**1. Introduction**

The system structure is a fundamental topological constraint to the dynamical evolution, which determines how the attractor stretches in phase space. Symmetric systems give birth to attractors with a symmetrical face [1–5]. When symmetry is broken, the attractor splits into a symmetric pair of attractors [6–8] or is preserved by doubling coexisting attractors [9]. Asymmetric systems seem to give a single asymmetric attractor in most cases, although sometimes it hatches coexisting asymmetric attractors [10–14] under a set of combined parameters. However, many asymmetric systems have coexisting attractors of conditional symmetry with the new polarity balance from the offset boosting.

Furthermore, symmetric structure does not reject conditional symmetry. In this paper, the symmetry evolution in chaotic systems is analyzed, as shown in Figure 1. From the start of the variable polarity reversal, if a dynamical system can establish its own polarity balance from itself, the system is symmetric, or else losing the polarity balance indicates the asymmetric structure. If a system recovers its polarity balance from a step with offset boosting, the derived system is of conditional symmetry. From this observation, we can conclude that a system, whether it is symmetric or asymmetric, can be transformed to be of conditional symmetry. In Section 2, the early proposed chaotic systems of conditional symmetry are collected. In Section 3, conditional symmetry is coined in a symmetric system. In Section 4, the collapse of polarity balance is thoroughly explored in two directions, one of which is from the constant planting, and the other of which is from the dimension growth. Conditional symmetry is therefore in the primary road where the offset-boosting-induced polarity balance is well preserved. The conclusion is given in the last section.

**Figure 1.** Relationship among symmetry, asymmetry and conditional symmetry.

## **2. Conditional Symmetry from Asymmetry**

As we know, for a dynamical system . *<sup>X</sup>* = *<sup>F</sup>*(*X*) = (*f*1(*X*), *<sup>f</sup>*2(*X*), ... , *fN*(*X*))*T*, (*<sup>X</sup>* = (*x*1, *x*2, ... , *xN*) *<sup>T</sup>*), if there exists a variable substitution *ui*<sup>1</sup> <sup>=</sup> <sup>−</sup>*xi*<sup>1</sup> , *ui*<sup>2</sup> <sup>=</sup> <sup>−</sup>*xi*<sup>2</sup> , ··· , *uik* <sup>=</sup> <sup>−</sup>*xik* , *ui* <sup>=</sup> *xi*, (here 1 <sup>≤</sup> *<sup>i</sup>*1, ··· , *ik* <sup>≤</sup> *<sup>N</sup>*, *<sup>i</sup>*1, ··· , *ik* are not identical, *<sup>i</sup>* <sup>∈</sup> {1, 2, ... , *<sup>N</sup>*}\{*i*1, ··· , *ik*}) satisfying . *U* = *F*(*U*) (*<sup>U</sup>* <sup>=</sup> (*u*1, *<sup>u</sup>*2, ... , *uN*)), then the system . *X* = *F*(*X*) (*X* = (*x*1, *x*2, ... , *xN*)) is symmetric. Conditional symmetry is a new terminology to describe the polarity balance from offset boosting [15–18]. For a differential dynamical system, . *<sup>X</sup>* = *<sup>F</sup>*(*X*) = (*f*1(*X*), *<sup>f</sup>*2(*X*), ... , *fN*(*X*))*T*, (*<sup>X</sup>* = (*x*1, *<sup>x</sup>*2, ... , *xN*) *T*), the substitution *ui*<sup>0</sup> = *xi*<sup>0</sup> + *c* (*i*<sup>0</sup> ∈ {1, 2, ... , *N*} (*c* is an arbitrary constant) brings the offset boosting in the variable *xi*<sup>0</sup> , where the new constant *c* will change the average value of the variable *xi*<sup>0</sup> . For a dynamical system, if there exists a variable substitution, *ui*<sup>0</sup> = *xi*<sup>0</sup> + *c*0, *ui* = *xi* (here *c*<sup>0</sup> is a non-zero constant, then *i*<sup>0</sup> ∈ {1, 2, ... , *N*}, and *i* ∈ {1, 2, ... , *N*}\{*i*0}), which makes the deduced system . *<sup>U</sup>* <sup>=</sup> *<sup>F</sup>*∗(*U*) <sup>=</sup> - *f*∗ <sup>1</sup> (*U*), *<sup>f</sup>*<sup>∗</sup> <sup>2</sup> (*U*), ... , *<sup>f</sup>*<sup>∗</sup> *<sup>N</sup>*(*U*) (*U* = (*u*1, *u*2, ... , *uN*)) asymmetric, but when *f*<sup>∗</sup> *j*0 (*U*) (1 ≤ *j*<sup>0</sup> ≤ *N*, *j*<sup>0</sup> *<sup>i</sup>*0) is revised, the system becomes symmetric, and then system . *X* = *F*(*X*) (*X* = (*x*1, *x*2, ... , *xN*)) is conditionally symmetric. Some early proposed chaotic systems of conditional symmetry [19,20] are listed in Table 1. All the coexisting attractors of conditional symmetry are shown in Figure 2. As we can see, all these systems are asymmetric ones but give twin attractors.

**Figure 2.** Coexisting twin attractors in chaotic systems in Table 1: (**a**) CS1, (**b**) CS2, (**c**) CS3, (**d**) CS4, (**e**) CS5, (**f**) CS6.


**Table 1.** Early explored typical chaotic systems of conditional symmetry.
