Conservative chaos in a simple oscillatory system with non-smooth nonlinearity

On a High-Precision Method for Studying Attractors of Dynamical Systems and Systems of Explosive Type

“…Therefore, the task of developing the high-precision numerical methods for constructing approximate solutions to systems of the form ( 21) is important. Note that this idea can also be transferred to models with hysteresis which were considered in [24][25][26].…”

“…Similarly, the phase space can be divided into regions of smoothness for systems with hysteresis. For example, for systems with a non-smooth potential [25]. The joint point coordinates are…”

Natural Exponential and Three‐Dimensional Chaotic System

“…Besides, the chaos edge theory [ 12 ] is also proposed in resolving the Smale paradox, where complex behaviors including the static and dynamic pattern formulation emerging in the bio‐inspired array can reproduce shocking phenomenon when investigating a chaotic model from cellular biology. To overcome the chaos disappearance caused by parameter disturbance in practical applications, many new chaotic models and nonlinear systems [ 13,14 ] are proposed including the transient chaos, [ 15 ] spatial chaos, [ 16 ] topological chaos, [ 17 ] mixed multi‐chaos, [ 18 ] 3D pulsed chaos. [ 19 ] Although the existing exponential or logarithmic chaotic model is proposed to produce one‐dimensional (1D) chaotic maps with robust chaos, [ 20 ] higher dimensional chaotic system is still needed to bring more information in understanding the dynamic motions and chaos phenomena.…”

“…b) Chordal graph for approximate entropy (AE). xL and xr represent the AE computation results of x(t) when the parameters of L and r vary within the scope of[2,4,6,8,10,12,14,16,18,20] and [0.01, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9]. c) Chordal graph for Shannon entropy (SE).…”

“…c) Chordal graph for Shannon entropy (SE). xq and xw mean the SE computation results of x(t) when the parameters of q and w varies within the scope of[2,4,6,8,10,12,14,16,18,20] and [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. xa, xb, xc, and xd in (a)-(c) express the corresponding entropy of x(t) when the system parameters of a, b, c, and d varies within the scope of [−5, −1, −0.5, −0.3, −0.2, −0.1, −0.01, 0, 0.01, 0.1], [0.1, 0.2, 0.3, 0.5, 1, 2, 3, 5, 7, 10], [−10, −5, −1, 0, 1, 5, 10, 20, 30, 50], and [−7, −5, −1, 0, 1, 3, 5, 15, 20, 50], respectively.…”