2015 AIChE Annual Meeting Proceedings

(508d) Bifurcation and Stability Analysis of Nonlinear Dynamic Systems Using Complete Search

Authors

Mario E. Villanueva - Presenter, Imperial College London
Benoit Chachuat - Presenter, Imperial College London
Jai Rajyaguru - Presenter, Imperial College London

Dynamic systems found in chemical and biological engineering are typically governed by parametric nonlinear dynamic systems of the form

$\displaystyle \dot{x}(t) = f(x(t),p)$    

where
$ x(t)\in\mathbb{R}^{n_x}$ and
$ p\in\mathbb{R}^{n_p}$ denote the state and parameter vectors, respectively. An important aspect of the asymptotic study of such systems involves computing the equilibrium manifold
$ \Gamma:=\{(\hat x,\hat p)\in \mathbb{R}^{n_x+n_p} \vert f(\hat x,\hat p) = 0\}$, and analyzing its stability. Another related aspect is the computation of bifurcation points, whereby qualitative change in the dynamics occur. Typically, stability and bifurcation analysis are performed using continuation methods [1]. Such methods use numerical integration starting from a grid of initial conditions to identify stable equilibrium points. Then, parameter perturbation and local search are used to reconstruct the equilibrium manifold. Due to their local nature, the results obtained using numerical continuation may be incomplete, which may lead to missing unconnected branches for instance.

The focus of this paper is on using complete-search methods to rigorously enclose the equilibrium manifold and all the bifurcation points on a domain
$ Z_0 := X_0\times P_0 \subseteq \mathbb{R}^{n_x+n_p}$. A set-inversion algorithm [3] is used in order to compute partitions
$ Z_0 := X_0\times P_0 \subseteq \mathbb{R}^{n_x+n_p}$
satisfying

$\displaystyle \bigcup_{Z\in\mathbb{Z}_\text{in}}Z\subseteq\Gamma\subseteq\bigcup_{Z\in\mathbb{Z}_\text{bnd}\cup\mathbb{Z}_\text{in}}Z\,.$

Enclosures of the image of f under Z are computed using polynomial models, in particular Chebyshev model arithmetic [8], and optimization-based domain reduction is applied to accelerate the convergence of the set-inversion algorithm [5]. Furthermore, a set-valued implicit equation solver [7] is used to compute a Chebyshev model representation of the boundary of the set whenever possible. It is important to note that, in this proposed approach, bifurcation points can be computed without computing the equilibrium manifold first. This is done by appending algebraic conditions for bifurcation (steady-state or Hopf bifurcations) [2] to the equilibrium constraints. Another contribution of the paper is a validated a posteriori stability analysis of the equilibrium points. This analysis is based on an validated extension of the Neville elimination algorithm for Hurwitz matrices [6] using Chebyshev models. Bifurcation points can also be obtained as a by-product of this stability analysis.

The performance of the proposed algorithm is illustrated with several case studies of various complexity drawn from chemical engineering and systems biology. For illustration, the figure below shows the results of a bifurcation and stability analysis for a simple nutrient-resource-consumer model [4] with respect to the nutrient density parameter. Stable equilibrium points are shown in blue, unstable points in red, steady-state bifurcation points are shown as circles and Hopf bifurcations as squares.

Image NRC

References:

  1. E. L. Allgower and K. Georg. Numerical Continuation Methods. Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 1990.
  2. M. El Kahoui and A. Weber. Deciding Hopf bifurcations by quantifier elimination in a software-component architecture. Journal of Symbolic Computation, 30(2):161-179, 2000.
  3. L. Jaulin and E. Walter. Set inversion via interval analysis for nonlinear bounded-error estimation. Automatica, 29(4):1053-1064, July 1993.
  4. B. W. Kooi. Numerical bifurcation analysis of ecosystems in a spatially homogeneous environment. Acta Biotheoretica, 51(3):189-222, 2003.
  5. R. Paulen, M. E. Villanueva, and B. Chachuat. Guaranteed parameter estimation of nonlinear dynamic systems using high-order bounding techniques with domain and CPU-time reduction strategies. IMA Journal of Mathematical Control & Information, in press (doi: 10.1093/imamci/dnu055), 2015.
  6. J. M. Peña. Characterizations and stable tests for the Routh-Hurwitz conditions and for total positivity. Linear Algebra & Its Applications, 393:319-332, 2004.
  7. J. Rajyaguru and B. Chachuat. Taylor models in deterministic global optimization for Large-Scale systems with few degrees of freedom. Computer Aided Chemical Engineering, 32:973-978, 2013.
  8. J. Rajyaguru, M. E. Villanueva, B. Houska, and B. Chachuat. Higher-order inclusions of nonlinear systems by Chebyshev models. In Proc. AIChE Annual Meeting 2014, Paper 385510, 2014.