The review presents a parameter switching algorithm and his applications which allows numerical approximation of any attractor of a class of continuous-time dynamical systems depending linearly on a real parameter. The considered classes of systems are modeled by a general initial value problem embedding dynamical systems which are continuous and discontinuous with respect to the state variable, and of integer and fractional order. The numerous results, presented in several papers, are systematized here on four representative known examples representing the four classes. The analytical proof of the algorithm convergence for the systems belonging to the continuous class is briefly presented, while for the other categories of systems the convergence is numerically verified via computational tools. The utilized numerical tools necessary to apply the algorithm are contained in five appendices.
In this paper, the problem of approximating hidden chaotic attractors of a general class of nonlinear systems is investigated. The Parameter Switching (PS) algorithm is utilized, which switches the control parameter within a given set of values with the initial value problem numerically solved. The PS-generated generated attractor approximates the attractor obtained by averaging the control parameter with the switched values, which represents the hidden chaotic attractor. The hidden chaotic attractors of a generalized Lorenz system and the Rabinovich-Fabrikant system are simulated for illustra…
For low-dimensional chaotic attractors there is usually a single number of unstable dimensions for all of its periodic orbits and we can say such attractors exhibit "mono-chaos". In high-dimensional chaotic attractors, trajectories are prone to travel through quite different regions of phase space, some far more unstable than others. This heterogeneity makes predictability even more difficult than in low-dimensional homogeneous chaotic attractors. A chaotic attractor is "multi-chaotic" if every point of the attractor is arbitrarily close to periodic points with different numbers of unstable di…
For a model system defined as combination of sequentially applied continuous transformations of a sphere, the question of arrangement of the parameter space around the domain of existence of the Plykin-type attractor is considered. Results of numerical calculations are presented, including charts of dynamical regimes and Lyapunov exponents on the parameter plane, as well as portraits of attractors in characteristic regions of chaotic and regular dynamics. The Plykin attractor region is determined and depicted using the computational procedure for checking hyperbolicity, which consists in analy…
Continuous and discrete time systems possessing strange non-chaotic attractors are under investigation. It is demonstrated that unpredictable trajectories exist in the dynamics. A recent numerical technique, the sequential test, is utilized to show the presence of unpredictability.
In this paper, a continuous approximation to studying a class of PWC systems of fractionalorder is presented. Some known results of set-valued analysis and differential inclusions are utilized. The example of a hyperchaotic PWC system of fractional order is analyzed. It is found that without equilibria, the system has hidden attractors.
In this paper, it is presented a novel method for increasing the number of scrolls in a hybrid nonlinear switching system. Using the definition of the "Round to the Nearest Integer Function", as a generalization of a PWL function, which is capable of generating up to a thousand of scrolls. An equation that characterizes the grown in the number of scrolls is calculated, which fits to the behavior of the system measured by means of the coefficient of determination, denoted $R^{2}$, and pronounced "R squared". The proposed equation is based on obtaining as many scrolls as desired, based on the co…
Crisis-induced intermittency due to attractor-widening in a buoyancy-driven solar dynamo
在最近的一篇论文[M。A. J. H. Ossendrijver，A&A { \bf 359}，364（2000）]的二维平均场模型的数值模拟，其中显示产生大极小值，典型的太阳磁活动的长期行为。该模型由发电机组成，发电机的特点是基于磁通量管浮力不稳定性的$α$效应，这会导致从极大极小值到"正常“太阳行为的来回切换。在这封信中，我们报告的证据，从一个时间序列分析的模型存在的危机-引起的不稳定性，由于吸引-扩大。我们支持这一发现，显示的平均持续时间的最小值，$<τ>$，遵循理论预测的缩放$<τ> \sim (C_{δα}-C_{δα}^*)^{-γ}$，其中$C_{δα}$是分叉参数的兴趣，连同其他统计证据。据我们所知，这是第一次产生具体和详细的证据，证明这种类型的危机的发生-由于吸引子扩大-这种发电机模型引起的不稳定性。
