Dynamical system, a description

The peroxidas-oxidase (PO) reaction is an important bridge between the chemical excitable oscillators Beluzov Zabotinsky reaction and biological oscillators such as intracellular $\textrm{Ca}^{2+}$ oscillators [6]. It is now clear that PO reactions shows a wide spectrum of interesting behaviors including simple oscillations, bistability, quasiperiodicity and chaos [7]. These behaviors have all been observed in vitro under well controlled laboratory conditions. This reaction appears in plants as part of the process of lignifications [8] with nicotinamide adenine dinucleotide (NADH) as electron donor. Its estoqueometric formulae is: $2 \, \mathtt{NADH} + \mathtt{O}_2 + 2 \, \mathtt{H} \to 2 \, \mathtt{NAD}^{+} + 2 \, \mathtt{H}_2\mathtt{O}$. In 1983, a model of PO reaction, now commonly referred as the Olsen's Model [9], was proposed. Simulations with the Olsen model quantitatively reproduce both the simple and chaotic oscillations of PO reactions. Studies with this model showed that increasing the parameter $k_3$ caused the system to undergo a transition from simple oscillations to chaos via a cascade of periodic doubling bifurcations. The Olsen's model involves four variables, molecular oxygen A, NADH B and two intermediate species. One of the intermediates is very likely $\textrm{NAD}^{+}$ X, while another one is oxyferrous peroxidase Y, also known as compound III. The complete mechanism corresponds system of four differential equations are given below, with $k_3$ as control parameter which induces a transition to chaos type I [10].

$$\frac{d A}{d t} = k_7-k_{-7} A-k_3 A B Y$$ (5)

$$\frac{d B}{d t} = k_8-k_1 B X-k_3 A B Y$$ (6)

$$\frac{d X}{d t} = k_1 B X-2 k_2 X^2 +3 k_3 A B Y -k_4 X +k_6$$ (7)

$$\frac{d Y}{d t} = 2 k_2 X^2-k_5 Y -k_3 A B Y$$ (8)