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Abstract The Hilbert numberH(n) is defined as the maximum number of limit cycles of a planar autonomous system of ordinary differential equations (ODEs) with right-hand sides containing polynomials of degree at most$$n \in {{\mathbb {N}}}$$ $n\in N$. The dynamics of chemical reaction systems with two chemical species can be (under mass-action kinetics) described by such planar autonomous ODEs, wherenis equal to the maximum order of the chemical reactions in the system. Analogues of the Hilbert number H(n) for three different classes of chemical reaction systems are investigated: (i) chemical systems with reactions up to then-th order; (ii) systems with up ton-molecular chemical reactions; and (iii) weakly reversible chemical reaction networks. In each case (i), (ii) and (iii), the question on the number of limit cycles is considered. Lower bounds on the modified Hilbert numbers are provided for both algebraic and non-algebraic limit cycles. Furthermore, given a general algebraic curve$$h(x,y)=0$$ $h(x,y)=0$of degree$$n_h \in {{\mathbb {N}}}$$ $n_h\in N$and containing one or more ovals in the positive quadrant, a chemical system is constructed which has the oval(s) as its stable algebraic limit cycle(s). The ODEs describing the dynamics of the constructed chemical system contain polynomials of degree at most$$n=2\,n_h+1.$$ $n=2n_h+1.$Considering$$n_h \ge 4,$$ $n_h\ge 4,$the algebraic curve$$h(x,y)=0$$ $h(x,y)=0$can contain multiple closed components with the maximum number of ovals given by Harnack’s curve theorem as$$1+(n_h-1)(n_h-2)/2$$ $1+(n_h-1)(n_h-2)/2$, which is equal to 4 for$$n_h=4.$$ $n_h=4.$Algebraic curve$$h(x,y)=0$$ $h(x,y)=0$with$$n_h=4$$ $n_h=4$and the maximum number of four ovals is used to construct a chemical system which has four stable algebraic limit cycles. 

Abstract We considertoric dynamical systems, which are also calledcomplex-balanced mass-action systems. These are remarkably stable polynomial dynamical systems that arise from the analysis of mathematical models of reaction networks when, under the assumption of mass-action kinetics, they can give rise tocomplex-balanced equilibria. Given a reaction network, we study theset of parameter valuesfor which the network gives rise to toric dynamical systems, also calledthe toric locusof the network. The toric locus is an algebraic variety, and we are especially interested in its topological properties. We show that complex-balanced equilibriadepend continuouslyon the parameter values in the toric locus, and, using this result, we prove that the toric locus has a remarkableproduct structure: it is homeomorphic to the product of the set of complex-balanced flux vectors and the affine invariant polyhedron of the network. In particular, it follows that the toric locus is acontractible manifold. Finally, we show that the toric locus is invariant with respect to bijective affine transformations of the generating reaction network. 

ABSTRACT Mathematical models of reaction networks are ubiquitous in applications, especially in chemistry, biochemistry, chemical engineering, ecology, and population dynamics. Under the standard assumption ofmass‐action kinetics, reaction networks give rise to general dynamical systems with polynomial right‐hand side. These depend on many parameters that are difficult to estimate and can give rise to complex dynamics, including multistability, oscillations, and chaos. On the other hand, a special class of reaction systems calledcomplex‐balanced systemsare known to exhibit remarkably stable dynamics; in particular, they have unique positive fixed points and no oscillations or chaotic dynamics. One difficulty, when trying to take advantage of the remarkable properties of complex‐balanced systems, is that the set of parameters where a network satisfies complex balance may have positive codimension and therefore zero measure. To remedy this we are studyingdisguised complex balanced systems(also known asdisguised toric systems), which may fail to be complex balanced with respect to an original reaction network , but are actually complex balanced with respect to some other network , and therefore enjoy all the stability properties of complex‐balanced systems. This notion is especially useful when the set of parameter values for which the network gives rise to disguised toric systems (i.e., thedisguised toric locusof ) has codimension zero. Our primary focus is to compute the exact dimension (and therefore the codimension) of this locus. We illustrate the use of our results by applying them to Thomas‐type and circadian clock models.