Narrow Results

The characteristic of the fixed points of the Carotid–Kundalini (C–K) map is investigated and the boundary equation of the first bifurcation of the C–K map in the parameter plane is given. Based on the studies of the phase graph, the power spectrum, the correlation dimension and the Lyapunov exponents, the paper reveals the general features of the C–K map transforming from regularity. Meanwhile, using the periodic scanning technology proposed by Welstead and Cromer, a series of Mandelbrot–Julia (M–J) sets of the complex C–K map are constructed. The symmetry of M–J set and the topological inflexibility of distributing of periodic region in the Mandelbrot set are investigated. By founding the whole portray of Julia sets based on Mandelbrot set qualitatively, we find out that Mandelbrot sets contain abundant information of structure of Julia sets.

This paper introduces and analyzes new hyperchaotic complex Lorenz systems. These systems are 6-dimensional systems of real first order autonomous differential equations and their dynamics are very complicated and rich. In this study we extend the idea of adding state feedback control and introduce the complex periodic forces to generate hyperchaotic behaviors. The fractional Lyapunov dimension of the hyperchaotic attractors of these systems is calculated. Bifurcation analysis is used to demonstrate chaotic and hyperchaotic behaviors of our new systems. Dynamical systems where the main variables are complex appear in many important fields of physics and communications.

A lattice Boltzmann model using upstream finite volume scheme has been employed in the investigation of bifurcation and transition of flow through suddenly-expanded channels. To enhance the stability and accuracy of simulation, a fifth-order Runge–Kutta method is used for the time-marching and upwind biasing factors based on pressure are used as flux correctors in the lattice Boltzmann equation. In the range of Reynolds numbers investigated, the laminar flow through the expansion underwent a symmetry-breaking bifurcation. The critical Reynolds number evaluated from the experiments and the simulations were compared to other values reported in the literature. Comparisons are found to be quantitatively accurate. Furthermore, the results show that the critical Reynolds number decreases with increasing channel expansion ratio. At a fixed supercritical Reynolds number, the location at which the jet first impinges on the channel wall grows with the expansion ratio.

Gursey Model is the only possible 4D conformally invariant pure fermionic model with a nonlinear self-coupled spinor term. It has been assumed to be similar to the Heisenberg's nonlinear generalization of Dirac's equation, as a possible basis for a unitary description of elementary particles. Gursey Model admits particle-like solutions for the derived classical field equations and these solutions are instantonic in character. In this paper, the dynamical nature of damped and forced Gursey Nonlinear Differential Equations System (GNDES) are studied in order to get more information on spinor type instantons. Bifurcation and chaos in the system are observed by constructing the bifurcation diagrams and Poincaré sections. Lyapunov exponent and power spectrum graphs of GNDES are also constructed to characterize the chaotic behavior.

We propose a special opinion model on Internet users' social platform selections where users only view those converging platforms as tools to maintain their communications with all of their friends or partners and one may use more than one platform at the same time. We construct the time evolution differential equations, seek the fixed points, and study their attractability and repellency by analyzing those equations. Then, we verify the analytical results and observe their accuracy by numerical simulation. The conclusion shows that in any practical system described by our model, one platform will completely eliminate its competitor sooner or later, and when the average degree of the interaction network is relatively low, the laggard may have more chance to turn the tide, but when that average degree is high, that chance is extremely limited.

The shapes of rotating drops of liquid bound by gravitation is a classical subject of study. Inspired by this classical theory, we have simulated numerically a related problem, the rotating rigid configurations of a finite number of point particles. Two particles at a distance $r$ have a long range attractive potential $-1∕r$ and a short range repulsive potential $1∕r^2$ preventing collapse. We take the angular momentum to be conserved, but not the energy.This system has a variable density, unlike the classical liquid drops. When the number of particles is small, it is more rigid than a liquid drop, implying that many different stable equilibrium configurations may exist with the same angular momentum but different energies. When the number of particles becomes large, our system should resemble a self-gravitating polytropic gas.We present here the results of a preliminary study with only 3, 4, and 5 particles. The methods used could be applied to the study of rotating molecules.

This short paper is focused on the bifurcation theory found in map functions called evolution functions that are used in dynamical systems. The most well-known example of discrete iterative function is the logistic map that puts into evidence bifurcation and chaotic behavior of the topology of the logistic function. We propose a new iterative function based on Lorentizan function and its generalized versions, based on numerical study, it is found that the bifurcation of the Lorentzian function is of second-order where it is characterized by the absence of chaotic region.

A new methodology was introduced to investigate the pattern formation in time series systems due to their viscoelastic behavior. Four stochastic processes, uniform distribution, normal distribution, Nasdaq-100 stock market index, and a melody were studied within this context. The time series data were converted into vectorial forms in a scattering diagram. The sequential vectors can be split into its in-line (or conservative) and out-of-line (or dissipative) components. Thus, one can define the storage and loss modulus for conservative, and dissipative components, respectively. Instead of using the geometric Brownian equation which involves Wiener noise term, the changes were taken into consideration at every step by introducing “lethargy” concept and the deviation from it. Thus, the mathematics is somehow simplified, and the dynamical behavior of time series systems can be elucidated at every step of change. The viscoelastic behavior of time series systems reveals patterns of the viscoelastic parameters such as storage and loss modulus, and also of thermodynamic work-like and heat-like properties. Besides, there occur some minima and maxima in the distribution of the angles between the sequential vectors in the scattering diagram. The same is true for the change of entropy of the system.

This paper formulates a new hyperchaotic system for particle motion. The continuous dependence on initial conditions of the system’s solution and the equilibrium stability, bifurcation, energy function of the system are analyzed. The hyperchaotic behaviors in the motion of the particle on a horizontal smooth plane are also investigated. It shows that the rich dynamic behaviors of the system, including the degenerate Hopf bifurcations and nondegenerate Hopf bifurcations at multiple equilibrium points, the irregular variation of Hamiltonian energy, and the hyperchaotic attractors. These results generalize and improve some known results about the particle motion system. Furthermore, the constraint of hyperchaos control is obtained by applying Lagrange’s method and the constraint change the system from a hyperchaotic state to asymptotically state. The numerical simulations are carried out to verify theoretical analyses and to exhibit the rich hyperchaotic behaviors.

Dynamical behavior of nonlinear wave solutions of the perturbed and unperturbed generalized Newell–Whitehead–Segel (GNWS) equation is studied via analytical and computational approaches for the first time in the literature. Bifurcation of phase portraits of the unperturbed GNWS equation is dispensed using phase plane analysis through symbolic computation and it shows stable oscillation of the traveling waves. Chaotic behavior of the perturbed GNWS equation is obtained by applying different computational tools, like phase plot, time series plot, Poincare section, bifurcation diagram and Lyapunov exponent. A period-doubling bifurcation behavior to chaotic behavior is shown for the perturbed GNWS equation and again it shows chaotic to periodic motion through inverse period-doubling bifurcation. The perturbed GNWS equation also shows chaotic motion through a sequence of periodic motions (period-1, period-3 and period-5) depending on the variation of the parameter of linear coefficient. Thus, the parameter of linear coefficient plays the role of a controlling parameter in the chaotic dynamics of the perturbed GNWS equation.

A computational analysis has been performed to study the flow instability of two-parallel wall motions in a Cuboidal enclosure incorporated by a cylinder under different radii sizes. A numerical methodology based on the Finite Volume Method (FVM) and a full Multigrid acceleration is utilized in this paper. Left and right parallel walls of the cavity are maintained driven and all the other walls completing the domain are motionless. Different radii sizes ($R=0.075$, 0.1, 0.125, 0.15 and 0.175) are employed encompassing descriptive Reynolds numbers that range three orders of magnitude 100, 400 and 800 for the steady state. The obtained results show positions $R=0.15$ and $R=0.175$ of the inner cylinder promote cell distortion. Also, when the radius equates to $R=0.15$, it may lead to the birth of tertiary cells at $\mathrm{\text{Re}}=400$ which are more developed for $\mathrm{\text{Re}}=800$. Thereafter, analysis of the flow evolution shows that with increasing Re beyond a certain critical value, the flow becomes unstable and undergoes a Hopf bifurcation. A nonuniform variation with the radius size of the inner cylinder is observed. Otherwise said, elongating the radius of the cylinder leads to decrease in the critical Reynolds number. Hence, the acceleration of the unsteadiness is realized. On the other hand, by further increasing Reynolds number more than the critical value from 1200 to 2100, we note that the kinetic energy is monotonously increasing with Reynolds number and a stronger motion in the velocity near the rear wall of the enclosure is observed. Furthermore, the symmetry of flow patterns observed in the steady state has been lost. Therefore, a systematic description of various effects illuminating the optimum geometrical parameters to achieve effective flow behavior in those systems has been successfully established through this paper.

The bifurcation phenomenon generated in neurons is considered to be the cause of many neurological diseases, thus, lots of investigators have been dedicated to bifurcation control, but most of them only focus on the integer-order neuronal models and control methods. Recently, increasing number of research works have shown that the differential equations of fractional-order are more appropriate for description of the electrical properties of certain nerve cell membranes. In this paper, we employ a fractional-order washout filter to realize bifurcation control for fractional-order Morris–Lecar neuronal model. Computer simulation verifies the effectiveness and improved performance of the proposed method than the integer-order washout filter. In addition, the analysis of the fractional-order controller’s impacts on the model is also presented.