Planar chemical reaction systems with algebraic and non-algebraic limit cycles

The Hilbert number H(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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi></mml:mrow> </mml:math> . The dynamics of chemical reaction systems with two chemical species can be (under mass-action kinetics) described by such planar autonomous ODEs, where n is 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 the n-th order; (ii) systems with up to n-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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>h</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn></mml:mrow> </mml:math> of degree <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>n</mml:mi> <mml:mi>h</mml:mi></mml:msub> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi></mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mspace></mml:mspace> <mml:msub><mml:mi>n</mml:mi> <mml:mi>h</mml:mi></mml:msub> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>.</mml:mo></mml:mrow> </mml:math> Considering <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>n</mml:mi> <mml:mi>h</mml:mi></mml:msub> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo></mml:mrow> </mml:math> the algebraic curve <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>h</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn></mml:mrow> </mml:math> can contain multiple closed components with the maximum number of ovals given by Harnack's curve theorem as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mrow><mml:mo>(</mml:mo> <mml:msub><mml:mi>n</mml:mi> <mml:mi>h</mml:mi></mml:msub> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo></mml:mrow> <mml:mrow><mml:mo>(</mml:mo> <mml:msub><mml:mi>n</mml:mi> <mml:mi>h</mml:mi></mml:msub> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo></mml:mrow> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:math> , which is equal to 4 for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>n</mml:mi> <mml:mi>h</mml:mi></mml:msub> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> <mml:mo>.</mml:mo></mml:mrow> </mml:math> Algebraic curve <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>h</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn></mml:mrow> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>n</mml:mi> <mml:mi>h</mml:mi></mml:msub> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn></mml:mrow> </mml:math> and the maximum number of four ovals is used to construct a chemical system which has four stable algebraic limit cycles.