Narrow Results

In this paper we analytically and numerically investigate the dynamics of a nonlinear three-dimensional autonomous first-order ordinary differential equation system, obtained from paradigmatic Lorenz system by suppressing the y variable in the right-hand side of the second equation. The Routh–Hurwitz criterion is used to decide on the stability of the nontrivial equilibrium points of the system, as a function of the parameters. The dynamics of the system is numerically characterized by using diagrams that associate colors to largest Lyapunov exponent values in the parameter-space. Additionally, phase-space plots and bifurcation diagrams are used to characterize periodic and chaotic attractors.

Short-time correlations in multivariate time series are notoriously difficult to detect. Extending the classical concept of correlation, we present a method to tackle this problem with little computational cost. In essence, the method uncovers multidimensional phase-locking and is especially useful for revealing changes of the dynamical coupling between different brain areas. The approach also permits us to estimate the shortest time-window in which the coupling occurs. Furthermore, the coupling can be quantified by a measure that defines a metric space. Therefore it may be used to identify task-dependent coupling hierarchies between two brain areas. We illustrate the analysis technique by studying the dynamical coupling between two brain regions involved in olfactory processing in the honeybee. We show that the neural activity in both areas is coupled with each other and that the coupling increases significantly during odor processing.

In this paper, we report a new four-dimensional autonomous hyperchaotic system, constructed from a Chua system where the piecewise-linear function usually taken to describe the nonlinearity of the Chua diode has been replaced by a cubic polynomial. Analytical and numerical procedures are conducted to study the dynamical behavior of the proposed new hyperchaotic system.

We analytically investigate the dynamics of the generalized Lorenz equations obtained by Stenflo for acoustic gravity waves. By using Descartes' Rule of Signs and Routh–Hurwitz Test, we decide on the stability of the fixed points of the Lorenz–Stenflo system, although without explicit solution of the eigenvalue equation. We determine the precise location where pitchfork and Hopf bifurcation of fixed points occur, as a function of the parameters of the system. Parameter-space plots, Lyapunov exponents, and bifurcation diagrams are used to numerically characterize periodic and chaotic attractors.

In this letter we report a new four-dimensional autonomous system, constructed from a chaotic Lorenz system by introducing an adequate feedback controller to the third equation. We show that when parameters are conveniently chosen, the control method can drive the chaotic Lorenz system to hyperchaotic regions. Analytical and numerical procedures are conducted to study the dynamical behaviors of the proposed new system.

This paper discusses the thermodynamics of a black hole with respect to Hawking radiation and the entropy. We look at a unified picture of black hole entropy and curvature and how this can lead to the usual black hole luminosity due to Hawking radiation. It is also shown how the volume inside the horizon, apart from the surface area (hologram!), can play a role in understanding the Hawking flux. In addition, holography implies a phase space associated with the interior volume and this happens to be just a quantum of phase space, filled with just one photon. The generalized uncertainty principle can be incorporated in this analysis. These results hold for all black hole masses in any dimension.

In the background dynamics of a spatially flat FLRW model of the universe, we investigate an interacting dark energy (DE) model in the context of Lyra’s geometry. Pressure-less dust is considered as dark matter, mass of which varies with time via scalar field in the sense that decaying of dark matter particles reproduces the scalar field. Here, quintessence scalar field is adopted as DE candidate which evolves in exponential potential. Mass of the dark matter particles is also considered to be evolved in exponential function of the scalar field. Cosmological evolution equations are studied in the framework of dynamical systems analysis. Dimension-less variables are chosen properly so that the cosmological evolution equations are converted into an autonomous system of ordinary differential equations. Linear stability is performed to find the nature of critical points by perturbing the system around the critical points in the phase-space. Classical stability is also executed by finding out the speed of sound. Dynamical systems explore several viable results which are physically interested in some parameter regions. Late-time scalar field-dominated attractors are found by critical points, corresponding to the accelerating universe. Scalar field-displacement vector field scaling solutions are realized that represent late-time decelerated universe. Dark energy-dark matter scaling solutions are also exhibited by critical points which correspond to accelerated attractors possessing similar order of energy densities of dark energy and dark matter, that provides the possible solutions of coincidence problem.

Applications of nonlinear dynamic tools for studying air pollution are gaining attention. Studies on ozone concentration in urban areas have reported the presence of low-dimensional deterministic natures and thus the possibility of good predictions of air pollution dynamics. In light of these encouraging results, a nonlinear deterministic approach is employed herein to study air pollution dynamics in a rural, and largely agricultural, setting in California. Specifically, air quality index (AQI) data observed at the University of California, Davis/National Oceanic and Atmospheric Administration (UCD/NOAA) climate station are studied. Four different daily AQI types of data are analyzed: maximum, minimum, difference (between maximum and minimum), and average. The correlation dimension method, a nonlinear dynamic technique that uses phase–space reconstruction and nearest neighbor concepts, is employed to identify the nature of the underlying dynamics, whether high-dimensional or low-dimensional. Correlation dimensions of 5.12, 6.20, 6.68, and 5.84 obtained for the above four series, respectively, indicate the presence of low-dimensional deterministic behavior, with six or seven dominant governing variables in the underlying dynamics. The dimension results and number of variables are in reasonable agreement with those reported by past studies, even though the studied data are different: rural versus urban, and AQI versus ozone concentration. Future efforts will focus on strengthening the present results on the nature of air pollution dynamics, identifying the actual governing variables, and predictions of air pollution dynamics.

After a cursory introduction of the basic ideas behind Born’s Reciprocal Relativity theory, the geometry of the cotangent bundle of spacetime is studied via the introduction of nonlinear connections associated with certain nonholonomic modifications of Riemann–Cartan gravity within the context of Finsler geometry. A novel gauge theory of gravity in the $8D$ cotangent bundle $T^{\ast }M$ of spacetime is explicitly constructed and based on the gauge group $SO(6,2){\times }_s{R}^8$ which acts on the tangent space to the cotangent bundle $T_{(x,p)}{T}^{\ast }M$ at each point $(x,p)$. Several gravitational actions involving curvature and torsion tensors and associated with the geometry of curved phase-spaces are presented. We conclude with a brief discussion of the field equations, the geometrization of matter, quantum field theory (QFT) in accelerated frames, T-duality, double field theory, and generalized geometry.

The Special Affine Fourier Transform (SAFT) generalizes a number of well-known unitary transformations as well as signal processing and optics related mathematical operations. Unlike the Fourier transform, the SAFT does not work well with the standard convolution operation. Recently, Q. Xiang and K. Y. Qin introduced a new convolution operation that is more suitable for the SAFT and by which the SAFT of the convolution of two functions is the product of their SAFTs and a phase factor. However, their convolution structure does not work well with the inverse transform insofar as the transform of the product of two functions is not equal to the convolution of the transforms. In this chapter we introduce a new convolution operation that works well with both the SAFT and its inverse leading to an analogue of the convolution and product formulas for the Fourier transform. Furthermore, we introduce a second convolution operation that leads to the elimination of the phase factor in the convolution formula obtained by Q. Xiang and K. Y. Qin. We conclude the chapter by introducing a convolution operation associated with the SAFT that will enable one to convolve a function and a sequence of numbers. Such an operation is needed in the study of shift-invariant spaces associated with the SAFT.

Real-time principal component analysis of an atmospheric pressure plasma electrical (current, voltage and drive frequency) parameters is reported. Lissajous figure phase-space projection of the current and voltage waveforms and linear transformation of the electrical parameters on a Loading plot is also performed. Using LabVIEW graphical programming it is shown that the Lissajous figure captures a snap-shot of the plasma chemistry in the form of current waveform micro-scale events. The Loading plot provides a time-evolving view of the process that enables differentiation of plasma modes (bifurcation) and chemistry.