Dynamic systems found in chemical and biological engineering are typically governed by parametric nonlinear dynamic systems of the form and denote the state and parameter vectors, respectively. An important aspect of the asymptotic study of such systems involves computing the equilibrium manifold , 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 . 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 . A set-inversion algorithm  is used in order to compute partitions satisfying
Enclosures of the image of f under Z are computed using polynomial models, in particular Chebyshev model arithmetic , and optimization-based domain reduction is applied to accelerate the convergence of the set-inversion algorithm . Furthermore, a set-valued implicit equation solver  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)  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  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  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.
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