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In this work, the influences of geometry and domain size on spatiotemporal pattern formation are investigated to establish the parameter spaces for a cross-diffusive reaction–diffusion model on an annulus. By applying the linear stability theory, we derive the conditions which can give rise to Turing, Hopf and transcritical types of diffusion-driven instabilities. We explore whether the selection of a sufficiently large domain size, together with the appropriate selection of parameters, can give rise to the development of patterns on nonconvex geometries, e.g. annulus. Hence, the key research methodology and outcomes of our studies include a complete analytical exploration of the spatiotemporal dynamics in an activator-depleted reaction–diffusion system; a linear stability analysis to characterize the dual roles of cross-diffusion and domain size of pattern formation on an annulus region; the derivation of the instability conditions through the lower and upper bounds of the domain size; the full classification of the model parameters; and a demonstration of how cross-diffusion relaxes the general conditions for the reaction–diffusion system to exhibit pattern formation. To validate the theoretical findings and predictions, we employ the finite element method to reveal the spatial and spatiotemporal patterns in the dynamics of the cross-diffusive reaction–diffusion system within a two-dimensional annular domain. These observed patterns resemble those found in ring-shaped cross-sectional scans of hypoxic tumors. Specifically, the cross-section of an actively invasive region in a hypoxic tumor can be effectively approximated by an annulus.

In this paper, we study the parameter space of the quadratic polynomial family f_λ,μ(z, w) = (λz + w², μw + z²), which exhibits interesting dynamics. Two distinct subsets of the parameter space are studied as appropriate analogs of the one-dimensional Mandelbrot set and some of their properties are proved by using Lyapunov exponents. In the more general context of holomorphic families of regular maps, we show that the sum of the Lyapunov exponents is a plurisubharmonic function of the parameter, and pluriharmonic on the set of expanding maps. Moreover, for the family f_λ,μ, we prove that the sum of the Lyapunov exponents is continuous.

To explain the matter–antimatter asymmetry, a supersymmetric extension of the Standard Model (SM) is proposed where baryon and lepton numbers are local-gauged (BLMSSM), and exotic superfields are introduced when gauge group is enlarged to $\mathrm{\text{SU}}{(3)}_C\otimes \mathrm{\text{SU}}{(2)}_L\otimes \mathrm{\text{U}}{(1)}_Y\otimes \mathrm{\text{U}}{(1)}_B\otimes \mathrm{\text{U}}{(1)}_L$. Owing to the consistency of the SM prediction and the observation of large hadron collider (LHC), the parameter space that related to the masses of new particles is stringently constrained. By diagonalizing the mass-squared matrices for neutral scalar sectors and the mass-squared matrices for exotic quarks, we obtain the mass of these particles, then present the contour plot of mass varying from different parameters with some assumptions, so the constraints on model parameter can be obtained with different lower limits of particle mass.

A class of cellular neural networks, containing the input and output terms, is investigated in this paper. The input and output synaptic weights are taken as the parameter in conjunction with the thresholds, and parameter spaces are constructed from them. Based on geometrical method, the parameter spaces are decomposed into many finite regions, and the value assignment range of stationary solutions is given when system parameters are determined on some regions.

In previous work, the authors explored the parameter space for Chua's circuit and its generalizations, discovering new routes to chaos, and nearly a thousand new attractors. These were obtained by varying the parameters of the physical circuit and of systems derived from it. Here, we present a novel class of computational system that does not respect the classical constraints in Chua's circuit, and that generates chaotic dynamics via an iterative process based on discrete versions of the equations for Chua's circuit and its variants. We call these systems Chua Machines. After presenting the chaotic dynamics, we provide a formal description of Chua Machines and a Gallery of 222 3D images, illustrating their dynamics. We discuss the method used to discover these systems and the metrics applied in the exploration of their parameter space and offer examples of highly complex bifurcation maps, together with images showing how patterns can evolve with time, or vary significantly changing the values of one of the parameters. Finally, we present a detailed analysis of qualitative changes in a Chua Machine as it traverses the parameter space of the bifurcation map. The evidence suggests that these dynamics are even richer and more complex than their counterpart in the continuous domain.

The stability analysis is used in order to identify and to map different dynamics of Chua’s circuit in full four-parameter spaces. The study is performed using describing functions that allow to identify fixed point, periodic orbit, and unstable states with relative accuracy, as well as to predict route to chaos and hidden dynamics that conventional computational methods do not detect. Numerical investigations based on the computation of eigenvalues and Lyapunov exponents partially support the predictions obtained from the theoretical analysis since they do not capture the multiple dynamics that can coexist in the operation of Chua’s circuit. Attractors obtained from initial conditions outside of neighborhoods of the equilibrium points confirm the multiplicity of dynamics in the operation of Chua’s circuit and corroborate the theoretical analysis.

Formation and study of periodic orbits in phase space in the case of nonlinear oscillators have been a topic of much interest in the recent past. In the current work, a method to go deep inside the limit cycle zone on one side of the bifurcation curve of a 2D non-Lienard biochemical oscillator has been introduced. It is discussed how such an introduction facilitates predicting the boundaries of limit cycles at various points of parameter space, nearly accurately, by the use of perturbative Renormalization Group. Sel’kov model of Glycolytic oscillator has been chosen as the base model to introduce the method.

In this paper, we study the approximation orders of a real number $x\in (0,1)$ by the partial sums of its $\beta$-expansions as $\beta$ varies in the parameter space $\{\beta \in ℝ:\beta >1\}$. More precisely, letting $S_n(x,\beta )$ be the partial sum of the first $n$ items of the $\beta$-expansion of $x$, we prove that for any real number $x\in (0,1)$, the approximation order of $x$ by $S_n(x,\beta )$ is ${\beta }^{-n}$ for Lebesgue almost all $\beta >1$. Moreover, we obtain the size of the set of $\beta >1$ for which $x$ can be approximated with a more general order ${\beta }^{-\phi (n)}$, where $\phi :\mathbb{N}\to {ℝ}^{+}$ is a positive function. We also determine the Hausdorff dimension of the set

This paper reports numerical examples for a 3 individual 2 good CES/LES pure exchange economy in which 5 equilibria exist. We explore the size of the regions of the parameter space for which 5 equilibria persist, and show these ranges to be very small in each parameter. Other features of the equilibrium manifolds are examined. The number of equilibria can jump from 5 to 3 to 1, or even from 5 to 1. We also find changes of (Kehoe) equilibrium indices from -1 to +1 or from +1 to -1. Then, we provide parameter changes simultaneously in a large range that preserve given 5 equilibria.

Motion estimation is the crucial step for video stabilization algorithm. There are many literatures focus on it and various methods have been proposed. The solution of this problem is now sophisticated when only replacement is taken into account. But if the rotation between two successive frames can't be ignored, conventional methods may lose their valid. In this paper, we aim to estimate motion parameters when replacement and rotation both exist in the video sequence. A framework based on feature points is employed. In order to find corresponding feature points in successive frames to estimate global motion parameters, a novel feature point match algorithm based on parameter space is proposed. It can endure great changes of the distribution of feature points and the processing speed is rather fast.