The solution of
 s
 exp(
 s
 ) =
 a
 is not always the lambert
 w
 function of
 a

Corless, Robert M.; Ding, Hui; Higham, Nicholas J.; Jeffrey, David J.

doi:10.1145/1277548.1277565

Cited by 16 publications

(14 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Because we can decouple the equations for the characteristic roots, we can avoid the use of the matrix version of the Lambert W -function [25], which, moreover, lacks [27][28][29] a property similar to Property 1.…”

Section: A Characteristic Equation Without Diffusionmentioning

confidence: 99%

“…We thus obtained the same condition, D v > D u , one can find in the framework without delay, but now the working hypothesis is on the sign of derivatives f uτ and g vτ related to the delayed concentrations of the activator and inhibitor. We can determine a critical value for the ratio of the diffusion coefficients D = Dv Du , for which the interval given by (28) reduces to a single point, the condition being ∆ = 0, namely…”

Section: Stationary Turing Patternsmentioning

confidence: 99%

“…Combining the previous equation with (28) gives the corresponding critical value for Λ α for a given D. If D = D crit , the rightmost eigenvalue µ α 2 has zero value [9]. If D > D crit , the system with diffusion is unstable.…”

Section: Stationary Turing Patternsmentioning

confidence: 99%

See 2 more Smart Citations

Pattern formation in a two-component reaction–diffusion system with delayed processes on a network

Petit

Asllani

Fanelli

et al. 2016

Physica A: Statistical Mechanics and its Applications

View full text Add to dashboard Cite

Reaction-diffusion systems with time-delay defined on complex networks have been studied in the framework of the emergence of Turing instabilities. The use of the Lambert W -function allowed us get explicit analytic conditions for the onset of patterns as a function of the main involved parameters, the time-delay, the network topology and the diffusion coefficients. Depending on these parameters, the analysis predicts whether the system will evolve towards a stationary Turing pattern or rather to a wave pattern associated to a Hopf bifurcation. The possible outcomes of the linear analysis overcome the respective limitations of the single-species case with delay, and that of the classical activator-inhibitor variant without delay. Numerical results gained from the Mimura-Murray model support the theoretical approach.

show abstract

Section: A Characteristic Equation Without Diffusionmentioning

confidence: 99%

Section: Stationary Turing Patternsmentioning

confidence: 99%

See 1 more Smart Citation

Pattern formation in a two-component reaction–diffusion system with delayed processes on a network

Petit

Asllani

Fanelli

et al. 2016

Physica A: Statistical Mechanics and its Applications

View full text Add to dashboard Cite

show abstract

“…In the matrix case, for A ∈ C n×n any solution of the equation W e W = A can be called a matrix Lambert W function [10]. The solutions of this matrix equation break into two types.…”

Section: ) (O) the Colors Of The Curves That Separate Two Adjacent Rmentioning

confidence: 99%

An Algorithm for the Matrix Lambert $W$ Function

Fasi

Higham

Iannazzo³

2015

SIAM J. Matrix Anal. & Appl.

Self Cite

View full text Add to dashboard Cite

Abstract. An algorithm is proposed for computing primary matrix Lambert W functions of a square matrix A, which are solutions of the matrix equation W e W = A. The algorithm employs the Schur decomposition and blocks the triangular form in such a way that Newton's method can be used on each diagonal block, with a starting matrix depending on the block. A natural simplification of Newton's method for the Lambert W function is shown to be numerically unstable. By reorganizing the iteration a new Newton variant is constructed that is proved to be numerically stable. Numerical experiments demonstrate that the algorithm is able to compute the branches of the matrix Lambert W function in a numerically reliable way.

show abstract

“…Recently, a special function named Lambert W function, defined as the solution w=W(z) of e = , w w z has been frequently used for the stability analysis of time-delay system [22][23][24][25][26][27][28]. Though the attempt of generalizing the scalar Lambert W function to matrix Lambert W function for more complicated delay differential equations has been shown not always successful [29], the use of Lambert W function does work effectively for the stability analysis of some special time-delay systems.…”

Section: Introductionmentioning

confidence: 99%

A simple algorithm for testing the stability of periodic solutions of some nonlinear oscillators with large time delay

Zhang

Wang

2011

Sci. China Technol. Sci.

View full text Add to dashboard Cite

Periodic solutions occur commonly in linear or nonlinear dynamical systems. In some cases, the stability of a periodic solution holds if the rightmost characteristic root has negative real part. Based on the Lambert W function, this paper presents a simple algorithm for locating the rightmost characteristic root of periodic solutions of some nonlinear oscillators with large time delay. As application, the proposed algorithm is used to study the primary resonance and 1/3 subharmonic resonance of the Duffing oscillator under harmonic excitation and delayed feedback, as well as the control problem of the van der Pol oscillator under harmonic excitation by using delayed feedback, with a number of case studies. The main advantage of this algorithm is that though very simple in implementation, it works effectively with high accuracy even if the delay is large. time-delay system, weakly nonlinear, stability, periodic solution, characteristic function, Lambert W function Citation:Li J Y, Zhang L, Wang Z H. A simple algorithm for testing the stability of periodic solutions of some nonlinear oscillators with large time delay. Sci

show abstract

The solution of s exp( s ) = a is not always the lambert w function of a

Cited by 16 publications

References 11 publications

Pattern formation in a two-component reaction–diffusion system with delayed processes on a network

Pattern formation in a two-component reaction–diffusion system with delayed processes on a network

An Algorithm for the Matrix Lambert $W$ Function

A simple algorithm for testing the stability of periodic solutions of some nonlinear oscillators with large time delay

Contact Info

Product

Resources

About