1$, $\nu_2$ and $\nu_3$, and this defines hyper-surfaces in the 4-D parameter space. The Hopf bifurcation equation defining $h_{hf}$ is :

$$4 \nu_1^2 \nu_2^2 + h^2((\nu_1 - \nu_2)^2 + (\nu_1 + \nu_2)\nu_3) + h(\nu_1 - \nu_2)(\nu_3 + \nu_1 + \nu_2)\sqrt{h^2 - \nu_1\nu_2} = 0 \quad (\text{with } \nu_2 > \nu_1 + \nu_3) \quad (3)$$

As for the homoclinic ($h = h_{ho}(\nu_1, \nu_2, \nu_3)$) and the heteroclinic ($h = h_{he}(\nu_1, \nu_2, \nu_3)$) bifurcation curves, we cannot calculate them analytically and so we must approximate them numerically (see figure 10). Nevertheless we will introduce algebraic bounds to the homoclinic curve in the parameter space. For $\nu_1 = 1$, $\nu_2 = 4$, $\nu_3 = 1$, we have $h_{pk} = 2$, $h_{hf} \simeq 5$. Thanks to numerical integration, we find $h_{he} \simeq 4.8$, $h_{ho} \simeq 3.99$.

<footnote n="1">¹In fact, the chaotic attractor of the PRT system (2) and the Lorenz attractor look similar. So do the routes to chaos of these two systems. Nevertheless, the PRT system has one more nonlinearity and is more symmetric in $x \leftrightarrow y$</footnote>
<figure><img src="image_4.png" /><figcaption>Figure 3: Double heteroclinic trajectory for system (2) with $\nu_1 = 1$, $\nu_2 = 4$, $\nu_3 = 1$, $h \simeq 4.8$. We have drawn 3 projections of the trajectory as well. For greater values of $h$, the system exhibits stable chaotic dynamics.</figcaption></figure>

### 3 Integrals of motion and semi-permeable surfaces

We now turn to the 'integrability information' we have on system (2). We will show how to use this information to study the chaotic features of the system. There are seven known integrals of motion for system [3]:

$$1- I_1 = (x^2 + y^2 - 4 h z) e^{2\nu t} \quad \text{when } \nu = \nu_1 = \nu_2 = \frac{\nu_3}{2}$$

$$2- I_2 = (x^2 - y^2 + 2 z^2) e^{2\nu t} \quad \text{when } \nu_1 = \nu_2 = \nu_3 = \nu$$

$$3- I_3 = (x^2 + y^2) e^{2\nu t} \quad \text{when } h = 0, \nu_1 = \nu_2 = \nu$$

$$4- I_4 = y^2 - (z + h)^2 \quad \text{when } \nu_2 = \nu_3 = 0$$

$$5- I_5 = x^2 + (z - h)^2 \quad \text{when } \nu_1 = \nu_3 = 0$$

$$6- I_6 = (y^2 + z^2) e^{2\nu t} \quad \text{when } \nu_2 = \nu_3 = \nu \text{ and } h = 0$$

$$7- I_7 = (x^2 - z^2) e^{2\nu t} \quad \text{when } \nu_1 = \nu_3 = \nu \text{ and } h = 0$$

Integrals of motion of higher degree have been searched, but none were found [3]. Thanks to a rescaling², we can set $\nu_3 = 1$ (in fact they was no $\nu_3$ in system (1), it has been introduced to enable to existence of $I_4$ and $I_5$). In [5], it has been shown that the existence of an integral of motion for certain value of the parameters generally comes together with the existence of transverse sections

$$^2(x, y, z) \rightarrow (x, y, z)/\nu_3, h \rightarrow h/\nu_3, t \rightarrow t \nu_3, \nu_{1,2} \rightarrow \nu_{1,2}/\nu_3$$
that exist for a much wider range of the parameters. These transverse sections, also called semi-permeable surfaces (in a 3D phase space they are surfaces, crossed in one way by the trajectories), yield important exact information about the asymptotic behavior of the system.

Hence the existence of the integral $I_1$ when $\nu_1 = \nu_2 = \frac{1}{2}$ leads us to seek semi-permeable surfaces with the following algebraic form:

$$R_1(x, y, z) = z - a(x^2 + y^2) - b = 0 \quad (4)$$

Surfaces $R_1$ are paraboloïds of revolution about the $z$ axis. As explained in [5, 6, 7], we compute the scalar product between the normal vector of $R_1$ and the vector field and we evaluate this scalar product on the surface $R_1 = 0$:

$$\dot{R}_{1|R_1=0}(x, y) = a(2\nu_2 - 1)y^2 + (1 - 4ah)xy + a(2\nu_1 - 1)x^2 - b \quad (5)$$

$\dot{R}_{1|R_1=0}$ is a quadratic polynomial in $y$, it has constant sign (and hence surfaces (4) are semi-permeable) in the three following cases:

$$(1A) \quad \text{when } \nu_1 > \frac{1}{2}, \nu_2 > \frac{1}{2}, h > \sqrt{(\nu_1 - \frac{1}{2})(\nu_2 - \frac{1}{2})}, \forall b \le 0,$$

$$0 < a_1 = \frac{1/4}{h + \sqrt{(\nu_1 - \frac{1}{2})(\nu_2 - \frac{1}{2})}} < a < \frac{1/4}{h - \sqrt{(\nu_1 - \frac{1}{2})(\nu_2 - \frac{1}{2})}} = a_2$$

in this case, the chaotic attractor, when it exists, is compelled to evolve above the uppermost surface (4) ($b = 0, a = a_2$), see figure (4). This case also establishes that, for these values of $\nu_1, \nu_2, h$, all the asymptotic motion (chaotic or not) takes place in the $z > 0$ half space.

(1B) $\nu_1 > \frac{1}{2}, \nu_2 > \frac{1}{2}, h < \sqrt{(\nu_1 - \frac{1}{2})(\nu_2 - \frac{1}{2})} < \sqrt{\nu_1 \nu_2}, \forall b < 0 \text{ if } a \in ]-\infty; a_2] \text{ or } \forall b > 0 \text{ if } a \in [a_1; +\infty[$. The origin $O(0, 0, 0)$ is the only equilibrium point in this case, and the semi-permeable surfaces (4) establish its asymptotic stability (i.e. all trajectories in phase space eventually stabilize on the origin).

(1C) $\nu_1 < \frac{1}{2}, \nu_2 < \frac{1}{2}, \forall (h, b \ge 0)$ and $a \in [a_1, a_2]$. The surfaces prevent any homoclinic trajectory from returning to the origin by strictly positive $z$ (see figure 5). So for these values of the parameters $\nu_1, \nu_2$ there is no homoclinic bifurcation and hence no chaotic motion $\forall h$.
<figure><img src="image_5.png" /><figcaption>Figure 4: Chaotic attractor of system (2) bounded by the upper-most semi-permeable surface (4) with $b = 0$ and $a = a_2$.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5: Semi-permeable surfaces (4) prevent the homoclinic bifurcation from taking place when $\nu_1 < \frac{1}{2}$, $\nu_2 < \frac{1}{2}$, $\forall h$. The coefficient $s$ is chosen in order that the three equilibrium points are on the projection.</figcaption></figure>
The existence of integrals of motion $I_4$ and $I_5$ lead us to propose :

$$R_2(x, y, z) = x^2(\alpha + h) + y^2(\alpha - h) + 2h(z - \alpha)^2 - \beta \quad (6)$$

Calculating the scalar product, one finds :

$$\dot{R}_{2|R_2=0}(y, z) = 2h z^2(\nu_1 - 1) + 2z \alpha h(1 - 2\nu_1) + y^2(h - \alpha)(\nu_2 - \nu_1) + \nu_1(2h\alpha^2 - \beta) \quad (7)$$

when eliminating the $x$ variable, or

$$\dot{R}_{2|R_2=0}(x, z) = 2h z^2(\nu_2 - 1) + 2z \alpha h(1 - 2\nu_2) + x^2(h + \alpha)(\nu_2 - \nu_1) + \nu_2(2h\alpha^2 - \beta) \quad (8)$$

when eliminating the $y$ variable. Depending on $\alpha$ and $\beta$, the surfaces (6) can be ellipsoïds or hyperboloïds of revolution (with $y$ or $z$ axis) with one or two sheets (see figure (6)).

Case A of figure (6) proves that $\forall (h, \nu_1, \nu_2)$ the asymptotic motion is bounded in phase space. The attractor(s) must lie inside the smallest ellipsoïd. We have thus to consider the ellipsoïd with the smallest radius ($\beta$). If we do so, we get something which still depends on $\alpha$, for example when $\nu_1 > \frac{1}{2}$ and $\nu_2 > \frac{1}{2}$ :

$$\hat{R}_2(x, y, z, \alpha) \stackrel{\text{def}}{=} R_2(\beta = 2h\alpha^2) = x^2(\alpha + h) + y^2(\alpha - h) + 2h(z - \alpha)^2 - 2h\alpha^2 = 0 \quad (9)$$

The centre of these semi-permeable ellipsoïds (9) and their size both depend on $\alpha$. So one has to consider the envelope of all the ellipsoïds (9) when $h < \alpha$, solving :

$$\left\{ \begin{array}{l} \hat{R}_2(x, y, z, \alpha) = 0 \\ \frac{d\hat{R}_2}{d\alpha}(x, y, z, \alpha) = 0. \end{array} \right. \quad (10)$$

One finds :

$$E_1 : \left\{ \begin{array}{l} 4h z = x^2 + y^2 \\ x^2 + (z - h)^2 = h^2. \end{array} \right. \quad (11)$$

This corresponds to the inner intersection of a paraboloïd and a cylinder. The cylinder is the same as the one we find in case (AB). It establishes that all asymptotic motion for $\nu_1 > \frac{1}{2}$, $\forall (\nu_2, h)$ takes place in the $z > 0$ half space. As for the parabola, it does not introduce any improvement to surface (4) with $a = a_2$ and $b = 0$.

In case B, the surfaces (6) with $\alpha = 0$ show that there can be no homoclinic bifurcation for $\nu_1 > \nu_2$ and $\nu_2 < 1$, $\forall h$ because the 1-D unstable manifold of the origin is separated from the 2-D stable manifold by the semi-permeable surfaces.
<figure><img src="image_7.png" /><figcaption>Figure 6: Shape of surfaces (6) and conditions under which they are semi-permeable. $\Delta_i = \frac{\alpha^2 h}{2\nu_i(1-\nu_i)}$ with $i = 1, 2$.</figcaption></figure>
Case (BC) shows that there can be no homoclinic bifurcation for $\nu_2 < \frac{1}{2}$, $\forall (\nu_1, h)$ for the same reason as in the case (1C).

Case C shows that there can be no homoclinic bifurcation for $\nu_1 < \frac{1}{2}$ and $\nu_2 < \frac{1}{2}$, $\forall h$ for the same reason as in the previous case.

Case D yields bounds for the chaotic attractor. Here for each $\nu_1, \nu_2$, we have to consider the surface with the smallest $\beta$. Then as we still have one free parameter $\alpha$, we calculate the envelope of the family of surfaces for $\alpha < -h$. This yields :

$$E_2 = \nu_i(\nu_i - 1)(x^2 + y^2)(8hz - (x^2 + y^2)) + 2h^2((1 - 2\nu_i)^2(x^2 - y^2) + 2z^2) = 0 \quad (12)$$

with $i = 1, 2$ and the restriction :

$$4hz - (x^2 + y^2) < -h^2 \frac{(2\nu_i - 1)^2}{\nu_i(\nu_i - 1)} < 0. \quad (13)$$

For points $(x, y, z)$ for which the inequality (13) does not hold, the closest surface from the attractor is the surface (6) with $\alpha = -h$ and $\beta = \Delta_i$. Yet better bounds are found in the next case E where the envelope (12) is to be considered with $i = 2$ and for $-h < \alpha < h$ which yields the restrictions :

$$-h^2 \frac{(2\nu_2 - 1)^2}{\nu_2(\nu_2 - 1)} < 4hz - (x^2 + y^2) \left( < h^2 \frac{(2\nu_2 - 1)^2}{\nu_2(\nu_2 - 1)} \right). \quad (14)$$

The parentheses mean that the upper inequality is of no use because the envelope (12) do not reach $4hz > (x^2 + y^2) + h^2 \frac{(2\nu_2 - 1)^2}{\nu_2(\nu_2 - 1)}$. Besides, thanks to surfaces (4), we know that the chaotic attractor lies in the zone where $z > \frac{1/4}{h - \sqrt{(\nu_1 - \frac{1}{2})(\nu_2 - \frac{1}{2})}}(x^2 + y^2) > \frac{1}{4h}(x^2 + y^2) > \frac{1}{4h}(x^2 + y^2) - \frac{h}{4} \frac{(2\nu_2 - 1)^2}{\nu_2(\nu_2 - 1)}$. Hence surface (12) with $i = 2$ in case E is a bound for the chaotic attractor with no restriction (see figure (7)).

One naturally wonders whether surface (12) is entirely semi-permeable or not (and if yes under what conditions). For $\nu_1 = \nu_2 > 1$, we can be sure the answer is yes because in this case surfaces (6) are semi-permeable both in case D and E and so is their envelope. Now if surface (12) is a semi-permeable surface, we could wonder if there is an integral of motion (with the same algebraic form) attached to it. Taking $\nu_i = 0$ or $\nu_i = 1$ in (12), we find $E_2 = 2z^2 - y^2 + x^2$ which correspond to $I_2$. And taking $\nu_i = \frac{1}{2}$, we find $E_2 = (4hz - (x^2 + y^2))^2$ which corresponds to $I_1$.

The existence of integrals of motion $I_2, I_3, I_6$ and $I_7$ lead us to propose :

$$R_3(x, y, z) = x^2 A + y^2 + (A - 1) z^2 - B \quad (15)$$
<figure><img src="image_8.png" /><figcaption>Figure 7: Chaotic attractor of system (2) with $\nu_1 = 1$, $\nu_2 = 4$, $\nu_3 = 1$ and $h = 6$ bounded by the envelope of surfaces (6) in case E defined by (12) with $i = 2$ and $z > 0$.</figcaption></figure>

The scalar product on the surface is:

$$ \dot{R}_{3|R_3=0} = x^2 A (1 - \nu_1) + (1 - \nu_2)y^2 + h(A + 1)xy - B \quad (16) $$

Depending on A and B the surfaces (15) can be ellipsoïds or hyperboloïds of revolution (with y or z axis) with one or two sheets (see figure (8)).

$$ A_{1\pm} = \frac{-(2h^2 + (\nu_1 - \nu_2)^2) \pm \sqrt{(\nu_1 - \nu_2)^2(4h^2 + (\nu_1 - \nu_2)^2)}}{2h^2} \quad (17) $$

$$ A_{2\pm} = \frac{-(2h^2 - 4(\nu_1 - 1)(\nu_2 - 1)) \pm 4\sqrt{(\nu_1 - 1)(\nu_2 - 1)((\nu_1 - 1)(\nu_2 - 1) - h^2)}}{2h^2} \quad (18) $$

$$ A_{3\pm} = \frac{-(2h^2 - 4\nu_1\nu_2) \pm \sqrt{\nu_1\nu_2 - h^2}}{2h^2} \quad (19) $$

In case F of figure (8), the semi-permeable ellipsoïds state that for these parameter values ($h^2 < \nu_1\nu_2$) the origin is asymptotically stable.

Case G is of no interest since here the origin, which is the only equilibrium point, is stable.

Case H provides a bound for the chaotic attractor (see figure 9) when $\nu_2 > 1$ and $\nu_2 > \nu_1$.

Case J proves that there is no homoclinic bifurcation (and hence no chaotic motion) for $\nu_1 > \nu_2$ and $\nu_2 < 1$ for the same reason as in case B.

In Case K, the surfaces (15) are semi-permeable when $h < \sqrt{(\nu_1 - 1)(\nu_2 - 1)}$ and this prevents the existence of an homoclinic curve tangent to the z axis in its finishing part. But for $h < \sqrt{(\nu_1 - 1)(\nu_2 - 1)}$, as we saw earlier, the finishing part of a possible homoclinic curve would not be tangent to z axis. So this case does not yield any new information.

We have drawn in figure (10), the curve $h_{hf}(\nu_1, \nu_2) = +\infty$ which is the line $\nu_2 = \nu_1 + 1$ (cf. eq. (3)):
<figure><img src="image_9.png" /><figcaption>Figure 8: Shape of the surfaces (15) and conditions for them to be semi-permeable. The values of $A_{1\pm}$, $A_{2\pm}$, $A_{3\pm}$ are given by equations (17), (18) and (19).</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 9: Semi-permeable surfaces (15) in case H ($d = \sqrt{-A_{1+}x^2 - (A_{1+}-1)z^2}$ with $A_{1+} < 0$) and chaotic attractor of system (2) with $\nu_1 = 1$, $\nu_2 = 4$, $\nu_3 = 1$ and $h = 6$.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 10: Elements of the bifurcation diagram for system (2). Note that there is chaotic motion possible under the line $\nu_2 = \nu_1 + 1$, which means that the system can be chaotic for some $h$ even if the equilibrium points remain stable for all $h$. The curves $h_{ho} = 100$ and $h_{he} = 100$ were obtained numerically by a continuation method using Mathematica.</figcaption></figure>
* For each $(\nu_1, \nu_2)$ above this line, there is a value of $h$ for which the equilibrium points $M_{\pm}$ lose their stability in a sub-critical Hopf bifurcation. There are also two different values of $h$ for which the system undergoes homoclinic and heteroclinic bifurcations.

* For $(\nu_1, \nu_2)$ under this line, the equilibrium points $M_{\pm}$ never lose their stability as $h$ is increased but there may still be values of $h$ for which the homoclinic and heteroclinic bifurcation take place.

Nevertheless we know from surfaces (4), (6) and (15) that for $(\nu_1, \nu_2)$ is the zone where $(\nu_2 < 1$ and $\nu_2 < \nu_1$) or $(\nu_2 < 1/2)$ there can be no homoclinic (nor heteroclinic) bifurcation $\forall h$. Hence in between this zone and the line $\nu_2 = \nu_1 + 1$, there must be curves for which $h_{he} = +\infty$ and $h_{ho} = +\infty$. We have drawn, thanks to numerical integration, the curves $h_{he} = 100$ and $h_{ho} = 100$ which are supposed to be very near to the '$\infty$' curves. These results on the parameter space drawn in fig. (10) can be used to understand the different behaviours of the three waves in the plasma. Our method enables us to state that for certain values of the damping decrements $\nu_1$ and $\nu_2$:

$$ (\nu_2 < 1 \cap \nu_2 < \nu_1) \cup \nu_2 < \frac{1}{2}, \qquad (20) $$

the amplitudes of the three waves will not become chaotic for any value of the pump $h$. Above this region in the plane $(\nu_1, \nu_2)$, we have to rely on numerical integration to draw a frontier $h_{he} = +\infty$ (which seems to be a line) between two regions: in the lower region (which contains our exact region (20)) the instability in the plasma will never lead to chaotic behaviour as in the upper region there will be a value of the pump $h = h_{he}$ for which the amplitudes of the waves will have a chaotic behaviour.

# 4 Bounding the Lyapunov dimension

Let us consider now the three Lyapunov exponents along the attractor: $\mu_1 > 0 \ge \mu_2 > \mu_3$. Thanks to numerical integration, we know that $\mu_1 + \mu_2 > 0$, hence the Kaplan-Yorke formula for the Lyapunov dimension reads:

$$ D_L = 2 + \frac{\mu_1 + \mu_2}{-\mu_3}. \qquad (21) $$

Using the relation $\mu_1 + \mu_2 + \mu_3 = -(\nu_1 + \nu_2 + \nu_3)$, we can write $D_L$ as:

$$ D_L = 2 + \frac{\mu_1 + \mu_2}{\mu_1 + \mu_2 + \nu_1 + \nu_2 + \nu_3}. \qquad (22) $$
An upper bound of the (positive) sum of the first two Lyapunov exponents may be calculated by considering the maximum real part of the eigenvalues of the matrix $M(t) = (\vec{\nabla} \cdot \vec{F}) \mathbb{I} - L(t)$, where $L(t)$ is the Jacobian matrix of the vector field and $\mathbb{I}$ is the 3-D identity matrix [8]. For system (2), $\mu_1 + \mu_2$ is bounded by the maximum real part of the eigenvalues of :

$$ M(t) = \begin{pmatrix} -\nu_2 - \nu_3 & -h + z(t) & y(t) \\ -h - z(t) & -\nu_1 - \nu_3 & -x(t) \\ -y(t) & -x(t) & -\nu_1 - \nu_2 \end{pmatrix} \quad (23) $$

At first sight, this trick seems to be of no help since one still needs numerical integration to evaluate the eigenvalues of the matrix $M(t)$. But if we consider that the values of $x(t)$, $y(t)$, $z(t)$ on the chaotic attractor are bounded (thanks to the semi-permeable surfaces), we may bound the eigenvalues of $M(t)$. The set of points $Z$, which lie above surface (4) with $b = 0$ and $a = a_2$, inside surface (6) in case (AB), above surface (12) with $i = 2$ and outside of the cone defined by surface (15) in case H, is a rather tight bound for the chaotic attractor :

$$ (x, y, z) \in Z \quad \text{iff} : \quad z \ge \frac{1}{4} \frac{x^2 + y^2}{h - \sqrt{(\nu_1 - \frac{1}{2})(\nu_2 - \frac{1}{2})}} \quad (24) $$

$$ h^2 \ge x^2 + (z - h)^2 \quad (25) $$

$$ 0 \le \nu_2(\nu_2 - 1)(x^2 + y^2)(8hz - (x^2 + y^2)) + 2h^2((1 - 2\nu_2)^2(x^2 - y^2) + 2z^2) \quad (26) $$

$$ 0 \ge A_{1+}x^2 + y^2 + (A_{1+} - 1)z^2 \quad (27) $$

Setting $\nu_1 = 1, \nu_2 = 4, \nu_3 = 1, h = 6$ and looking for the maximum real part of the eigenvalues of $M(t)$ for $(x, y, z) \in Z$ (for these values of parameters, $A_{1+} \simeq -0.6$), we found (unlike in [9]) that the largest real part is realized for $x = y \simeq 5.82, z \simeq 4.8$. Hence one finds that $\mu_1+\mu_2 \lesssim 4.31$ which yields $D_L \lesssim 2.418$. Numerical integration yields $\mu_1 \simeq 0.39, \mu_2 \simeq -0.001, \mu_3 \simeq -6.39 : D_L \simeq 2.061$.

# 5 Conclusion

In this work, we have studied the dynamics of a 3D dissipative system which arises in the study of a parametric instability in a plasma.

We have established that the analytic information we have on the integrability of the system can be used to get information on the chaotic dynamics of this system. More specifically, we have shown that one can use the algebraic form of the integrals of motion (existing for specific parameters values) to bound the chaotic attractor in phase space and to bound the chaotic dynamics in
the parameter space (by introducing analytic bounds to the homoclinic bifurcation curves). These
results enable us to give information on the range of parameters for which the instability can lead
to chaos.

We have also shown that one can use the geometric bounds introduced for the chaotic attractor
to derive an upper bound for its Lyapunov dimension. We believe that this method can be used
on any system with a constant divergence, regardless of its dimension.

# 6 Acknowledgement

The author would like to thank Y. Elskens for useful comments.

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