Approximate Solutions to Lane-Emden Equation for Stellar Configuration

In this paper, we consider the Lane-Emden equation of the first kind which arises in the study of stellar structures. We use multiple algorithms, based on Homotopy Analysis Method (HAM) to find the convergent series solutions to the singular, non-linear, initial value problem. It is found that the radius of convergence for the solutions is affected by three factors: the choice of initial value, the order and type of non-linearity, and the linear operator used. We then compare analytical results to the 20th order series solution, the Pade approximant to the series, and the approximate solution obtained via the Runge-Kutta-Fehlberg method (RKF45)

Analytical Approach to Study the Generalized Lane-Emden Model Arises in the Study of Stellar Configuration

In this paper, we consider the generalized Lane-Emden model which arises in the study of steller configuration. We came up with the nonlinear, multi-singular, initial value ordinary differential equations. Mathematical induction is used to verify the generalized non-iterative higher-order Lane-Emden type equation. We use various Homotopy Analysis Method (HAM) algorithms to find the convergent series solutions of the model. It is observed how the choice of initial value, increasing values of M in the polynomial nonlinearity yM, and different choices of HAM algorithms impact the solution radius of convergence. Convergent series solutions obtained from HAM algorithms are compared with the traditional power series and Runge-Kutta-Fehlberg method (RKF45). The traditional series solution follows the actual solution in the domain where the actual solution is positive while HAM does not require domain restriction

Data from: Huygens' clocks revisited

In 1665, Huygens observed that two identical pendulum clocks, weakly coupled through a heavy beam, soon synchronized with the same period and amplitude but with the two pendula swinging in opposite directions. This behaviour is now called anti-phase synchronization. This paper presents an analysis of the behaviour of a large class of coupled identical oscillators, including Huygens' clocks, using methods of equivariant bifurcation theory. The equivariant normal form for such systems is developed and the possible solutions are characterized. The transformation of the physical system parameters to the normal form parameters is given explicitly and applied to the physical values appropriate for Huygens' clocks, and to those of more recent studies. It is shown that Huygens' physical system could only exhibit anti-phase motion, explaining why Huygens observed exclusively this. By contrast, some more recent researchers have observed in-phase or other more complicated motion in their own experimental systems. Here, it is explained which physical characteristics of these systems allow for the existence of these other types of stable solutions. The present analysis not only accounts for these previously observed solutions in a unified framework, but also introduces behaviour not classified by other authors, such as a synchronized toroidal breather and a chaotic toroidal breather

DOUBLE ZERO BIFURCATION WITH HUYGENS SYMMETRY

Abstract. This paper presents a study of the effects of symmetry on the generic bifurcation at a double-zero eigenvalue that was first investigated by Bogdanov and Takens. Two different symmetry groups are considered: Huygens symmetry and odd-Huygens symmetry. Here Huygens symmetry means that the system is equivariant under permutation of the two state variables. Using Hilbert-Weyl theory, normal forms are given for each symmetry group. The normal forms are further simplified using Gavrilov's transformation, and formulae are presented that allow identification of the normal form parameters in terms of the coefficients of the original system. Complete sets of codimension-two bifurcation diagrams with representative phase portraits are presented, for both symmetries. These diagrams exhibit codimension-one bifurcations including saddlenode, pitchfork, Hopf and heteroclinic. The effects of symmetry-breaking perturbations on these codimension-one bifurcations are analyzed. The results presented here contrast strongly with the classical results of Bogdanov and Takens

An Efficient Algorithm for Eigenvalue Problem of Latin Squares in a Bipartite Min-Max-Plus System

In this paper, we consider the eigenproblems for Latin squares in a bipartite min-max-plus system. The focus is upon developing a new algorithm to compute the eigenvalue and eigenvectors (trivial and non-trivial) for Latin squares in a bipartite min-max-plus system. We illustrate the algorithm using some examples. The proposed algorithm is implemented in MATLAB, using max-plus algebra toolbox. Computationally speaking, our algorithm has a clear advantage over the power algorithm presented by Subiono and van der Woude. Because our algorithm takes 0 . 088783 sec to solve the eigenvalue problem for Latin square presented in Example 2, while the compared one takes 1 . 718662 sec for the same problem. Furthermore, a time complexity comparison is presented, which reveals that the proposed algorithm is less time consuming when compared with some of the existing algorithms