Is a hyperchaotic attractor superposition of two multifractals?

Little seems to be known about the five-dimensional (5D) differential dynamical system with infinitely many hyperchaotic attractors, which have three positive Lyapunov exponents under no or infinitely many equilibria. This article presents a 5D dynamical system that can generate infinitely many hyperchaotic attractors. Of particular interest is the system exists not only infinitely many hyperchaotic attractors but also infinitely many periodic attractors in the following three cases: (i) no equilibria, (ii) only infinitely many non-hyperbolic equilibria, (iii) only infinitely many hyperbolic equilibria. By numerical analysis, one finds the 5D system could generate infinitely many coexisting hyperchaotic or chaotic or periodic attractors in the three kinds of equilibria cases. And one obtains the global dynamical behavior of the system, such as the Lyapunov exponential spectrum, bifurcation diagram. To study the hyperchaotic complexity of the 5D system, we rigorously show the stability of hyperbolic equilibria and some mathematical characterization for 5D Hopf bifurcation. In particular, the existence of an infinite number of isolated bifurcated periodic orbits is strictly proven. These complex dynamics studies in this paper may further contribute to a deep understanding of the hyperchaotic systems with infinitely many attractors.

We examine the dynamics of three-dimensional cells with square planform in dissipative Rayleigh-Benard convection. By applying a triple Fourier series ansatz up to second order, we obtain a system of nine nonlinear ordinary differential equations from the governing hydrodynamic equations. Depending on two control parameters, namely the Rayleigh number and the Prandtl number, the asymptotic behaviour can be stationary, periodic, quasiperiodic or chaotic. A period-doubling cascade is identified as a route to chaos. Hereafter, the asymptotic behaviour progressively evolves towards a hyperchaotic attractor. For given values of control parameters beyond the accumulation point, we observe a low-dimensional chaotic attractor as is currently done for dissipative systems. Although the correlation dimension strongly suggests that this attractor could be embedded in a three-dimensional space, a topological characterization reveals that a higher-dimensional space must be used. Thus, we reconstruct a four-dimensional model which is found to be in agreement with the properties of the original dynamics. The nine-dimensional Lorenz model could therefore play a significant role in developing tools to characterize chaotic attractors embedded in phase space with a dimension greater than 3.

The investigation of chaotic systems containing hidden and coexisting attractors has attracted extensive attention. This paper presents a four-dimensional (4D) novel hyperchaotic system, evolved by adding a linear state feedback controller to a 3D chaotic system with two stable node-focus points. The proposed system has no equilibrium point or two lines of equilibria, depending on the value of the constant term. Complex dynamical behaviors such as hidden chaotic and hyperchaotic attractors and five types of coexisting attractors of the simple 4D autonomous system are investigated and discussed, and are numerically verified by analyzing phase diagrams, Poincaré maps, the Lyapunov exponent spectrum, and its bifurcation diagram. The short unstable cycles in the hyperchaotic system are systematically explored via the variational method, and symbol codings of the cycles with four letters are realized based on the topological properties of the trajectory projection on the 2D phase space. The bifurcations of the cycles are explored through a homotopy evolution approach. Finally, the novel 4D system is implemented by an analog electronic circuit and is found to be consistent with the numerical simulation results.

Dynamics analysis, FPGA realization and image encryption application of a 5D memristive exponential hyperchaotic system

Chaotic signals generated by chaotic oscillators based on memory elements are suitable for use in the field of confidential communications because of their very good randomness. But often their maximum Lyapunov exponent is not high enough, so the degree of randomness is not enough. It can be chaos enhanced by transforming it to fractional order using the Caputo differential definition. In this paper, based on the proposed hyperchaotic oscillator, it is extended to a fractional-order form to obtain a chaos-enhanced fractional-order memcapacitor meminductor system, in which several different styles of chaotic and hyperchaotic attractors are found. The dynamical behaviour of the system is studied using bifurcation diagrams, Lyapunov exponent spectrums and Lyapunov dimensions. The multistability of the system is explored in different initial orbits, and the spectral entropy complexity of this system is examined. Finally, a hardware implementation of the memcapacitor meminductor system is given, which demonstrates the effectiveness of the system. This study provides a reference for the study of chaos-enhanced.

This paper presents a new four-dimensional non-Hamiltonian conservative hyperchaotic system. In the absence of equilibrium points in the system, the phase trajectories generated by the system have hidden features. The conservative features that vary with the parameter have been analyzed in detail by Lyapunov exponent spectrum, bifurcation diagram, the sum of Lyapunov exponents, and the fractional dimensions, and during the analysis, multiple quasi-periodic four-dimensional tori as well as hyperchaotic attractors have been observed. The Poincaré sections confirm these dynamic behaviors. Amidst the process of studying the dynamical behavior of the system with initial values, the hidden extreme multistability, and the initial offset boosting behavior, the results have been witnessed for the very first time in a conservative chaotic system. The phase diagram and attraction basin also confirm this assertion, while two complex transient transition behaviors have been observed. Moreover, through the introduction of a spectral entropy algorithm, the complexity analysis of the time sequences generated by the system have been performed and compared with the existing literature. The results show that the system has a high degree of complexity. The design and construction of the analog circuit of the system for simulation, the circuit experimental results are consistent with the numerical simulation, further verifying the physical realizability of the newly proposed system. This lays a good foundation for its practical application in engineering.

A note on “Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system” in [Appl. Math. Comput. 291 (2016) 323–339] by Amin Zarei and Saeed Tavakoli

A dual-parameter hyperchaotic system with constant Lyapunov exponent and its circuit emulation

In hyperbolic systems, transient chaos is associated with an underlying chaotic saddle in phase space. The structure of the chaotic saddle of a class of piecewise linear, area-preserving, two-dimensional maps with overall constant Lyapunov exponents has been observed by a scattering method. The free energy obtained in this way displays a “phase transition” at β<0 in spite of the fact that no phase transition occurs in the free energy dedcued from the spectrum of Lyapunov exponents. This is possible because pruning introduces a second effective scaling exponent by creating, at each level of the approximation, particular small pieces in the incomplete Cantor set approximating the saddle. The second scaling arises for a subset of values of the control parameter that is dense in the parameter interval.

This report discusses Saffman-Taylor instability; cylinder wake; Levy walk and turbulent channel flow; bubble motion and bubble streams; spinal turbulent and wetting; collective behavior of a coupled map system with a conserved quantity; stability of temporally periodic states; generic nonergodic behavior in continuous systems; characterization of unstable periodic orbits; in low-dimensional chaotic attractors and repellers; and Ginzburg-Landau theory for oil-water-surfactant mixture.

Correlating dimension for a gas-liquid contactor

We examine the nonlinear dynamical properties of the monthly smoothed group sunspot number R-g and find that the solar activity underlying the time series of R-g, is globally governed by a low-dimensional chaotic attractor. This finding is consistent with the nonlinear; study results of the monthly Wolf sunspot numbers. We estimate the maximal Lyaponuv exponent (MLE) for the R-g series to be positive and to equal approximately 0.0187 +/- 0.0023 (month(-1)). Thus, the Lyaponuv time or predictability time of the chaotic motion is obtained to be about 4.46 +/- 0.5 years, which is slightly different with the predictability time obtained from R-z. However, they both indicate that solar activity forecast should be,done only for,a short to medium term due to the intrinsic complexity of the time behavior concerned.

Evidence of low dimensional chaos in renal blood flow control in genetic and experimental hypertension

Can the start of an epileptic seizure be described as a transition to a low dimensional chaotic attractor?

Georg Cantor (1848-1914) was a German mathematician who worked on the foundations of set theory. He published the Cantor set in 1883. The Cantor set, along with the other fractals discussed in this chapter, are referred to as mathematical monsters by Mandelbrot because of their ambiguous features. The Cantor set is produced by starting with a unit interval [0, 1] and taking out the center open interval (1/3, 2/3). An open interval does not include the end points and is shown with rounded parenthesis. A closed interval includes the end points and is shown with squared parenthesis. This process leaves two closed intervals [0,1/3] and [2/3,1]. The same procedure is repeated on each of these two intervals. This procedure is repeated an infinite number of times (Peitgen, Jurgens, and Saupe, 1992). The result is an infinite number of clustered points. Figure 2.1 shows the construction process for the Cantor set through a few of the infinite steps.