We propose and solve an extended fractional-order Oregonator model for the Belousov–Zhabotinsky reaction by extending the classical autocatalytic core with explicit proton balance and buffering. The resulting Caputo system <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${}^{C}D_t^{\alpha }\textbf{X}=\textbf{F}(\textbf{X})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mrow /> <mi>C</mi> </mmultiscripts> <msubsup> <mi>D</mi> <mi>t</mi> <mi>α</mi> </msubsup> <mi mathvariant="bold">X</mi> <mo>=</mo> <mi mathvariant="bold">F</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$0<\alpha \le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, is analyzed analytically via the fractional differential transform method and numerically using the Adams–Bashforth–Moulton predictor–corrector scheme. In the proposed model, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textbf{X}=(X,Y,Z,H,P)^{\top }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">X</mi> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo>,</mo> <mi>H</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mi>⊤</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> represents the autocatalyst (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$X\sim \hbox {HBrO}_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>∼</mo> <msub> <mtext>HBrO</mtext> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>), inhibitor (<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$Y\sim \hbox {Br}^{-}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Y</mi> <mo>∼</mo> <msup> <mtext>Br</mtext> <mo>-</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>), oxidized catalyst (Z), free protons (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$H\sim \hbox {H}^{+}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo>∼</mo> <msup> <mtext>H</mtext> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>), and buffered protons (P). The reservoir variable P introduces delayed proton exchange and additional feedback, thereby capturing proton-regulated oscillations observed experimentally. Semi-analytical solutions obtained via the fractional differential transform method yield convergent fractional power series, while numerical integration using Adams–Bashforth–Moulton schemes recovers classical Runge–Kutta dynamics as <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha \rightarrow 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">→</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Using comparison principles and Lyapunov-type functionals, positivity <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\textbf{X}(t)\in \mathbb {R}^{5}_{\ge 0}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">X</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msubsup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mo>≥</mo> <mn>0</mn> </mrow> <mn>5</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation> and uniform dissipativity are established, ensuring the existence of a compact absorbing set. The equilibrium set contains a trivial state and a unique chemically admissible equilibrium <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\textbf{E}^{+}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="bold">E</mi> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation>. Spectral analysis of the block Jacobian <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$J(E^{+})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>J</mi> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mo>+</mo> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> shows that the trivial equilibrium is unstable, while for <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$0<\alpha <1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, Matignon’s criterion <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$|\arg (\lambda )|>\alpha \pi /2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mo>arg</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mo>></mo> <mi>α</mi> <mi>π</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> confirms that fractional memory deforms stability sectors without suppressing oscillatory (<InlineEquation ID="IEq13"> <EquationSource Format="TEX">$ {\text {HBrO}}_{2} - {\text {Br}}^{ - } - {\text {H}}^{ + } $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>HBrO</mtext> <mn>2</mn> </msub> <mo>-</mo> <msup> <mrow> <mtext>Br</mtext> </mrow> <mo>-</mo> </msup> <mo>-</mo> <msup> <mrow> <mtext>H</mtext> </mrow> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>) instabilities. Phase portraits, nullclines, equilibrium manifolds, bifurcation diagrams, and sensitivity indices demonstrate how proton buffering (<InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\hbox {H}^{+}\rightleftarrows \hbox {H}^{+}_{buf}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mtext>H</mtext> <mo>+</mo> </msup> <mo>⇄</mo> <msubsup> <mtext>H</mtext> <mrow> <mi mathvariant="italic">buf</mi> </mrow> <mo>+</mo> </msubsup> </mrow> </math></EquationSource> </InlineEquation>), kinetic rates, and fractional memory jointly regulate oscillation amplitude, stability, and long-time chemical dynamics.

Extended fractional-time oregonator model with proton buffering and recombination: stability and bifurcation analysis

期刊论文
发表日期：

2026年7月21日

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Sarita Pippal

摘要

We propose and solve an extended fractional-order Oregonator model for the Belousov–Zhabotinsky reaction by extending the classical autocatalytic core with explicit proton balance and buffering. The resulting Caputo system ${}^{C}D_t^{\alpha }\textbf{X}=\textbf{F}(\textbf{X})$ $^{C} D_{t}^{α} X = F (X)$ , $0<\alpha \le 1$ $0 < α \leq 1$ , is analyzed analytically via the fractional differential transform method and numerically using the Adams–Bashforth–Moulton predictor–corrector scheme. In the proposed model, $\textbf{X}=(X,Y,Z,H,P)^{\top }$ $X = {(X, Y, Z, H, P)}^{⊤}$ represents the autocatalyst ( $X\sim \hbox {HBrO}_{2}$ $X \sim {HBrO}_{2}$ ), inhibitor ( $Y\sim \hbox {Br}^{-}$ $Y \sim {Br}^{-}$ ), oxidized catalyst (Z), free protons ( $H\sim \hbox {H}^{+}$ $H \sim H^{+}$ ), and buffered protons (P). The reservoir variable P introduces delayed proton exchange and additional feedback, thereby capturing proton-regulated oscillations observed experimentally. Semi-analytical solutions obtained via the fractional differential transform method yield convergent fractional power series, while numerical integration using Adams–Bashforth–Moulton schemes recovers classical Runge–Kutta dynamics as $\alpha \rightarrow 1$ $α \to 1$ . Using comparison principles and Lyapunov-type functionals, positivity $\textbf{X}(t)\in \mathbb {R}^{5}_{\ge 0}$ $X (t) \in R_{\geq 0}^{5}$ and uniform dissipativity are established, ensuring the existence of a compact absorbing set. The equilibrium set contains a trivial state and a unique chemically admissible equilibrium $\textbf{E}^{+}$ $E^{+}$ . Spectral analysis of the block Jacobian $J(E^{+})$ $J (E^{+})$ shows that the trivial equilibrium is unstable, while for $0<\alpha <1$ $0 < α < 1$ , Matignon’s criterion $|\arg (\lambda )|>\alpha \pi /2$ $| arg (λ) | > α π / 2$ confirms that fractional memory deforms stability sectors without suppressing oscillatory ( $ {\text {HBrO}}_{2} - {\text {Br}}^{ - } - {\text {H}}^{ + } $ ${HBrO}_{2} - {Br}^{-} - H^{+}$ ) instabilities. Phase portraits, nullclines, equilibrium manifolds, bifurcation diagrams, and sensitivity indices demonstrate how proton buffering ( $\hbox {H}^{+}\rightleftarrows \hbox {H}^{+}_{buf}$ $H^{+} ⇄ H_{buf}^{+}$ ), kinetic rates, and fractional memory jointly regulate oscillation amplitude, stability, and long-time chemical dynamics.