**3. Constructing Conditional Symmetry from Symmetry**

Interestingly, a symmetric structure also gives the chance for hosting an offset-boosting-assisted polarity balance and leading to conditional symmetry. Taking the diffusionless Lorenz system [21,22], for example,

$$\begin{cases} \dot{\mathbf{x}} = \mathbf{y} - \mathbf{x} + \mathbf{n}, \\ \dot{\mathbf{y}} = -\mathbf{x}\mathbf{z} + \mathbf{m}, \\ \dot{\mathbf{z}} = \mathbf{x}\mathbf{y} - \mathbf{R}. \end{cases} \tag{1}$$

where the parameters *m* and *n* are introduced for later discussion. When *m* = *n* = 0, *R* = 1, the system has a chaotic attractor with Lyapunov exponents (0.2101, 0, −1.2101) and a corresponding Kaplan–Yorke dimension DKY = 2.1736 under initial conditions (−1, 0, −1). In this work, for obtaining representative Lyapunov exponents rather than absolute ones [23–25], all the finite-time Lyapunov exponents (LEs) are computed for the time interval [0, 107] for the initial points on the attractor based on the Wolf algorithm. It is a simple matter to determine the Kaplan–Yorke dimension from the spectrum of Lyapunov exponents by *k* + (LE1 + ... + LE*k*)/|LE*k*+1| (here LE1 + ... + LEk ≥ 0, and LE1 + ... + LEk<sup>+</sup><sup>1</sup> ≤ 0). System (1) is of rotational symmetry since the system is invariant under the transformation (*x*, *y*, *z*) → (−*x*, −*y*, *z*) when *m* = *n* = 0, corresponding to a 180◦ rotation about the *z*-axis. In this case, system (1) has a symmetric oscillation or a symmetric pairs of twin attractors under different initial condition (IC), as shown in Figure 3.

**Figure 3.** Symmetric attractor or symmetric pairs of attractors of system (1) with *m* = *n* = 0, IC = (1, 1, 1) is red and IC = (1, −1, 1) is green: (**a**) *R* =1, (**b**) *R* =4.9, (**c**) *R* = 5.2, (**d**) *R* = 5.4.

Taking a further function introducing,

$$\begin{cases} \dot{\mathbf{x}} = F(\mathbf{y}) - \mathbf{x} + \mathbf{n}, \\ \dot{\mathbf{y}} = -\mathbf{x}G(\mathbf{z}) + m, \\ \dot{\mathbf{z}} = \mathbf{x}F(\mathbf{y}) - \mathbf{R}. \end{cases} \tag{2}$$

where *<sup>F</sup>*(*y*) = *y* <sup>−</sup> 6, *<sup>G</sup>*(*z*) <sup>=</sup> <sup>|</sup>*z*<sup>|</sup> <sup>−</sup> 8, *<sup>m</sup>* <sup>=</sup> *<sup>n</sup>* <sup>=</sup> 0, *<sup>R</sup>* <sup>=</sup> 1, system (2) gives birth to twin coexisting attractors of conditional symmetry, as shown in Figure 4. Compared with the rotational symmetry with system (1), system (2) is of conditional reflection symmetry since it is invariant under the transformation (*x*, *y*, *z*) → (−*x*, *y* + *c*1, *z* + *c*2) (*c*1, *c*<sup>2</sup> stand for calling a polarity reverse from the absolute value function). We can compare these twin attractors; each one is symmetrically different from the above cases.

**Figure 4.** Coexisting twin attractors of system (2) with *<sup>F</sup>*(*y*) <sup>=</sup> *y* <sup>−</sup> 6, *<sup>G</sup>*(*z*) <sup>=</sup> <sup>|</sup>*z*<sup>|</sup> <sup>−</sup> 8, *<sup>m</sup>* <sup>=</sup> *<sup>n</sup>* <sup>=</sup> 0, *<sup>R</sup>* <sup>=</sup> 1, IC = (1, 7, 9) is red, and IC = (−1, −6, −7) is green.

#### **4. Recovering Conditional Symmetry from Destroyed Symmetry**

#### *4.1. Symmetry Destroyed by the Constant Planting*

For observing the effect to conditional symmetry owing to the symmetric structure, two additional constants are introduced in the diffusionless Lorenz system. The constant term, like a polarity fire extinguisher, revises the polarity balance. As shown in Figures 5 and 6, when *m* and *n* vary, system (1) switches between symmetric attractors and asymmetric ones for the compound structure with Lorenz

attractor. Note that any constant *m* or *n* removes the polarity balance, which identifies that system (1) loses symmetry when *m* - 0, or *n* - 0. However, for system (2), the situation is different. If *m* = 0, *n* - 0, system (2) does not keep conditional symmetry. However, if *n* = 0, *m* - 0, system (2) maintains conditional symmetry, giving two coexisting bifurcations, as shown in Figure 7. Unlike the attractors shown in Figures 3 and 4, now all the coexisting attractors of conditional symmetry reside in the asymmetric structure, as shown in Figure 8. Two typical pairs of chaotic signals are shown in Figure 9, where the signals lose symmetry but stand steadily in the form of conditional symmetry.

**Figure 5.** Dynamical evolvement in system (1) with *n* = 0, *R* = 1 and initial conditions (1, 1, 1): (**a**) Lyapunov exponents (LEs), and (**b**) bifurcation diagram.

**Figure 6.** Dynamical evolvement in system (1) with *m* = 0, *R* = 1 and initial conditions (1, 1, 1): (**a**) Lyapunov exponents, and (**b**) bifurcation diagram.

**Figure 7.** Dynamical evolvement in system (2) with *n* = 0, *R* = 1: (**a**) Lyapunov exponents, and (**b**) bifurcation diagram.

**Figure 8.** Conditional symmetric pairs of attractors in system (2) with *n* = 0, *R* = 1, IC = (1, 7, 9) is red and (−1, −6, −7) is green: (**a**) *m* = 0.25, (**b**) *m* = 0.45, (**c**) *m* = 0.55, (**d**) *m* = 0.7.

**Figure 9.** Conditional symmetric pairs of signals in system (2) with *n* = 0, *R* = 1, IC = (1, 7, 9) is red and (−1, −6, −7) is green: (**a**) *m* = 0.25, (**b**) *m* = 0.45.

#### *4.2. Symmetry Evolution Induced by the Dimension Growth*

The influence of dimension growth to polarity balance is complicated, some of which may preserve or destroy the polarity balance of the original system. Taking the following system, for example,

$$\begin{cases} \dot{\mathbf{x}} = \mathbf{y} - \mathbf{x}, \\ \dot{y} = -\mathbf{x}z, \\ \dot{z} = \mathbf{x}y - \mathbf{R} + a\mathbf{x}u, \\ \dot{u} = b\mathbf{x}. \end{cases} \tag{3}$$

In this case, system (3) is still symmetric, since it is invariant under the transformation (*x*, *y*, *z*, *u*) → (−*x*, −*y*, *z*, −*u*). Now system (3) has a symmetric chaotic attractor with Lyapunov exponents (0.2609, 0, −0.0079, −1.2530) and corresponding Kaplan–Yorke dimension DKY = 3.2019, is shown in Figure 10.

**Figure 10.** Symmetric attractor of system (3) with *a* = 0.5, *b* = 0.1, *R* = 3, IC = (1, 1, 1, 2) is red, IC = (−1, −1, 1, −2) is green: (**a**) *x-y-z* space, (**b**) *x-z-u* space.

System (3) is also a seed system for hosting conditional symmetry,

$$\begin{cases} \dot{\mathbf{x}} = F(y) - \mathbf{x}, \\ \dot{y} = -\mathbf{x}G(z), \\ \dot{z} = \mathbf{x}F(y) - \mathbf{R} + a\mathbf{x}u, \\ \dot{u} = b\mathbf{x}. \end{cases} \tag{4}$$

where *<sup>F</sup>*(*y*) = *y* <sup>−</sup> 15, *<sup>G</sup>*(*z*) <sup>=</sup> <sup>|</sup>*z*<sup>|</sup> <sup>−</sup> 15, *<sup>a</sup>* <sup>=</sup> 0.5, *<sup>b</sup>* <sup>=</sup> 0.1, *<sup>R</sup>* <sup>=</sup> 3; system (4) gives birth to twin coexisting attractors of conditional symmetry, as shown in Figure 11. System (4) is of conditional rotational symmetry since it is invariant under the transformation (*x*, *y*, *z*, *u*) → (–*x*, *y*+*c*1, *z*+*c*2, –*u*) (*c*1, *c*<sup>2</sup> stand for calling a polarity reverse from the absolute value function).

**Figure 11.** Coexisting conditional symmetric attractors in system (4) with *<sup>F</sup>*(*y*) <sup>=</sup> *y* −15, *<sup>G</sup>*(*z*) <sup>=</sup> <sup>|</sup>*z*|−15, *a* = 0.5, *b* = 0.1, *R* = 3, IC = (1, 16, 16, 2) is red, IC = (−1, −14, −14, −2) is green.

The dimension growth sometimes changes the polarity balance of the original system.

$$\begin{cases} \dot{\mathbf{x}} = \mathbf{y} - \mathbf{x} - a \mathbf{x}u, \\ \dot{y} = -\mathbf{x}z, \\ \dot{z} = \mathbf{x}y - \mathbf{R}, \\ \dot{u} = b\mathbf{x}. \end{cases} \tag{5}$$

System (5) becomes asymmetric since it is changed under the polarity transformation. When *a* = 0.1, *b* = 0.1, *R* = 3, system (5) has chaotic attractor with Lyapunov exponents (0.0432, 0, −0.1083, −2.8978) and corresponding Kaplan–Yorke dimension *D*KY = 2.3989 under initial conditions (1, 1, 1, 2). Interestingly, this time the variable *u* is positive, and therefore the absolute value symbol of *u* can be introduced for hatching coexisting attractors, as shown in Figure 12.

$$\begin{cases} \dot{\mathbf{x}} = \mathbf{y} - \mathbf{x} - ax \|\mathbf{u}\|,\\ \dot{\mathbf{y}} = -\mathbf{x}z, \\ \dot{z} = xy - \mathbf{R}, \\ \dot{u} = b\mathbf{x}. \end{cases} \tag{6}$$

where *a* = 0.1, *b* = 0.1, *R* = 3; system (6) has a symmetric pair of coexisting chaotic attractors. Interestingly, here these coexisting attractors are unlike the cases shown in reference [9]. In the fourth dimension of system (6), the polarity balance is recovered by the out variable *x* rather than by an extra imported signum function.

**Figure 12.** Symmetric attractor of system (6) with *a* = 0.5, *b* = 0.1, *R* = 3, IC = (1, 1, 1, 2) is red, IC = (−1, −1, 1, −2) is green.

Furthermore, based on the above case, the dimension growth also leaves the possibility for hosting conditional-symmetry-like coexisting attractors. Taking a further function introducing to system (6).

$$\begin{cases} \dot{\mathbf{x}} = F(\mathbf{y}) - \mathbf{x} - \alpha \mathbf{x} |H(\boldsymbol{u})|\_{\prime} \\ \dot{\mathbf{y}} = -\mathbf{x} \mathbf{G}(\boldsymbol{z}), \\ \dot{\mathbf{z}} = \mathbf{x} F(\boldsymbol{y}) - \mathbf{R}, \\ \dot{\boldsymbol{u}} = \boldsymbol{b} \mathbf{x}. \end{cases} \tag{7}$$

where *<sup>F</sup>*(*y*) = *y* <sup>−</sup> 15, *<sup>G</sup>*(*z*) <sup>=</sup> <sup>|</sup>*z*<sup>|</sup> <sup>−</sup> 15, *<sup>H</sup>*(*u*) <sup>=</sup> <sup>|</sup>*u*<sup>|</sup> <sup>−</sup> 10, *<sup>a</sup>* <sup>=</sup> *<sup>b</sup>* <sup>=</sup> 0.1, *<sup>R</sup>* <sup>=</sup> 3; system (7) gives birth to twin coexisting attractors, which have the features of conditional symmetry, as shown in Figure 13. However, system (7) is not of conditional rotational symmetry since it seems not invariant under the transformation (*x*, *y*, *z*, *u*) → (−*x*, *y* + *c*1, *z* + *c*2, *u* + *c*3) (*c*1, *c*2, *c*<sup>3</sup> stand for calling a polarity reverse from the absolute value function). The mechanism of the coexistence of attractors hides in the same balance ability from the structure (6).

**Figure 13.** Coexisting attractors in systems (7) with *<sup>F</sup>*(*y*) <sup>=</sup> *y* <sup>−</sup> 15, *<sup>G</sup>*(*z*) <sup>=</sup> <sup>|</sup>*z*<sup>|</sup> <sup>−</sup> 15, *<sup>H</sup>*(*u*) <sup>=</sup> <sup>|</sup>*u*<sup>|</sup> <sup>−</sup> 10, *a* = *b* = 0.1, *R* = 3, IC = (1, 16, 16, 11) is red, IC = (−1, −14, −14, −10) is green.

## **5. Conclusions**

Conditional symmetry is a more flexible symmetry, which can be derived from both symmetry and asymmetry. In fact, in the physical world symmetric structure is prone to be destroyed by a newly introduced constant or by the dimension growth. However, asymmetric systems have enough space for conditional symmetry if the offset-boosting assisted polarity balance is established. Conditional symmetric systems are more promising than symmetric ones, which have reliable twin attractors rather than a broken butterfly. In those chaos-based communications, conditional symmetry or symmetry usually indicates that the corresponding system has double monopolar chaotic signals, which meets the needs of engineering application to a large extent.

**Author Contributions:** Conceptualization, C.L.; Data curation, C.L. and J.S.; formal analysis, T.L. (Tianai Lu); funding acquisition, C.L.; investigation, C.L., J.S., T.L. (Tianai Lu) and T.L. (Tengfei Lei); methodology, C.L.; project administration, C.L.; resources, C.L.; software, T.L. (Tianai Lu); supervision, C.L.; validation, C.L. and T.L. (Tengfei Lei); visualization, J.S.; writing—original draft, C.L.; writing—review and editing, C.L. and T.L. (Tengfei Lei). All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported financially by the National Natural Science Foundation of China (Grant No.: 61871230), the Natural Science Foundation of Jiangsu Province (Grant No.: BK20181410), and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

**Conflicts of Interest:** The authors declare no conflict of interest.
