## 2 A Nonequilibrium System with Ten Terms and Its Feasibility  
Recently, there has been considerable critical attention on nonequilibrium systems with chaos [40; 41; 42; 43; 44]. Such chaotic systems are totally different from common chaotic ones, in which there are some unstable equilibrium points. Investigating new chaotic systems without equilibria is a continuing concern [45]. For studying new nonequilibrium systems, we consider a general three-dimensional form:  
\[\left\{\begin{array}{l}\dot{x}=a_{1}y,\\ \dot{y}=a_{2}x+a_{3}z+a_{4}xz,\\ \dot{z}=a_{5}x+a_{6}y+a_{7}z+a_{8}xy+a_{9}xz+a_{10},\end{array}\right. \tag{1}\]  
in which parameters are denoted as \(a_{i}\) (\(i=1,\ldots,10\)), and \(a_{i}\
eq 0\). The origin of the system (1) is based on the published work [42].  
By used a search procedure [42], we have found the parameter set (2), for which there is no any real equilibrium in the three-dimensional form (1),  
\[\left\{\begin{array}{l}a_{1}=a,\\ a_{2}=a_{7}=a_{8}=-1,\\ a_{3}=b,\\ a_{4}=a_{5}=a_{10}=1,\\ a_{6}=c,\\ a_{9}=d,\end{array}\right. \tag{2}\]  
with \(a,b,c,d>0\).  
Thus, form (1) becomes the following system:  
\[\left\{\begin{array}{l}\dot{x}=ay,\\ \dot{y}=-x+bz+xz,\\ \dot{z}=x+cy-z-xy+dxz+1.\end{array}\right. \tag{3}\]  
It is trivial to confirm that for \(a=2\), \(b=2.5\), \(c=0.2\), and \(d=0.3\) system (3) does not have any real equilibrium. It is noted that there are some sets of values which make a system without real equilibria. We have selected such a set of values because in this case the system is elegant [46; 47; 48]. Especially, the system exhibits chaos (see Figure 1). Chaotic dynamics was verified by calculated Lyapunov exponents \(L_{1}=0.2376\), \(L_{2}=0\), \(L_{3}=-0.5231\) as presented in Figure 2. For calculating the Lyapunov exponents, we applied the known [PERSON]'s algorithm [49]. Entropy was used to estimate the transfer rate of information in a particular system [50]. Moreover, entropy indicates the level of chaos in a dynamical system [51; 52]. Here we measured the approximate entropy (ApEn) [53; 54] for system (3). The calculated approximate entropy of the system without equilibrium (3) is 0.1503, which presents the level of chaotic behavior.  
Figure 1: Different attractors without equilibrium in (**a**) \(x-y\) plane, (**b**) \(x-z\) plane, and (**c**) \(y-z\) plane. The parameter set is \(a=2\), \(b=2.5\), \(c=0.2\), \(d=0.3\) and initial conditions are \((x(0),y(0),z(0))=(1,1,1)\).  
Figure 2: Calculated Lyapunov exponents (\(L_{1}\) (red), \(L_{2}\) (green), and \(L_{3}\) (blue)) of the system when \(a=2\), \(b=2.5\), \(c=0.2\), \(d=0.3\) and \((x(0),y(0),z(0))=(1,1,1)\). It is noted that chaos of the system is indicated by \(L_{1}>0\).  
Circuit implementation provides an effective tool for verifying the feasibility of theoretical models [55; 56; 57; 58; 59]. Therefore, we have designed and realized the nonequilibrium system via a real circuit as shown in Figures 3 and 4. The circuit was implemented with selected components: R1 = 200 k\(\Omega\), R2 = R3 = R7 = R8 = R15 = R16 = 100 k\(\Omega\), R4 = R9 = R11 = 400 k\(\Omega\), R5 = 160 k\(\Omega\), R6 = R14 = 8 k\(\Omega\), R10 = 2 M\(\Omega\), R12 = 30 M\(\Omega\), R13 = 26.6 k\(\Omega\), and C1 = C2 = C3 = 1 nF. The phase portraits are displayed by the oscilloscope as reported in Figure 5. Figure 1 is a theoretical figure obtained by solving the system at nonequilibrium conditions, while Figure 5 was obtained experimentally by using the implemented electronic circuit.  
Figure 3: Circuit schematic including electronic components.