A New System with a Self-Excited Fully-Quadratic Strange Attractor and Its Twin Strange Repeller

In nonlinear dynamics, the study of chaotic systems has attracted the attention of many researchers around the world due to the exciting and peculiar properties of such systems. In this regard, the present paper introduces a new system with a self-excited strange attractor and its twin strange repeller. The unique characteristic of the presented system is that the system variables are all in their quadratic forms; therefore, the proposed system is called a fully-quadratic system. This paper also elaborates on the study of the bifurcation diagram, the interpretation of Lyapunov exponents, the representation of basin of attraction, and the calculation of connecting curves as the employed method for investigating the system’s dynamics. The investigation of 2D bifurcation diagrams and Lyapunov exponents indicated in this paper can better recognize the system’s dynamics since they are plotted considering simultaneous changes of two parameters. Moreover, the connecting curves of the proposed system are calculated. The system’s connecting curves help identify the system’s different behaviors by providing general information about the nature of the flows.

Strange attractor in earthquake swarms near Valsad (Gujarat), India

Valsad district in south Gujarat near the western coast of the peninsular India experienced earthquake swarms since early February 1986. Seismic monitoring through a network of micro earthquake seismographs showed a well concentrated seismic activity over an area of 7 × 10 km2 with the depth of foci extending from 1 to 15 km. A total number of 21,830 earthquakes were recorded during March 1986 to June 1988. The daily frequency of earthquakes for this period was utilized to examine deterministic chaos through evaluation of dimension of strange attractor and Lyapunov exponent. The low dimension of 2.1 for the strange attractor and positive value of the largest Lyapunov exponent suggest chaotic dynamics in Valsad earthquake swarms with at least 3 parameters for earthquake predictability. The results indicate differences in the characteristics of deterministic chaos in intraplate and interplate regions of India.

Method of attractor reconstruction in modeling the integral risk indicator

This study examines a mathematical model for forecasting the dynamics of changes in the industrial safety risk indicator of an enterprise. The results of the development of a method for reconstructing the attractor of the integral risk indicator based on the Grassberger-Prokacchia algorithm, which is currently the most popular for analyzing time series and allowing automating the process of calculating a parameter that determines the number of points, for making a forecast. There are results of estimating the forecast of the dynamics of the behavior for a strange attractor, obtained on the basis of real data.

Freedom and necessity from the perspective of the fractality concept

The article presents a view of the problem of freedom through the prism of the author’s concept of fractality. The importance of theoretical understanding of the concept of freedom in its foundation is noted; by the example of a series of experiments of Libet it is shown that without such an understanding errors and distortions of interpretative conclusions in solving the problem of freedom are inevitable, which is reflected in a contradiction in the worldview position of man as a free being. The article establishes connection between logical problems and ways of solving contradiction of freedom and necessity. It shows that disregard of monistic logic of wholeness which is represented by Spinoza’s position leads to transformed (fractal) form of freedom — freedom of choice as subjective ability to act independently of external circumstances. The concept of a strange attractor is considered and a conclusion about fractality as the essence of its «strange» nature is made. The connection between fractality and freedom in the being dimension of its development as a universal human definition is shown.

Numerical Analysis of a Novel 3D Chaotic System with Period-Subtracting Structures

In this work, we present a novel three-dimensional chaotic system with only two cubic nonlinear terms. Dynamical behavior of the system reveals a period-subtracting bifurcation structure containing all [Formula: see text]th-order ([Formula: see text]) periods that are found in the dynamical evolution of the novel system concerning different values of parameters. The new system could be evolved into different states such as point attractor, limit cycle, strange attractor and butterfly strange attractor by changing the parameters. Also, the system is multistable, which implies another feature of a chaotic system known as the coexistence of numerous spiral attractors with one limit cycle under different initial values. Furthermore, bifurcation analysis reveals interesting phenomena such as period-doubling route to chaos, antimonotonicity, periodic solutions, and quasi-periodic motion. In the meantime, the existence of periodic solutions is confirmed via constructed Poincaré return maps. In addition, by studying the influence of system parameters on complexity, it is confirmed that the chaotic system has high spectral entropy. Numerical analysis indicates that the system has a wide variety of strong dynamics. Finally, a message coding application of the proposed system is developed based on periodic solutions, which indicates the importance of studying periodic solutions in dynamical systems.

A New Circumscribed Self-Excited Spherical Strange Attractor

Studying new chaotic flows with specific characteristics has been an open-ended field of exploring nonlinear dynamics. Investigation of chaotic flows is an area of research that has been taken into consideration for many years; thus, it helps in a better understanding of the chaotic systems. In this paper, an original chaotic 3D system, which has not been investigated yet, is presented in spherical coordinates. A unique feature of the proposed system is that its velocity becomes zero for a specific value of the radius variable. Hence, the system’s attractor is expected to be stuck on one side of a plane in spherical coordinates and inside or outside a sphere in the corresponding Cartesian coordinates. It means that the attractor cannot pass through the sphere or even touch it. The introduced system owns two unstable equilibria and a self-excited strange attractor. The 1D and 2D system’s bifurcation diagrams concerning the alteration of two bifurcation parameters are plotted to investigate the system’s dynamical properties. Moreover, the system’s Lyapunov exponents in the corresponding period of bifurcation parameters are calculated. Then, two 2D basins of attraction for two different third dimension values are explored. Based on the basin of attraction, it can be found that the sphere has attraction itself, partially, and some initial conditions are led to the sphere, not to the strange attractor. Ultimately, the connecting curves of the proposed system are explored to find an informative 1D set in addition to the system’s equilibria.

Stability Analysis of the Chaotic Lorenz System with a State-Feedback Controller

The Lorenz model is considered a benchmark system in chaotic dynamics in that it displays extraordinary sensitivity to initial conditions and the strange attractor phenomenon. Even though the system tends to amplify perturbations, it is indeed possible to convert a strange attractor to a non-chaotic one using various control schemes. In this work it is shown that the chaotic behavior of the Lorenz system can be suppressed through the use of a feedback loop driven by a quotient controller. The stability of the controlled Lorenz system is evaluated near its equilibrium points using Routh-Hurwitz testing, and the global stability of the controlled system is established using a geometric approach. It is shown that the controlled Lorenz system has only one globally stable equilibrium point for the set of parameter values under consideration.

"Large" strange attractors in the unfolding of a heteroclinic attractor

<p style='text-indent:20px;'>We present a mechanism for the emergence of strange attractors in a one-parameter family of differential equations defined on a 3-dimensional sphere. When the parameter is zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between two saddles-foci with different Morse indices. After slightly increasing the parameter, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We will show that, for a set of parameters close enough to zero with positive Lebesgue measure, the dynamics exhibits strange attractors winding around the "ghost'' of a torus and supporting Sinai-Ruelle-Bowen (SRB) measures. We also prove the existence of a sequence of parameter values for which the family exhibits a superstable sink and describe the transition from a Bykov network to a strange attractor.</p>