<p>This paper focuses on the normal form computation of the pitchfork&nbsp;bifurcation for&nbsp;the&nbsp;fractional&nbsp;differential&nbsp;system with Caputo-Hadamard&nbsp;derivative. Recently, it has been proved that Hadamard<br>
derivative also plays a significant role in the practical problems&nbsp;related to mechanics and engineering, especially in the crack problem&nbsp;and the Lomnitz logarithmic creep law of viscoelastic materials. Bifurcation analysis is the important topic of&nbsp;the nonlinear differential system with parameter. When the parameter changes&nbsp;to a critical value, the topological structure of the system will change&nbsp;suddenly, which implies the appearance of bifurcation. Bifurcation phenomenon&nbsp;is of great significance in the analysis of practical dynamic problems,&nbsp;so it is necessary to discuss the bifurcation of fractional differential system&nbsp;with parameter. Furthermore, the normal form is one of the useful tools for&nbsp;bifurcation analysis, and it is the simplest differential system of nonlinear&nbsp;system with essential characteristics near bifurcation point. Hence, we consider the&nbsp;normal form of the pitchfork&nbsp;bifurcation for the&nbsp;fractional&nbsp;differential&nbsp;system with Caputo-Hadamard&nbsp;derivative. First, by constructing homeomorphism, we present a system&nbsp;that is locally topologically equivalent to the Caputo-Hadamard fractional&nbsp;differential system. Then, based on Taylor expansion and the Implicit&nbsp;Function Theorem, we study the normal form of the pitchfork bifurcation&nbsp;for the fractional differential system with Caputo-Hadamard derivative.</p>

Normal Form of Bifurcation for Caputo-Hadamard Fractional Differential System With a Parameter

Paper Proceedings: https://asmedigitalcollection.asme.org/IDETC-CIE/IDETC-CIE2021/volume/85369

ASME 2021 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (IDETC/CIE2021)

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