Nonlinear Dynamical Model and Analysis of Emotional Propagation Based on Caputo Derivative
Conventional integer-order models fail to adequately capture non-local memory effects and constrained nonlinear interactions in emotional dynamics. To address these limitations, we propose a coupled framework that integrates Caputo fractional derivatives with hyperbolic tangent–based interaction fun...
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MDPI AG
2025-06-01
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| author | Liang Hong Lipu Zhang |
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| collection | DOAJ |
| description | Conventional integer-order models fail to adequately capture non-local memory effects and constrained nonlinear interactions in emotional dynamics. To address these limitations, we propose a coupled framework that integrates Caputo fractional derivatives with hyperbolic tangent–based interaction functions. The fractional-order term quantifies power-law memory decay in affective states, while the nonlinear component regulates connection strength through emotional difference thresholds. Mathematical analysis establishes the existence and uniqueness of solutions with continuous dependence on initial conditions and proves the local asymptotic stability of network equilibria (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>W</mi><mrow><mi>i</mi><mi>j</mi></mrow><mo>*</mo></msubsup><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mi>δ</mi></mfrac></mstyle><msup><mrow><mi>sec</mi><mi mathvariant="normal">h</mi></mrow><mn>2</mn></msup><mrow><mo>(</mo><mo>∥</mo></mrow><msub><mi mathvariant="normal">E</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="normal">E</mi><mi>j</mi></msub><mrow><mo>∥</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, e.g., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>W</mi><mo>*</mo></msup><mo>≈</mo><mn>1.40</mn></mrow></semantics></math></inline-formula> under typical parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>=</mo><mn>0.5</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>=</mo><mn>0.3</mn></mrow></semantics></math></inline-formula>). We further derive closed-form expressions for the steady-state variance under stochastic perturbations (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Var</mi><mrow><mo>(</mo><msub><mi>W</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msubsup><mi>σ</mi><mi>ζ</mi><mn>2</mn></msubsup><mrow><mn>2</mn><mi>η</mi><mi>δ</mi></mrow></mfrac></mstyle></mrow></semantics></math></inline-formula>) and demonstrate a less than 6% deviation between simulated and theoretical values when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>σ</mi><mi>ζ</mi></msub><mo>=</mo><mn>0.1</mn></mrow></semantics></math></inline-formula>. Numerical experiments using the Euler–Maruyama method validate the convergence of connection weights toward the predicted equilibrium, reveal Gaussian features in the stationary distributions, and confirm power-law scaling between noise intensity and variance. The numerical accuracy of the fractional system is further verified through L1 discretization, with observed error convergence consistent with theoretical expectations for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo>=</mo><mn>0.5</mn></mrow></semantics></math></inline-formula>. This framework advances the mechanistic understanding of co-evolutionary dynamics in emotion-modulated social networks, supporting applications in clinical intervention design, collective sentiment modeling, and psychophysiological coupling research. |
| format | Article |
| id | doaj-art-a5dba898396d4ebea0c90e26ece57e42 |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-a5dba898396d4ebea0c90e26ece57e422025-08-20T03:17:07ZengMDPI AGMathematics2227-73902025-06-011313204410.3390/math13132044Nonlinear Dynamical Model and Analysis of Emotional Propagation Based on Caputo DerivativeLiang Hong0Lipu Zhang1College of Media Engineering, Communication University of Zhejiang, Hangzhou 310018, ChinaCollege of Media Engineering, Communication University of Zhejiang, Hangzhou 310018, ChinaConventional integer-order models fail to adequately capture non-local memory effects and constrained nonlinear interactions in emotional dynamics. To address these limitations, we propose a coupled framework that integrates Caputo fractional derivatives with hyperbolic tangent–based interaction functions. The fractional-order term quantifies power-law memory decay in affective states, while the nonlinear component regulates connection strength through emotional difference thresholds. Mathematical analysis establishes the existence and uniqueness of solutions with continuous dependence on initial conditions and proves the local asymptotic stability of network equilibria (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>W</mi><mrow><mi>i</mi><mi>j</mi></mrow><mo>*</mo></msubsup><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mi>δ</mi></mfrac></mstyle><msup><mrow><mi>sec</mi><mi mathvariant="normal">h</mi></mrow><mn>2</mn></msup><mrow><mo>(</mo><mo>∥</mo></mrow><msub><mi mathvariant="normal">E</mi><mi>i</mi></msub><mo>−</mo><msub><mi mathvariant="normal">E</mi><mi>j</mi></msub><mrow><mo>∥</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, e.g., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>W</mi><mo>*</mo></msup><mo>≈</mo><mn>1.40</mn></mrow></semantics></math></inline-formula> under typical parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>=</mo><mn>0.5</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>=</mo><mn>0.3</mn></mrow></semantics></math></inline-formula>). We further derive closed-form expressions for the steady-state variance under stochastic perturbations (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Var</mi><mrow><mo>(</mo><msub><mi>W</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msubsup><mi>σ</mi><mi>ζ</mi><mn>2</mn></msubsup><mrow><mn>2</mn><mi>η</mi><mi>δ</mi></mrow></mfrac></mstyle></mrow></semantics></math></inline-formula>) and demonstrate a less than 6% deviation between simulated and theoretical values when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>σ</mi><mi>ζ</mi></msub><mo>=</mo><mn>0.1</mn></mrow></semantics></math></inline-formula>. Numerical experiments using the Euler–Maruyama method validate the convergence of connection weights toward the predicted equilibrium, reveal Gaussian features in the stationary distributions, and confirm power-law scaling between noise intensity and variance. The numerical accuracy of the fractional system is further verified through L1 discretization, with observed error convergence consistent with theoretical expectations for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo>=</mo><mn>0.5</mn></mrow></semantics></math></inline-formula>. This framework advances the mechanistic understanding of co-evolutionary dynamics in emotion-modulated social networks, supporting applications in clinical intervention design, collective sentiment modeling, and psychophysiological coupling research.https://www.mdpi.com/2227-7390/13/13/2044fractional derivativenonlinear dynamicsemotional propagationmathematical modelstochastic differential |
| spellingShingle | Liang Hong Lipu Zhang Nonlinear Dynamical Model and Analysis of Emotional Propagation Based on Caputo Derivative Mathematics fractional derivative nonlinear dynamics emotional propagation mathematical model stochastic differential |
| title | Nonlinear Dynamical Model and Analysis of Emotional Propagation Based on Caputo Derivative |
| title_full | Nonlinear Dynamical Model and Analysis of Emotional Propagation Based on Caputo Derivative |
| title_fullStr | Nonlinear Dynamical Model and Analysis of Emotional Propagation Based on Caputo Derivative |
| title_full_unstemmed | Nonlinear Dynamical Model and Analysis of Emotional Propagation Based on Caputo Derivative |
| title_short | Nonlinear Dynamical Model and Analysis of Emotional Propagation Based on Caputo Derivative |
| title_sort | nonlinear dynamical model and analysis of emotional propagation based on caputo derivative |
| topic | fractional derivative nonlinear dynamics emotional propagation mathematical model stochastic differential |
| url | https://www.mdpi.com/2227-7390/13/13/2044 |
| work_keys_str_mv | AT lianghong nonlineardynamicalmodelandanalysisofemotionalpropagationbasedoncaputoderivative AT lipuzhang nonlineardynamicalmodelandanalysisofemotionalpropagationbasedoncaputoderivative |