<p>The concept of chaos control through local stabilization in fixed and periodic points of discrete dynamical systems via feedback algorithms has emerged as a significant research focus in several domains, including traffic control, cardiac arrhythmia, reduction control, chemical chaos, spine-wave instability, and heat convection. This study introduces a chaos controlling mechanism based on the SP feedback algorithm. First, the formulation of the SP control mechanism is established, and stability theorems determining the effective regimes for control parameters <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> are proved. Crucially, our theoretical analysis reveals a structural limitation: under the restriction <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha ,\beta ,\gamma \in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>γ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the mechanism is specifically effective for attaining local asymptotic stability at points where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$g'(w^)<-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>w</mi> <mo>∗</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, while fixed points with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$g'(w^)>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>w</mi> <mo>∗</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> remain unstable. The Lyapunov exponent properties and MLE values are analyzed to verify the local stability intervals computed for different points. As an application, a generalized control-based traffic flow model is provided within the identified admissible stability ranges.</p>

On chaos controlling mechanism for discrete dynamical systems driven by SP feedback algorithm with applications

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发表日期：

2026年7月30日

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Renu,
Ashish,
Renu Chugh

摘要

The concept of chaos control through local stabilization in fixed and periodic points of discrete dynamical systems via feedback algorithms has emerged as a significant research focus in several domains, including traffic control, cardiac arrhythmia, reduction control, chemical chaos, spine-wave instability, and heat convection. This study introduces a chaos controlling mechanism based on the SP feedback algorithm. First, the formulation of the SP control mechanism is established, and stability theorems determining the effective regimes for control parameters $\alpha $ $α$ , $\beta $ $β$ , and $\gamma $ $γ$ are proved. Crucially, our theoretical analysis reveals a structural limitation: under the restriction $\alpha ,\beta ,\gamma \in (0,1)$ $α, β, γ \in (0, 1)$ , the mechanism is specifically effective for attaining local asymptotic stability at points where $g'(w^*)<-1$ $g^{'} (w^{*}) < - 1$ , while fixed points with $g'(w^*)>1$ $g^{'} (w^{*}) > 1$ remain unstable. The Lyapunov exponent properties and MLE values are analyzed to verify the local stability intervals computed for different points. As an application, a generalized control-based traffic flow model is provided within the identified admissible stability ranges.