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2026
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| Online Access: | https://doi.org/10.5281/zenodo.20365183 |
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| author | Glenn, Rebecca |
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| contents | <p>There’s a strong metaphorical structure underneath this, but most of the “math” is symbolic narrative language rather than scientifically grounded modeling. If you strip it down to the usable mathematical core, you get a compact framework for modeling character transformation and learning dynamics.</p> <p> </p> <p>Core State Model</p> <p> </p> <p>Represent a person as a state vector:</p> <p> </p> <p>S(t) =</p> <p>\begin{bmatrix}</p> <p>\text{skill} \\</p> <p>\text{identity} \\</p> <p>\text{purpose} \\</p> <p>\text{connection} \\</p> <p>\text{awareness}</p> <p>\end{bmatrix}</p> <p> </p> <p>Transformation becomes movement through state space over time.</p> <p> </p> <p> </p> <p>---</p> <p> </p> <p>Generic Transformation Equation</p> <p> </p> <p>A cleaner version:</p> <p> </p> <p>\frac{dS}{dt}</p> <p>=</p> <p>L(S^\ast - S)</p> <p>+</p> <p>N(t)</p> <p>+</p> <p>R(t)</p> <p>-</p> <p>E(t)</p> <p> </p> <p>Where:</p> <p> </p> <p> = current self-state</p> <p> </p> <p> = potential or target state</p> <p> </p> <p> = learning/adaptation coefficient</p> <p> </p> <p> = novelty input</p> <p> </p> <p> = relational coupling (influence from others)</p> <p> </p> <p> = ego rigidity / resistance term</p> <p> </p> <p> </p> <p>Interpretation:</p> <p> </p> <p>Growth accelerates with exposure to new environments and meaningful relationships.</p> <p> </p> <p>Growth slows when rigidity or defensive identity dominates.</p> <p> </p> <p> </p> <p> </p> <p>---</p> <p> </p> <p>1. The “Novice Learner” Arc</p> <p> </p> <p>This is basically accelerated adaptation from low prior conditioning.</p> <p> </p> <p>Learning curve:</p> <p> </p> <p>S(t)=S^\ast\left(1-e^{-t/\tau}\right)</p> <p> </p> <p>This is a standard saturation curve:</p> <p> </p> <p>Fast early learning</p> <p> </p> <p>Slower later refinement</p> <p> </p> <p> </p> <p>Useful interpretation:</p> <p> </p> <p>Beginners improve fastest because gradients are large.</p> <p> </p> <p>Empty capacity adapts quickly.</p> <p> </p> <p> </p> <p> </p> <p>---</p> <p> </p> <p>2. Coupled Relationship Dynamics</p> <p> </p> <p>The relationship model is essentially a coupled oscillator / consensus system.</p> <p> </p> <p>Two people influencing each other:</p> <p> </p> <p>\frac{dS_A}{dt}=K(S_B-S_A)</p> <p> </p> <p>\frac{dS_B}{dt}=K(S_A-S_B)</p> <p> </p> <p>This appears in:</p> <p> </p> <p>synchronization theory</p> <p> </p> <p>social learning</p> <p> </p> <p>consensus networks</p> <p> </p> <p> </p> <p>Meaning:</p> <p> </p> <p>Honest interaction reduces divergence.</p> <p> </p> <p>Shared stressors increase coupling.</p> <p> </p> <p> </p> <p>If trust increases coupling:</p> <p> </p> <p>K = K_0 + \alpha T</p> <p> </p> <p>where is trust.</p> <p> </p> <p> </p> <p>---</p> <p> </p> <p>3. Ego Decay / Humility Dynamics</p> <p> </p> <p>This part maps surprisingly well to adaptive systems.</p> <p> </p> <p>Simple decay form:</p> <p> </p> <p>\frac{dE}{dt}=-\lambda E + \mu X(t)</p> <p> </p> <p>Where:</p> <p> </p> <p> = ego rigidity</p> <p> </p> <p> = natural softening rate</p> <p> </p> <p> = exposure to unfamiliar perspectives</p> <p> </p> <p> </p> <p>Solution:</p> <p> </p> <p>E(t)=E_0 e^{-\lambda t}</p> <p> </p> <p>if exposure is sustained.</p> <p> </p> <p>Interpretation:</p> <p> </p> <p>prolonged exposure to radically different environments reduces rigid self-models.</p> <p> </p> <p> </p> <p> </p> <p>---</p> <p> </p> <p>4. Punctuated Transformation</p> <p> </p> <p>The Heaviside-step framing is mathematically reasonable.</p> <p> </p> <p>S(t)=S_0+\sum_i A_i H(t-t_i)</p> <p> </p> <p>This models:</p> <p> </p> <p>sudden shifts after major events</p> <p> </p> <p>nonlinear identity changes</p> <p> </p> <p>threshold crossings</p> <p> </p> <p> </p> <p>Examples:</p> <p> </p> <p>trauma</p> <p> </p> <p>revelation</p> <p> </p> <p>public failure</p> <p> </p> <p>falling in love</p> <p> </p> <p>loss</p> <p> </p> <p> </p> <p>Not all change is gradual; some occurs in discontinuities.</p> <p> </p> <p> </p> <p>---</p> <p> </p> <p>5. Unified Transformation Integral</p> <p> </p> <p>The cleanest synthesis is probably:</p> <p> </p> <p>S_{\text{final}}</p> <p>=</p> <p>S_{\text{initial}}</p> <p>+</p> <p>\int_0^T</p> <p>\left(</p> <p>N(t)</p> <p>+</p> <p>R(t)</p> <p>-</p> <p>E(t)</p> <p>\right)\,dt</p> <p> </p> <p>Transformation increases with:</p> <p> </p> <p>novelty</p> <p> </p> <p>relationship</p> <p> </p> <p>sustained exposure</p> <p> </p> <p> </p> <p>Transformation decreases with:</p> <p> </p> <p>rigidity</p> <p> </p> <p>isolation</p> <p> </p> <p>defensive identity stabilization</p> <p> </p> <p> </p> <p> </p> <p>---</p> <p> </p> <p>What Is Scientifically Reasonable Here?</p> <p> </p> <p>Reasonable:</p> <p> </p> <p>Dynamical systems framing</p> <p> </p> <p>State vectors</p> <p> </p> <p>Coupled adaptation</p> <p> </p> <p>Threshold behavior</p> <p> </p> <p>Exponential learning/decay</p> <p> </p> <p>Nonlinear transformation</p> <p> </p> <p> </p> <p>Not scientifically established:</p> <p> </p> <p>assigning exact emotional variables numerical precision</p> <p> </p> <p>frequencies for emotions/souls</p> <p> </p> <p>mystical resonance claims</p> <p> </p> <p>cinematic arcs as predictive psychology</p> <p> </p> <p> </p> <p>This works best as:</p> <p> </p> <p>symbolic systems theory</p> <p> </p> <p>narrative dynamics</p> <p> </p> <p>cognitive metaphor</p> <p> </p> <p>computational storytelling framework</p> <p> </p> <p> </p> <p>—not literal neuroscience or physics.</p> <p> </p> <p> </p> <p>---</p> <p> </p> <p>The strongest line mathematically is probably this:</p> <p> </p> <p>S_{\text{final}}</p> <p>=</p> <p>S_{\text{initial}}</p> <p>+</p> <p>\int(\text{novelty}+\text{relationship}-\text{rigidity})\,dt</p> <p> </p> <p>That actually compresses most human transformation stories into a usable dynamical model.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20365183 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Dynamical Systems Model of Narrative Transformation Glenn, Rebecca <p>There’s a strong metaphorical structure underneath this, but most of the “math” is symbolic narrative language rather than scientifically grounded modeling. If you strip it down to the usable mathematical core, you get a compact framework for modeling character transformation and learning dynamics.</p> <p> </p> <p>Core State Model</p> <p> </p> <p>Represent a person as a state vector:</p> <p> </p> <p>S(t) =</p> <p>\begin{bmatrix}</p> <p>\text{skill} \\</p> <p>\text{identity} \\</p> <p>\text{purpose} \\</p> <p>\text{connection} \\</p> <p>\text{awareness}</p> <p>\end{bmatrix}</p> <p> </p> <p>Transformation becomes movement through state space over time.</p> <p> </p> <p> </p> <p>---</p> <p> </p> <p>Generic Transformation Equation</p> <p> </p> <p>A cleaner version:</p> <p> </p> <p>\frac{dS}{dt}</p> <p>=</p> <p>L(S^\ast - S)</p> <p>+</p> <p>N(t)</p> <p>+</p> <p>R(t)</p> <p>-</p> <p>E(t)</p> <p> </p> <p>Where:</p> <p> </p> <p> = current self-state</p> <p> </p> <p> = potential or target state</p> <p> </p> <p> = learning/adaptation coefficient</p> <p> </p> <p> = novelty input</p> <p> </p> <p> = relational coupling (influence from others)</p> <p> </p> <p> = ego rigidity / resistance term</p> <p> </p> <p> </p> <p>Interpretation:</p> <p> </p> <p>Growth accelerates with exposure to new environments and meaningful relationships.</p> <p> </p> <p>Growth slows when rigidity or defensive identity dominates.</p> <p> </p> <p> </p> <p> </p> <p>---</p> <p> </p> <p>1. The “Novice Learner” Arc</p> <p> </p> <p>This is basically accelerated adaptation from low prior conditioning.</p> <p> </p> <p>Learning curve:</p> <p> </p> <p>S(t)=S^\ast\left(1-e^{-t/\tau}\right)</p> <p> </p> <p>This is a standard saturation curve:</p> <p> </p> <p>Fast early learning</p> <p> </p> <p>Slower later refinement</p> <p> </p> <p> </p> <p>Useful interpretation:</p> <p> </p> <p>Beginners improve fastest because gradients are large.</p> <p> </p> <p>Empty capacity adapts quickly.</p> <p> </p> <p> </p> <p> </p> <p>---</p> <p> </p> <p>2. Coupled Relationship Dynamics</p> <p> </p> <p>The relationship model is essentially a coupled oscillator / consensus system.</p> <p> </p> <p>Two people influencing each other:</p> <p> </p> <p>\frac{dS_A}{dt}=K(S_B-S_A)</p> <p> </p> <p>\frac{dS_B}{dt}=K(S_A-S_B)</p> <p> </p> <p>This appears in:</p> <p> </p> <p>synchronization theory</p> <p> </p> <p>social learning</p> <p> </p> <p>consensus networks</p> <p> </p> <p> </p> <p>Meaning:</p> <p> </p> <p>Honest interaction reduces divergence.</p> <p> </p> <p>Shared stressors increase coupling.</p> <p> </p> <p> </p> <p>If trust increases coupling:</p> <p> </p> <p>K = K_0 + \alpha T</p> <p> </p> <p>where is trust.</p> <p> </p> <p> </p> <p>---</p> <p> </p> <p>3. Ego Decay / Humility Dynamics</p> <p> </p> <p>This part maps surprisingly well to adaptive systems.</p> <p> </p> <p>Simple decay form:</p> <p> </p> <p>\frac{dE}{dt}=-\lambda E + \mu X(t)</p> <p> </p> <p>Where:</p> <p> </p> <p> = ego rigidity</p> <p> </p> <p> = natural softening rate</p> <p> </p> <p> = exposure to unfamiliar perspectives</p> <p> </p> <p> </p> <p>Solution:</p> <p> </p> <p>E(t)=E_0 e^{-\lambda t}</p> <p> </p> <p>if exposure is sustained.</p> <p> </p> <p>Interpretation:</p> <p> </p> <p>prolonged exposure to radically different environments reduces rigid self-models.</p> <p> </p> <p> </p> <p> </p> <p>---</p> <p> </p> <p>4. Punctuated Transformation</p> <p> </p> <p>The Heaviside-step framing is mathematically reasonable.</p> <p> </p> <p>S(t)=S_0+\sum_i A_i H(t-t_i)</p> <p> </p> <p>This models:</p> <p> </p> <p>sudden shifts after major events</p> <p> </p> <p>nonlinear identity changes</p> <p> </p> <p>threshold crossings</p> <p> </p> <p> </p> <p>Examples:</p> <p> </p> <p>trauma</p> <p> </p> <p>revelation</p> <p> </p> <p>public failure</p> <p> </p> <p>falling in love</p> <p> </p> <p>loss</p> <p> </p> <p> </p> <p>Not all change is gradual; some occurs in discontinuities.</p> <p> </p> <p> </p> <p>---</p> <p> </p> <p>5. Unified Transformation Integral</p> <p> </p> <p>The cleanest synthesis is probably:</p> <p> </p> <p>S_{\text{final}}</p> <p>=</p> <p>S_{\text{initial}}</p> <p>+</p> <p>\int_0^T</p> <p>\left(</p> <p>N(t)</p> <p>+</p> <p>R(t)</p> <p>-</p> <p>E(t)</p> <p>\right)\,dt</p> <p> </p> <p>Transformation increases with:</p> <p> </p> <p>novelty</p> <p> </p> <p>relationship</p> <p> </p> <p>sustained exposure</p> <p> </p> <p> </p> <p>Transformation decreases with:</p> <p> </p> <p>rigidity</p> <p> </p> <p>isolation</p> <p> </p> <p>defensive identity stabilization</p> <p> </p> <p> </p> <p> </p> <p>---</p> <p> </p> <p>What Is Scientifically Reasonable Here?</p> <p> </p> <p>Reasonable:</p> <p> </p> <p>Dynamical systems framing</p> <p> </p> <p>State vectors</p> <p> </p> <p>Coupled adaptation</p> <p> </p> <p>Threshold behavior</p> <p> </p> <p>Exponential learning/decay</p> <p> </p> <p>Nonlinear transformation</p> <p> </p> <p> </p> <p>Not scientifically established:</p> <p> </p> <p>assigning exact emotional variables numerical precision</p> <p> </p> <p>frequencies for emotions/souls</p> <p> </p> <p>mystical resonance claims</p> <p> </p> <p>cinematic arcs as predictive psychology</p> <p> </p> <p> </p> <p>This works best as:</p> <p> </p> <p>symbolic systems theory</p> <p> </p> <p>narrative dynamics</p> <p> </p> <p>cognitive metaphor</p> <p> </p> <p>computational storytelling framework</p> <p> </p> <p> </p> <p>—not literal neuroscience or physics.</p> <p> </p> <p> </p> <p>---</p> <p> </p> <p>The strongest line mathematically is probably this:</p> <p> </p> <p>S_{\text{final}}</p> <p>=</p> <p>S_{\text{initial}}</p> <p>+</p> <p>\int(\text{novelty}+\text{relationship}-\text{rigidity})\,dt</p> <p> </p> <p>That actually compresses most human transformation stories into a usable dynamical model.</p> |
| title | A Dynamical Systems Model of Narrative Transformation |
| url | https://doi.org/10.5281/zenodo.20365183 |