*3.3. Amplitude Control*

Besides the above two control knobs, the parameter *m* in the third dimension in system (2) is a single non-bifurcation knob for amplitude control. To understand this rescaling mechanism, we turn back to the initial system (2). Here, we take the transformation: *x* → *hx*, *y* → *hy*, *z* → *z*, *u* → *hu*(*h* > 0), which only leaves an additional coefficient in the third dimension:

$$\begin{cases} \dot{\mathbf{x}} = \ -ay - \mathbf{x}\mathbf{z} - \mathbf{u},\\ \dot{\mathbf{y}} = \ -\mathbf{c}\mathbf{x} + \mathbf{x}\mathbf{z},\\ \dot{\mathbf{z}} = \ -b - mh^2 \mathbf{x}y,\\ \dot{\mathbf{u}} = \ k\mathbf{x} - \mathbf{y}. \end{cases} \tag{5}$$

indicating that the amplitude of variables *x*, *y* and *u* can be controlled by the parameter *m*, with the signal *z* unchanged. It also has no effect on the frequency of the hyperchaotic chaotic signals.

The output signals are controlled by the non-bifurcation parameter *m* in system (2). As shown in Figure 9, the amplitude of the signals *x*, *y* and *u* are rescaled by the non-bifurcation parameter *m*. When *m* = 0.25, the amplitudes of the *x*, *y* and *u* signals are very large. The amplitudes of the *x*, *y* and *u* signals decrease with an inverse proportion to the parameter *m* without changing the amplitude of *z*. Figure 10 shows the phase trajectories on the planes of *x*-*u* and *y*-*z* when the control parameter *m* varies.

**Figure 9.** Rescaled variables in system (2) with *a* = 5, *b* = 4, *c* = 1, *k* = 0.5 under initial condition [1, −1, −1, 1]: (**a**) signal *x*(t), (**b**) signal *y*(t), (**c**) signal *u*(t), (**d**) signal *z*(t).

**Figure 10.** Phase trajectories of system (2) with *a* = 5, *b* = 4, *c* = 1, *k* = 0.5 under initial condition [1, −1, −1, 1]: (**a**) *x*-*u*, (**b**) *y*-*z*.

As we can see in Figure 11a, when the parameter *m* varies in [0, 5], the average of the absolute values of state variables *x, y* and *u* significantly decreases with an inverse proportion to *m*, while the average of signal *z* basically has no change. The corresponding Lyapunov exponent spectrum of parameter *m* varies in [0, 5] are shown in Figure 11b. It can be further proved that the parameter *m* of system (2) does not change the frequency of the signals.

**Figure 11.** Dynamical evolution of system (2) with *a* = 5, *b* = 4, *c* = 1, *k* = 0.5 and initial condition [1, −1, −1, 1]: (**a**) average values of the absolute value of chaotic signals, (**b**) invariable Lyapunov exponents.

### *3.4. O*ff*set Boosting*

Since the derivative of a constant is zero, when a constant is added to a variable in a dynamical system, the system exhibits the same dynamics. To understand this, we turn back to the initial system (2). Here, we take the transformation: *u* → *u* − *n*, which does not change the system equation but only leaves an additional constant in the first equation:

$$\begin{cases} \dot{\mathbf{x}} = \ -ay - \mathbf{x}z - \mathbf{u} + \mathbf{n}\_{\prime} \\ \dot{\mathbf{y}} = \ -c\mathbf{x} + xz\_{\prime} \\ \dot{\mathbf{z}} = \ -b - mxy\_{\prime} \\ \dot{\mathbf{u}} = k\mathbf{x} - y. \end{cases} \tag{6}$$

When changing the variable *u* with *u* − *n* (*n* is a constant), system (2) gives the same dynamics. Therefore, if the variable *u* does not show in the other equations in system (2), the introduced constant will give a boosting control of the variable *u.* The chaotic signal *u*(*t*) can be revised from unipolar to bipolar or vice versa.

When *a* = 5, *b* = 4, *c* = 1, *k* = 0.5 and *m* = 1, the signal *u* is boosted from a bipolar to a unipolar one, which is indicated by the red and blue attractors in Figure 12a. The waveform of chaotic signal *u*(*t*) is shown in Figure 12b. The change of parameter *n* causes the up and down translation of the signal *u*(*t*). Some monostable systems have relatively large areas of basins of attraction; therefore, the initial conditions do not need to modify according to the variable which makes the offset control simpler, as shown in Figure 13.

**Figure 12.** Typical chaotic oscillation of system (6) with *a* = 5, *b* = 4, *c* = 1, *k* = 0.5, *m* = 1 under initial condition [1, −1, −1, 1]: (**a**) phase trajectory in the plane of *x-u*, (**b**) waveform *u*(t).

**Figure 13.** Dynamical evolution of system (6) with *a* = 5, *b* = 4, *c* = 1, *k* = 0.5, *m* = 1 under initial conditions [1, −1, −1, 1]: (**a**) Lyapunov exponent spectra of *n*, (**b**) average values of the hyperchaotic signal.

Here the offset of the variable *u* is boosted along the *u*-axis according to the constant *n*. When *n* is positive, *u* is moved in the positive direction, and negative *n* causes the opposite direction. When the boosting controller *n* is changed from −30 to 30, system (6) has the same Lyapunov exponents, which is shown in Figure 13. The average value of variable *u* changes linearly with the increase of parameter *n*, while others remain unchanged.
