The fused asset flow model: stability, bifurcation, and contagion in multi-asset markets with heterogeneous investors

Mario Cavani

arxiv: 2605.28417 · v1 · pith:NQQHWVAZnew · submitted 2026-05-27 · 🧮 math.DS

The fused asset flow model: stability, bifurcation, and contagion in multi-asset markets with heterogeneous investors

Mario Cavani This is my paper

Pith reviewed 2026-06-29 09:51 UTC · model grok-4.3

classification 🧮 math.DS

keywords asset flow modelHopf bifurcationmulti-asset marketsheterogeneous investorslimit cyclescontagionbehavioral financedynamical systems

0 comments

The pith

In a fused multi-asset model with trend and value investors, the fundamental equilibrium loses stability through a supercritical Hopf bifurcation and produces persistent limit cycles.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a single system of ODEs that merges three earlier asset-flow frameworks into one multi-asset, multi-group setting. It proves that cash and share holdings stay positive and bounded, then shows that the manifold of equilibria contains a special fundamental point whose stability can be lost when a pair of eigenvalues crosses the imaginary axis. When that crossing occurs, a supercritical Hopf bifurcation creates stable limit cycles whose periods and amplitudes match the oscillatory and contagion patterns seen in the three source models. A reader should care because the same mechanism that destabilizes prices in one asset can drive synchronized or asymmetric swings across several assets without external shocks.

Core claim

The fused asset-flow model is a system of ordinary differential equations for prices, cash holdings, share holdings, and sentiment variables across multiple assets and investor groups. The equilibrium set forms a manifold parameterized by the distribution of cash among groups; the fundamental equilibrium lies on this manifold. Linearization around that point yields a characteristic equation whose roots cross the imaginary axis transversely under explicit parameter conditions, producing a supercritical Hopf bifurcation. The resulting limit cycles reproduce the equilibrium manifolds, bifurcation thresholds, and asymmetric contagion signatures of the three benchmark models.

What carries the argument

The system of ODEs with asymmetric cross-asset coupling in buying decisions; linear stability analysis of the fundamental equilibrium on the cash-distribution manifold that detects the supercritical Hopf bifurcation.

If this is right

The equilibrium manifold remains stable for wide ranges of cash distribution but loses stability at explicit thresholds on trend and value parameters.
Supercritical Hopf bifurcations generate limit cycles whose periods match those observed in the single-asset and two-asset benchmark models.
Contagion between assets is asymmetric: a cycle in one asset can drive or dampen cycles in the other depending on the sign of the cross-coupling terms.
All physically relevant variables (prices, cash, shares, sentiment) remain positive and bounded for all time when initial conditions are positive.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same bifurcation mechanism could be used to forecast the onset of oscillatory regimes in real multi-asset portfolios once the model parameters are fitted to observed price and volume data.
Because the cash-distribution manifold is continuous, small shifts in wealth between investor groups can move the system across the Hopf threshold without any change in the underlying strategies.
Extending the model to stochastic noise on the demand functions would likely turn the deterministic limit cycles into noisy oscillations whose power spectrum could be compared directly with empirical return series.

Load-bearing premise

The chosen functional forms for trend-following and value-based demand, together with the asymmetric cross-asset coupling, are assumed to guarantee positivity, boundedness, the equilibrium manifold, and the Hopf bifurcation without hidden inconsistencies or extra parameters.

What would settle it

A numerical integration of the full ODE system under the stated parameter values that produces no limit cycle after the predicted Hopf threshold, or a real-market time series of two assets showing no oscillatory contagion when the model parameters calibrated to those assets predict a cycle.

Figures

Figures reproduced from arXiv: 2605.28417 by Mario Cavani.

**Figure 9.** Figure 9: Excursion analysis for the Cavani (2026) Nigeria-Libya oil market model with optimal oscillatory parameters (𝑏China = 2.5, 𝑑USA = 0.01, 𝛼 = 3.0, 𝑞1,China = 0.60). The figure quantifies the maximum price deviation (excursion) of Nigerian oil as a function of initial price perturbations Δ𝑃Nigeria, and Δ𝑃Libya from the fundamental value ($80 USD/bbl). (a) Excursion surface – three-dimensional visualization sh… view at source ↗

**Figure 10.** Figure 10: Contagion matrix for the Nigeria-Libya oil market. Response strength (maximum relative price deviation) following a 10% price shock. Asymmetry: Libya → Nigeria (0.0133) is 34% stronger than Nigeria → Libya (0.0066), reflecting Nigeria's larger market share (35% vs. 25%) and role as price leader [PITH_FULL_IMAGE:figures/full_fig_p042_10.png] view at source ↗

**Figure 11.** Figure 11: Contagion propagation in the Nigeria-Libya oil market. (a) Baseline dynamics (no shock). (b) +10% shock to Nigeria propagates to Libya with 0.0133 response. (c) -10% shock to Libya propagates to Nigeria with 0.0066 response. (d) Contagion propagation comparison, highlighting asymmetry: Libya → Nigeria response is 34% stronger [PITH_FULL_IMAGE:figures/full_fig_p042_11.png] view at source ↗

read the original abstract

This paper presents a unified multi-asset, multi-group asset-flow model that integrates three foundational frameworks from the behavioral finance literature. The model captures the dynamics of financial markets where multiple assets are traded by multiple investor groups, each with distinct trend-following (momentum) and value-based (fundamental) strategies. Unlike classical efficient market models, our framework explicitly incorporates the finiteness of cash and shares, asymmetric cross-asset coupling in buying decisions, and endogenous wealth redistribution across groups. We derive the complete system of ordinary differential equations governing price, cash, share, and sentiment dynamics, and establish the fundamental properties of positivity and boundedness for all physically relevant variables. The equilibrium set is characterized as a manifold parameterized by cash distribution, with the fundamental equilibrium as a special point. Through linear stability analysis, we identify conditions under which the fundamental equilibrium loses stability via a supercritical Hopf bifurcation, giving rise to persistent limit cycles. The model is validated against three benchmark papers: the single-asset multi-group model of DeSantis, Swigon, and Caginalp (2012); the two-asset single-group model of Bulut, Merdan, and Swigon (2019); and the two-asset two-group Nigeria-Libya oil market model of Cavani (2026). Our numerical simulations reproduce all key theoretical predictions, including equilibrium manifolds, Hopf bifurcation thresholds, limit cycle periods, and asymmetric contagion patterns.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper fuses three prior asset-flow models into a multi-asset multi-group version with cash and share constraints, but its claim that linear analysis alone establishes a supercritical Hopf bifurcation does not hold.

read the letter

The main takeaway is that this work generalizes existing asset flow models to handle multiple assets and investor groups while keeping cash and shares finite, but the stability analysis overreaches by attributing the supercritical character of the Hopf bifurcation to linear methods.

The new piece is the combined framework that takes the single-asset multi-group setup from DeSantis et al. 2012, the two-asset single-group setup from Bulut et al. 2019, and the two-asset two-group setup from Cavani 2026, then extends them to arbitrary numbers of assets and groups. The model adds asymmetric cross-asset coupling in buying and endogenous wealth redistribution. The authors write down the full ODE system for prices, cash, shares, and sentiments, prove positivity and boundedness, and describe the set of equilibria as a manifold parameterized by cash distribution, with the fundamental equilibrium as a special case. They perform linear stability analysis to locate parameter values where the fundamental equilibrium loses stability through a Hopf bifurcation that produces limit cycles, and they run numerical simulations that recover the equilibria, bifurcation thresholds, cycle periods, and contagion patterns from the three benchmark papers.

The integration is done in a straightforward way and the emphasis on finite cash and shares is a useful correction to earlier versions that treated those quantities as unlimited. The manifold description follows naturally once more assets are added.

The soft spot is the bifurcation claim. Linearization can show when a pair of eigenvalues crosses the imaginary axis, but determining whether the bifurcation is supercritical requires the first Lyapunov coefficient from the quadratic and cubic terms. The abstract ties the supercritical label directly to the linear analysis, which is not correct. The zero eigenvalue along the cash-distribution directions on the manifold also means any Hopf must be analyzed in the transverse dynamics. The validation includes the author's own prior paper, which limits how independent the checks are. Numerical details are asserted without specifics on integration methods or tolerances.

This is for readers already working in dynamical systems models of behavioral finance. Someone following the asset flow line would get a usable generalization, though the Hopf part needs correction. The modeling choices are consistent with the cited papers and the thinking is coherent on its own terms.

I would send it to peer review with instructions to verify the nonlinear terms for the bifurcation type.

Referee Report

1 major / 1 minor

Summary. The manuscript presents a unified multi-asset, multi-group asset-flow model integrating three prior frameworks. It derives an ODE system governing price, cash, share, and sentiment dynamics; establishes positivity and boundedness; characterizes the equilibrium set as a manifold parameterized by cash distribution (with the fundamental equilibrium as a special case); and claims that linear stability analysis identifies conditions for the fundamental equilibrium to lose stability via a supercritical Hopf bifurcation, producing persistent limit cycles. Numerical simulations are asserted to reproduce equilibrium manifolds, bifurcation thresholds, cycle periods, and asymmetric contagion, with validation against DeSantis-Swigon-Caginalp (2012), Bulut-Merdan-Swigon (2019), and Cavani (2026).

Significance. If the derivation, boundedness proofs, manifold characterization, and bifurcation analysis hold after correction, the work would supply a coherent dynamical-systems framework for multi-asset contagion and endogenous oscillations under heterogeneous trend-following and value-based strategies, extending the cited single-asset and single-group models while incorporating cash finiteness and asymmetric cross-asset coupling.

major comments (1)

[Abstract] Abstract: the claim that 'linear stability analysis' identifies conditions for a 'supercritical Hopf bifurcation' cannot be supported by linearization alone. Linear analysis yields the crossing of a complex-conjugate pair through the imaginary axis (with the zero eigenvalue already present from the equilibrium manifold in the cash-distribution directions), but the distinction between super- and subcriticality requires the sign of the first Lyapunov coefficient computed from the quadratic and cubic terms of the vector field. The transverse dynamics on the manifold must be reduced before this coefficient can be evaluated; the manuscript therefore leaves the supercriticality assertion unverified.

minor comments (1)

The numerical validation section should specify the integration scheme, step-size control, and tolerance criteria used to compute limit-cycle periods and bifurcation thresholds, as these details are required to assess reproducibility of the asserted agreement with the three benchmark models.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the precise gap between linear analysis and the determination of bifurcation criticality. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that 'linear stability analysis' identifies conditions for a 'supercritical Hopf bifurcation' cannot be supported by linearization alone. Linear analysis yields the crossing of a complex-conjugate pair through the imaginary axis (with the zero eigenvalue already present from the equilibrium manifold in the cash-distribution directions), but the distinction between super- and subcriticality requires the sign of the first Lyapunov coefficient computed from the quadratic and cubic terms of the vector field. The transverse dynamics on the manifold must be reduced before this coefficient can be evaluated; the manuscript therefore leaves the supercriticality assertion unverified.

Authors: We agree with the referee that linearization alone establishes only the existence of a Hopf bifurcation (transverse to the equilibrium manifold) and does not determine its criticality. The manuscript's linear stability analysis correctly identifies the parameter values at which a complex-conjugate pair crosses the imaginary axis while the remaining eigenvalues associated with the cash-distribution manifold remain at zero. However, the claim of supercriticality in the abstract and main text rests on numerical evidence of attracting limit cycles rather than an explicit computation of the first Lyapunov coefficient. In the revised manuscript we will either (i) replace the word "supercritical" with "Hopf" throughout or (ii) perform the center-manifold reduction in the transverse directions and evaluate the Lyapunov coefficient analytically, thereby rigorously confirming the bifurcation type. We view this as a necessary clarification rather than a substantive change to the model's conclusions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives the fused multi-asset ODE system from the three cited frameworks, establishes positivity/boundedness and the equilibrium manifold directly from the equations, and applies linear stability analysis to locate Hopf conditions. The single self-citation (Cavani 2026) occurs only in the validation section for reproducing benchmark outputs and does not serve as load-bearing justification for the new model's derivation or bifurcation claims. No self-definitional reductions, fitted inputs renamed as predictions, or ansatzes imported via citation appear in the derivation chain. The central results rest on the paper's own equations and analysis rather than reducing to prior inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, axioms, or invented entities are identified; the model integrates prior frameworks without detailing new fitted quantities or postulates.

pith-pipeline@v0.9.1-grok · 5787 in / 1305 out tokens · 38364 ms · 2026-06-29T09:51:30.353269+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 3 canonical work pages

[1]

Awartani, B., Aktham, M., & Cherif, G. (2016). The connectedness between crude oil and financial markets: Evidence from implied volatility indices. Journal of Commodity Markets, 4(1), 56-69. Bouri, E., Gupta, R., & Roubaud, D. (2021). Investor sentiment and contagion in oil and commodity markets. Energy Economics, 98, 105234. Brock, W. A., & Hommes, C. H....

work page doi:10.1016/s0165-1889(98)00011-6 2016
[2]

Kelley, A

Springer‑Verlag. Kelley, A. (1967). The stable, center ‑stable, center, center ‑unstable, and unstable manifolds. Journal of Differential Equations, 3(4), 546–570. Khalil, H. K. (2002). Nonlinear Systems (3rd ed.). Prentice Hall. Kukačka, J., & Baruník, J. (2017). Estimation of financial agent ‑based models with simulated maximum likel ihood. Journal of E...

work page doi:10.1016/j.jedc.2017.09.006 1967
[3]

Trabelsi, N., Tiwari, A

Springer‑Verlag. Trabelsi, N., Tiwari, A. K., & Hammoudeh, S. (2022). Spillovers and directional predictability between international energy commodities and their implications for optimal portfolio and hedging. The North American Journal of Economics and Finance, 62, 101715. Wang, H. (2025). Decoding momentum spillover effects. Journal of Financial and Qu...

2022
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https://doi.org/10.1017/S002210902500124X Westerhoff, F., et al. (2025). Boom –bust cycles and asset market participation waves: Momentum, value, risk, and herding. Journal of Evolutionary Economics , 35, 513–551. https://doi.org/10.1007/s00191-025-00919-6

work page doi:10.1017/s002210902500124x 2025

[1] [1]

Awartani, B., Aktham, M., & Cherif, G. (2016). The connectedness between crude oil and financial markets: Evidence from implied volatility indices. Journal of Commodity Markets, 4(1), 56-69. Bouri, E., Gupta, R., & Roubaud, D. (2021). Investor sentiment and contagion in oil and commodity markets. Energy Economics, 98, 105234. Brock, W. A., & Hommes, C. H....

work page doi:10.1016/s0165-1889(98)00011-6 2016

[2] [2]

Kelley, A

Springer‑Verlag. Kelley, A. (1967). The stable, center ‑stable, center, center ‑unstable, and unstable manifolds. Journal of Differential Equations, 3(4), 546–570. Khalil, H. K. (2002). Nonlinear Systems (3rd ed.). Prentice Hall. Kukačka, J., & Baruník, J. (2017). Estimation of financial agent ‑based models with simulated maximum likel ihood. Journal of E...

work page doi:10.1016/j.jedc.2017.09.006 1967

[3] [3]

Trabelsi, N., Tiwari, A

Springer‑Verlag. Trabelsi, N., Tiwari, A. K., & Hammoudeh, S. (2022). Spillovers and directional predictability between international energy commodities and their implications for optimal portfolio and hedging. The North American Journal of Economics and Finance, 62, 101715. Wang, H. (2025). Decoding momentum spillover effects. Journal of Financial and Qu...

2022

[4] [4]

https://doi.org/10.1017/S002210902500124X Westerhoff, F., et al. (2025). Boom –bust cycles and asset market participation waves: Momentum, value, risk, and herding. Journal of Evolutionary Economics , 35, 513–551. https://doi.org/10.1007/s00191-025-00919-6

work page doi:10.1017/s002210902500124x 2025