*3.4. Type of Equilibrium Points*

Recall the following theorem:

**Theorem 1** (see, e.g., [36])**.** *Let <sup>A</sup>*(*t*)=[*aij*(*t*)] <sup>∈</sup> <sup>R</sup>*n*×*<sup>n</sup> be a continuous matrix-valued function on an interval <sup>I</sup> (i.e., each aij*(*t*) *is a real-valued continuous function on I). Let <sup>B</sup>*(*t*) <sup>∈</sup> <sup>R</sup>*<sup>n</sup> be a continuous vector-valued function on I. Then the following initial value problem*

$$X'(t) = A(t)X(t) + B(t), \quad X(0) = X\_{0\prime}$$

*has a unique solution X*(*t*) <sup>∈</sup> <sup>R</sup>*<sup>n</sup> on the interval I.*

This theorem guarantees the Equation (7) has a unique solution *X*(*t*) on any time interval (note that, in this case, *A* is a constant matrix). Thus in all cases of *x*, given an initial point (*x*(0), *y*(0), *z*(0)), the trajectory of (*x*(*t*), *y*(*t*), *z*(*t*)) is uniquely determined. The trajectory of (*x*(*t*), *y*(*t*), *z*(*t*)) in a neighborhood of each equilibrium point depends on the signs of the real/imaginary parts of the eigenvalues of the coefficient matrix *A*.

For Case 1: −1 < *x* < 1, we have the characteristic equation

$$\det\left(\lambda I - A\right) = \lambda^3 + \lambda^2 + 1 - m\_0 = 0.$$

Since all parameters of the equation are real and the equation degree is odd, we have that a root (says *λ*1) is real and other roots are a conjugate pair of complex numbers. Note that *m*<sup>0</sup> < 0 from Figure 6. Now, the product of all roots (eigenvalues) satisfies

$$
\lambda\_1 \lambda\_2 \lambda\_3 = m\_0 - 1 < \ 0.
$$

Since (*λ*2, *λ*3) is a complex conjugate pair, the real root *λ*<sup>1</sup> must be negative. Write *λ*<sup>2</sup> = *a* + *ib* and *<sup>λ</sup>*<sup>3</sup> <sup>=</sup> *<sup>a</sup>* <sup>−</sup> *ib*, where *<sup>a</sup>*, *<sup>b</sup>* <sup>∈</sup> <sup>R</sup>. Since the sum of products of two roots of the cubic equation satisfies

$$
\lambda\_1 \lambda\_2 + \lambda\_2 \lambda\_3 + \lambda\_3 \lambda\_1 = 1,
$$

we get

$$a^2 + 2\lambda\_1 a + b^2 = 1.$$

Solving this quadratic equation to obtain

$$a = -\lambda\_1 \pm \sqrt{\lambda\_1^2 - b^2 - 1}.$$

Since *<sup>λ</sup>*<sup>1</sup> <sup>&</sup>lt; 0 and *λ*2 <sup>1</sup> − *b*<sup>2</sup> − 1 < |*λ*1|, we get *a* > 0. Hence, an eigenvalue is a negative real and two other eigenvalues are a conjugate pair of complex numbers having positive real parts. Therefore, this equilibrium is a saddle focus, and the trajectory of (*x*(*t*), *y*(*t*)) diverges in a spiral form, but *z*(*t*) converges to the equilibrium point for any initial point (*x*(0), *y*(0), *z*(0)).

For Cases 2 and 3, the Jacobian matrices are the same and we have the characteristic equation

$$\det\left(\lambda I - A\right) = \lambda^5 + \lambda^2 + 1 - m\_1 = 0.$$

Since *m*<sup>1</sup> < 0 (from Figure 6), we obtain the same conclusion as in Case 1, i.e., the equilibrium point is a saddle focus.

We summarize the above discussion in the following theorem:

**Theorem 2.** *The system* (6) *has three equilibrium points, each of which is of type saddle focus. Moreover, the trajectory of* (*x*(*t*), *y*(*t*)) *diverges in a spiral form, but z*(*t*) *converges to the equilibrium point for any initial point* (*x*(0), *y*(0), *z*(0))*.*

Since the equilibrium points are saddle foci, our system has chaotic behavior.
