![]() ![]() Figure 2 shows basin of attraction of System ( 2) in the plane z = 0 for intervals x ∈, y ∈. The system has only one equilibrium, which is stable. Figure 1 shows three projections of this strange attractor. System ( 2) has a chaotic attractor for b = 0.4 (for e.g. It means that the origin is stable (spiral node, since it has three eigenvalues with negative real parts, and a pair of them is complex). ![]() In any case, better understanding of such systems may be of interest.Ĭhaotic systems with hidden attractors and multi-stability are very important in engineering and can be much challenging in applications like control, synchronization, anti-synchronization, image encryption, and so on. On the other hand, multistability makes systems flexible without tuning parameters. When it is important in a dynamical system to work in a specific state and not go out from it, then multistability in that system is a potential danger (because due to any disturbance the system can go to a new unwanted situation). In some occasions multistability is unwanted, while in some cases it is desired. Multistability is an important topic in nonlinear dynamics and chaos. Another important aspect about our new proposed system is that it is multi-stable. In that little part, only a few of them are systems with stable equilibria only. It should be noted that while there are many chaotic flows in the literature, only a little part of them are systems with hidden attractors. In this case the strange attractor is hidden, since the existence of an unstable equilibrium in its basin is impossible. In this paper, we propose a new three-dimensional chaotic flow with only one stable equilibrium. #STABLE EQUILIBRIUM 3D SERIES#Entropy measure is the best when only a short time series of system is available. Also, these measures quantify chaotic attractors. Features such as Lyapunov exponents, Entropy, fractal dimension and correlation dimension are some examples of them. There are many features which measure the complexity of dynamical systems. These attractors are called multi-stable since the final state of the system is dependent on its initial condition. In the case of chaotic attractors in systems with a stable equilibrium, at least there are two attractors simultaneously (strange attractor and stable equilibrium attractor). An attractor is called self-excited if its basin of attraction is associated with an unstable equilibrium. The second one is self-excited attractor. An attractor is called hidden if its basin of attraction does not intersect with a small neighborhood of any equilibrium point. From a point of view, attractors of dynamical systems have been categorized into two groups. It seems the relation between equilibria and their stable and unstable manifolds is unknown to us. The reason is that recently many new chaotic systems have been proposed without any equilibria, with one stable equilibria, with a line of equilibria, with a curve of equilibria and with a plane of equilibria. However it is clear now that the existence of a saddle point is not a necessary condition for existence of chaotic solutions. įor many years, researchers have believed that a chaotic attractor is related to a saddle equilibrium. This behavior is known as deterministic chaos, or simply chaos. In other words, the deterministic nature of these systems does not make them predictable. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general. Chaotic systems and their dynamical properties are interesting topics in nonlinear dynamics. ![]()
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