Pseudospectral Approximation of Hopf Bifurcation for Delay Differential Equations

Abstract

Pseudospectral approximation reduces delay differential equations (DDE) to ordinary differential equations (ODE). Next one can use ODE tools to perform a numerical bifurcation analysis. By way of an example we show that this yields an efficient and reliable method to qualitatively as well as quantitatively analyze certain DDE. To substantiate the method, we next show that the structure of the approximating ODE is reminiscent of the structure of the generator of translation along solutions of the DDE. Concentrating on the Hopf bifurcation, we then exploit this similarity to reveal the connection between DDE and ODE bifurcation coefficients and to prove the convergence of the latter to the former when the dimension approaches infinity.

Numerical Bifurcation Analysis of Physiologically Structured Population Models via Pseudospectral Approximation

“…We also remark that proving convergence of Hopf bifurcations requires not only the convergence of the eigenvalues, but also to verify that the transversality conditions are satisfied at the bifurcation point, and the direction of bifurcation is preserved as M → ∞ [1,2]. For DDE, this is done in [7]. To adapt the proofs to RE, one should obtain explicit formulas for the direction of Hopf and verify the convergence (see [1,Remark 2.22] and [7]).…”

“…For DDE, this is done in [7]. To adapt the proofs to RE, one should obtain explicit formulas for the direction of Hopf and verify the convergence (see [1,Remark 2.22] and [7]).…”

“…The nontrivial equilibrium undergoes a Hopf bifurcation as β 0 increases. An explicit parametrization of the Hopf bifurcation curves in the plane (µ, β 0 e −µ /µ) is computed in [7] for the DDE (2.3). Moreover, the numerical bifurcation analysis of (2.3) can be performed with standard software for DDE, like DDE-BIFTOOL for MATLAB [24,25].…”

“…In [4] the idea is launched to first reduce the infinite dimensional dynamical system corresponding to a delay equation to a finite dimensional one by pseudospectral approximation, and next use tools for ordinary differential equations (ODE) in order to perform a numerical bifurcation analysis. Several examples illustrate that this approach is promising (also see [5,6,7,8,9,10,11]). Note that we restrict to bounded maximal delay, as unbounded delays require to consider different (exponentially weighted) function spaces and corresponding results from weighted interpolation theory [11,12].…”

Study of time fractional proportional delayed multi‐pantograph system and integro‐differential equations

“…It is worthy to note that the evaluation of the exact solutions to the fractional differential equations is very tough task. In this consequence, a variety of analytical/numerical techniques have been developed for studying the behaviors of the physical models governed by FPDEs, [10][11][12][13] and so the study of fractional-order system of PDEs is very demanding. Being transcendental aspect of the delay concept makes these models more complex to reliably determine the behavior of numerical solutions.…”