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Nonlinear dynamic analysis and numerical continuation of periodic orbits in high-index differential–algebraic equation systems

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Abstract

In this paper, we present a novel methodology for nonlinear dynamic analysis of chemical processes that are posed as differential–algebraic equations (DAE) systems. With the proposed approach, for the first time, high-index systems, which are often the result of computer-aided modeling, can be treated “as is,” i.e., without the need for model reformulation in order to fit in particular structures (such as the Hessenberg forms) or a preconditioning procedure such as index reduction. This is a desirable feature because special forms cannot always be achieved, and reduced-index systems may present a different behavior than the original one due to the well-known drift-off effect or even result in misleading stability conclusions. The main problems addressed here are the direct computation of Hopf bifurcation points and the stability analysis and numerical continuation of steady-state and periodic solutions. The developed algorithms were packed together in ContiNum, a MATLAB toolbox with free distribution. In order to illustrate the methodology, an example of a high-index system is discussed in detail, including the analysis of its low-index counterpart, showing that bifurcation diagrams can be accurately built without index reduction.

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Data Availability Statement

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Notes

  1. https://github.com/asanet/continum

  2. It is also known as the transversality condition [1].

  3. For simplicity, the equilibrium constant \( (K_\text {eq}) \) is assumed to be invariant with temperature \( (x_3) \).

  4. The selection of the pair \( ({\varvec{\mathrm {x}}}_1, D_a) \) was made because it is the classic choice for this kind of problem (\( x_1 \) is the concentration of reactant A, which measures the reaction extent, and \( D_a \) is the Damköhler number, which is a measure of the relative importance of chemical reaction against the forced flow).

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Funding

This study was financed in part by the Coordination for the Improvement of Higher Education Personnel (CAPES), Finance Code 001, and the National Council for Scientific and Technological Development (CNPq), Grant Numbers 302893/2013-0 and 152572/2016-3.

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Authors and Affiliations

  1. Chemical Engineering Program, COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

    Ataíde S. Andrade Neto, Argimiro R. Secchi & Príamo A. Melo

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  1. Ataíde S. Andrade Neto
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  2. Argimiro R. Secchi
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  3. Príamo A. Melo
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Correspondence to Ataíde S. Andrade Neto.

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All the developed software applications are available at https://github.com/asanet/continum

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This study was financed in part by the Coordination for the Improvement of Higher Education Personnel (CAPES), Finance Code 001, and the National Council for Scientific and Technological Development (CNPq), Grant Numbers 302893/2013-0 and 152572/2016-3.

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Andrade Neto, A.S., Secchi, A.R. & Melo, P.A. Nonlinear dynamic analysis and numerical continuation of periodic orbits in high-index differential–algebraic equation systems. Nonlinear Dyn 108, 1495–1507 (2022). https://doi.org/10.1007/s11071-022-07254-4

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