*3.2. Bifurcation Analysis*

For system (2), the parameters modify the dynamics effectively. To make the demonstration simpler, we ignore the multistability caused by the special structure of symmetry. When *b* = 4, *c* = 1, *k* = 0.5, *m* = 1 under initial conditions (1, −1, −1, 1), Lyapunov exponent spectra and bifurcation diagram when *a* varies in [−10, 23.4] are shown in Figure 4, where a typical transition from period to chaos shows up and finally system (1) results in the state of hyperchaos. Typical phase trajectories are shown in Figure 5. Quasi-periodicity was not found in the examination interval of system (2). When *a* = 5, *c* = 1, *k* = 0.5, *m* = 1 and initial conditions are (1, −1, −1, 1), when *b* varies in [0, 15], system (2) heads to hyperchaos from chaos. Lyapunov exponent spectra and bifurcation diagrams are shown in Figure 6, which shows a robust hyperchaos. Both cases have almost linearly scaled Lyapunov exponents in specific regions indicating the function of frequency rescaling.

**Figure 4.** Dynamical behavior of system (2) with *b* = 4, *c* = 1, *k* = 0.5, *m* = 1 under initial conditions [1, −1, −1, 1]: (**a**) Lyapunov exponents, (**b**) bifurcation diagram.

**Figure 5.** Typical phase trajectories of system (2) with *b* = 4, *c* = 1, *k* = 0.5, *m* = 1 under initial condition [1, −1, −1, 1] in the plane *x-u*: (a) *a* = −5 (period), (**b**) *a* = −0.6 (chaos), (**c**) *a* = 3 (chaos), (**d**) *a* = 5 (hyperchaos).

**Figure 6.** Dynamical behavior of system (2) with *a* = 5, *c* = 1, *k* = 0.5, *m* = 1under initial condition [1, −1, −1, 1]: (**a**) Lyapunov exponents, (**b**) bifurcation diagram.

Comparing Figures 4 and 6, we can see that the parameter *a* or *b* visits chaos quickly but modifies the solution in its own way. The parameter *a* almost has positive correlation with amplitude in a limited range. Meanwhile parameter *b* has positive correlation with amplitude and frequency, which is distinct and different from other systems. Typical phase trajectories and waveforms are shown in Figure 7.

**Figure 7.** Chaotic oscillations of system (2) with *c* = 1, *k* = 0.5, *m* = 1 under initial condition [1, −1, −1, 1]: (**a**) phase trajectory in *x*-*z* (*b* = 4), (**b**) signal *x*(*t*), (**c**) phase trajectory in *y*-*u* plane (*a* = 5), (**d**) signal *y*(*t*).

Fix the parameters *a* = 5, *b* = 4, *k* = 0.5, *m* = 1, when parameter *c* varies in [0, 1.7]; the Lyapunov exponent spectra and bifurcation diagram are shown in Figure 8a,b. When *c* varies in [0, 1.4], system (2) exhibits hyperchaos, while when *c* varies in [1.4, 1.7], system (2) presents chaos. When *a* = 5, *b* = 4, *c* = 1 and *m* = 1, the parameter *k* varies in [0.15, 7.8]; the Lyapunov exponent spectra and bifurcation

diagram are shown in Figure 8c,d. When *k* varies in [0.15, 1.82], system (2) keeps chaos, and when *c* varies in [1.82, 7.8], system (2) exhibits hyperchaos. Comparing the Lyapunov exponents controlled by parameters *c* and *k*, system (2) has relatively robust hyperchaos under the parameters *c*.

**Figure 8.** Dynamical behavior of system (2) with *a* = 5, *b* = 4, *m* = 1 under initial conditions [1, −1, −1, 1]: (**a**,**b**): Lyapunov exponents and bifurcation diagram of *c* when *k* = 0.5, (**c**,**d**): Lyapunov exponents and bifurcation diagram of *k* when *c* = 1.
