MetabolicStateTransitionsModel

Code from paper: Modeling Metabolic State Transitions in Obesity Using a Time-Varying Lambda–Omega Framework

Lambda–Omega Dynamics: Time-Varying Stability and Limit-Cycle Adaptation

This repository contains MATLAB scripts for simulating and visualizing time-varying lambda–omega dynamical systems under three different scenarios. The code generates animated phase-plane trajectories together with a time-series summary of peak amplitudes and cumulative peak-to-peak changes, and can optionally export each simulation as an .mp4 movie.

The main goal of these scripts is to illustrate how changes in the radial growth term affect the system’s stability structure, nullclines, and limit-cycle behavior over time.

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Repository Overview

The repository includes three main simulation cases:

• case_1.m — sign-flip transition from stable oscillation to decay
• case_2.m — growing limit cycle
• case_3.m — shrinking limit cycle

Each script:

• solves a lambda–omega system using ode45
• animates the trajectory in the phase plane
• updates the vector field and nullclines over time
• tracks peaks of $x(t)$
• computes the cumulative absolute change between consecutive peaks
• highlights an adaptation window based on peak differences
• optionally saves the animation as a video

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Mathematical Background

The simulations are based on the planar lambda–omega system

$$ \dot{x} = \Lambda(r,t)x - \Omega(r,t)y $$

$$ \dot{y} = \Omega(r,t)x + \Lambda(r,t)y $$

where

$$ r = \sqrt{x^2 + y^2} $$

Here:

• $\Lambda(r,t)$ controls radial growth or decay
• $\Omega(r,t)$ controls angular velocity

Depending on how $\Lambda(r,t)$ is defined, the system can exhibit:

• a stable limit cycle
• a shrinking oscillation
• a growing oscillation
• a transition from oscillatory to non-oscillatory behavior

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Visualization Layout

Each simulation uses a two-panel figure.

Left Panel: Phase Plane

Shows:

• trajectory in $(x,y)$
• moving current state
• dynamic nullclines
• frozen nullcline snapshots at selected intervals
• vector field
• background shading indicating radial stability

Right Panel: Peak-Based Summary

Shows:

• peak values of $x(t)$
• cumulative absolute differences between consecutive peaks
• shaded adaptation region based on peak-change thresholding

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Case Descriptions

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Case 1 — Stability Flip / Oscillation Collapse

File: case_1.m

This case uses a lambda–omega model with a time-varying sign change in the radial term:

$$ \Lambda(r,t) = s(t)\lambda_{\text{base}}(r) $$

where

$$ \lambda_{\text{base}}(r) = 1 - r^2 $$

The sign function $s(t)$ transitions smoothly from $+1$ to $-1$ using a cosine function.

Interpretation

• Initially the system supports a stable oscillation
• Over time the sign flip reverses the radial stability
• The oscillation weakens and eventually collapses toward the origin

What this case demonstrates

• smooth transition in system stability
• deformation of nullclines during the transition
• reduction in oscillation amplitude over time
• detection of the adaptation period from peak differences

Output

• animated phase portrait
• peak evolution over time
• cumulative peak-change curve
• optional movie export: Movie/case_1_movie.mp4

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Case 2 — Growing Limit Cycle

File: case_2.m

This case uses an order-2 lambda–omega system

$$ \Lambda(r,t) = \lambda(t) - b r^2 $$

$$ \Omega(r,t) = \omega_0 + a r^2 $$

The target limit-cycle radius grows smoothly over time.

The desired radius evolves as

$$ R_{\text{target}}(t) = R_{\max} - (R_{\max}-R_{\min}) e^{-t/\tau_{\text{grow}}} $$

The time-dependent radial parameter is

$$ \lambda(t) = b , R_{\text{target}}(t)^2 $$

Interpretation

• the oscillation begins with a small radius
• the target radius increases gradually
• the trajectory expands outward until it reaches the larger limit cycle

What this case demonstrates

• gradual outward growth of oscillation amplitude
• time-varying nullclines associated with expanding amplitude
• increasing peak structure during transient adaptation
• stabilization onto a larger oscillatory orbit

Output

• animated phase portrait
• peak evolution over time
• cumulative peak-change curve
• optional movie export: Movie/case_2_movie.mp4

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Case 3 — Shrinking Limit Cycle

File: case_3.m

This case uses the same order-2 lambda–omega system as Case 2, but the target limit-cycle radius shrinks smoothly over time.

The radius evolves as

$$ R_{\text{target}}(t) = R_{\min} + (R_{\max}-R_{\min}) e^{-t/\tau_{\text{shrink}}} $$

The radial parameter remains

$$ \lambda(t) = b , R_{\text{target}}(t)^2 $$

Interpretation

• the oscillation starts on a large initial limit cycle
• the target radius decreases gradually
• the trajectory contracts inward toward a smaller stable cycle

What this case demonstrates

• inward contraction of oscillatory behavior
• time-varying nullcline motion during shrinking dynamics
• transient adaptation visible in peak amplitudes
• convergence onto a smaller oscillatory orbit

Output

• animated phase portrait
• peak evolution over time
• cumulative peak-change curve
• optional movie export: Movie/case_3_movie.mp4