3.1 Some Mathematical Properties

This section discusses some basic dynamical properties of Chen's equation (7). This system has the same complexity as the Lorenz equation -- they are both three-dimensional autonomous with only two quadratic terms. However, it is topologically not equivalent to the Lorenz equation. This can be rigorously proved by showing that a homeomorphism of the form

$$\left\{\begin{array}{l} x_C = \xi(x_L,y_L,z_L)\\ [3pt] y_C =... ...L,y_L,z_L)\\ [3pt] z_C = \zeta(x_L,y_L,z_L) \end{array}\right.$$ (8)

does not exist, where $(x_L,y_L,z_L)$ and $(x_C,y_C,z_C)$ are the variables of the Lorenz and Chen equations, respectively. The proof can be carried out by taking a time derivative on (8) and then performing some straightforward but tedious algebra, which leads to a system of algebraic equations without solutions.

It is easy to see that this equation has a trivial property of symmetry: the equation is invariant under the following transformation:

$$(x,y,z) \rightarrow (-x,-y,z)$$ (9)

Then, let us rewrite Eq. (7) in the following vector form:

$$\frac{d \mbox{\boldmath$ x $}}{dt} = \mbox{\boldmath$ f $}(\mbox{\boldmath$ x $})$$ (10)

where $\mbox{\boldmath$\ x $} = (x, y, z)\in \mbox{\boldmath$\ R $}^3$, and $\mbox{\boldmath$\ f $}(\mbox{\boldmath$\ x $})$ is the right-hand side of Eq. (7). The transformation (9) can now be denoted as

$$P\!\!: \mbox{\boldmath$ R $}^3 \rightarrow \mbox{\boldmath$ ... ...}{ccc} 0 & -1 & 0\\ -1 & 0 & 0\\ 0 & 0 & 1 \end{array} \right)$$ (11)

which satisfies

$$\mbox{\boldmath$ f $}(P\mbox{\boldmath$ x $}) = P\mbox{\boldmath$ f $}(\mbox{\boldmath$ x $})$$ (12)

The dynamics of the system is affected by this symmetry. For example, in the projection of its orbits onto the phase plane, variable $x$ and $y$ are symmetric with respect to the origin. Besides, this hints that pitchfork bifurcation for equilibria and periodic solutions are possible.

Let us examine the stability of Chen's system. It is easy to verify that the system is globally, uniformly and asymptotically stable about its zero equilibrium if $c<0<a$. This can be checked by using the Lyapunov function

$$V(x,y,z) = \frac{a-c}{2a}\, x^2 + y^2 + z^2$$

which gives

$$\dot V = -\, (a-c) x^2 + c\, y^2 - b z^2 < 0$$

Next, let us consider a volume in a certain domain $D_0$ of the state space. Notice that

$$\mathop{\rm div}\nolimits \mbox{\boldmath$ f $}(\mbox{\boldmath$ x $}) = -a - b + c.$$ (13)

We always use a set of parameters satisfying $-a-b+c < 0$, such as $(a,b,c)=(35,3,28)$ [Chen & Ueta, 1999]. This means that the dynamical system (7) is guaranteed to be dissipative, so the volume of any attractor of the system must be zero. The orbit flows into a certain bounded region as $t\rightarrow\infty$.