# New Chaotic Dynamical System with a Conic-Shaped Equilibrium Located on the Plane Structure
## 2 Mathematical Methods, Discovered Model and Numerical Analysis
### Numerical Analysis of Ordinary Differential Equations  
A numerical integration of a dynamical system (1) with initial conditions \(\mathbf{x}_{0}=(0\ 0\ 0)^{\mathrm{T}}\), final time \(t_{\mathrm{max}}=2000\ \mathrm{s}\) and time step \(\Delta_{t}=0.01\ \mathrm{s}\) by using Mathcad and the built-in fourth-order Runge-Kutta method is shown in Figure 1. The same method with final time \(t_{\mathrm{max}}\in(1,\ 10)\) is used to illustrate the local dynamical behavior near equilibrium, as demonstrated by means of Figure 2. Periodic solutions provided in this figure are excited by choosing the following sets of the initial conditions: \(\mathbf{x}_{0}=(-0.8\ \pm\ 0.1\ 0)^{\mathrm{T}}\), \(\mathbf{x}_{0}=(-0.8\ \pm\ 0.5\ 0)^{\mathrm{T}}\), \(\mathbf{x}_{0}=(-0.8\ \pm\ 0.8\ 0)^{\mathrm{T}}\), \(\mathbf{x}_{0}=(-0.2\ \pm\ 0.1\ 0)^{\mathrm{T}}\), \(\mathbf{x}_{0}=(-0.2\ \pm\ 0.5\ 0)^{\mathrm{T}}\) and \(\mathbf{x}_{0}=(-0.2\ \pm\ 0.8\ 0)^{\mathrm{T}}\). Numerical experiments show that the equilibrium structure can be seriously deformed while chaotic motion is still preserved. In fact, only a portion of hyperbolic function is responsible for the formation of the typical strange attractor as evident from the numerically integrated chaotic attractors provided in Figure 3. Of course, considering piecewise linear approximation of hyperbolic function does not ease the analysis of global behavior of the investigated dynamical system. Note, the degradation of the hyperbolic function into two parallel line segments leads to the characteristic in Equation (3) with the last term missing. This means that the local behavior of dynamical flow near the equilibrium plane \(z=0\) remains the same. However, deformation of equilibria into circle changes simplified the characteristic polynomial into  
\[\lambda\left(\lambda^{2}+2a\cdot y^{2}-2x\right)=0\quad\Rightarrow\quad \lambda_{1}=0\quad\lambda_{2,3}=\pm\sqrt{2x-2a(r^{2}-x^{2})} \tag{5}\]  
where \(r\) is a circle radius and numerical values of eigenvalues depend on the state variable \(x\). The regions of chaos can be visualized by plotting the largest LE as a function of the internal system parameters. The high-resolution graph having 201 \(\times\) 101 (step for both parameters is 0.01) points is plotted in Figure 4. Note that these regions are quite wide and surrounded by an unbounded solution, which means that the chaotic attractor at this moment seems to be experimentally observable. From the viewpoint of practical realization and potential applications, the basins of attraction for a chaotic attractor should be specified. So far, we admit existence of the attracting limit sets that are bounded into finite state space volume as well as orbits that tend to infinity. We can distinguish between these cases by the calculation of the final system energy. An example of dynamical system energy after 100 s of evolution as a function of the initial conditions is provided in Figure 5, where state space is slashed into parallel slices given by the planes \(x=-2\), \(x=-1\), \(x=0\) and \(x=1\). Huge system energy, that is, the module of distance in Euclidean space, at the end of the numerical integration process means that state trajectory is unbounded and tends to infinity. In a real electronic circuit, final state is limited in the cube given by output saturation levels of the active elements, that is, the size of this cube corresponds to supply voltages. Note, the original dynamical system, with hyperbolic equilibrium and its modification with a couple of line segments, has different energy patterns. Moreover, the dynamical system with plane \(z=0\) oriented double line equilibrium structure possesses a high but finite (light blue color) terminal energy state, which continuously surrounds a basin of attraction for the chaotic attractor and small limit cycles located near the equilibrium lines (dark blue color).