Self-Organization and Bifurcations in Coherent Flows

A three-dimensional reduced model is presented describing the interaction between a dominant flow mode (Galerkin projection of the Stokes problem) and two structural variables: the coherence λ(t) and the structural feedback force σ(t). The system analytically captures the spontaneous emergence of coherent structures through saddle-node bifurcations and possible Hopf bifurcations. The stationary points, the transcendental equilibrium equation, the full Jacobian, the cubic characteristic polynomial, and the exact Routh–Hurwitz conditions for asymptotic stability are explicitly derived. The origin is unstable when the structural force is autocatalytic (γ > 0), enabling self-organization. The dimensionless form of the system and a qualitative analysis of the dynamical regimes are included. The framework is applicable to viscoelastic fluids, biological active fluids, and elastic turbulence.