Higher-codimension points as organizing centers in nonreciprocal pattern-forming systems with O(2)-symmetry

TL;DR

This work analyzes nonreciprocal pattern formation in a two-field Swift–Hohenberg model with linear nonreciprocal coupling and $O(2)$-symmetry. By combining full PDE simulations, a one-mode amplitude reduction, and a normal-form analysis near a codimension-two Takens–Bogdanov point, it reveals that the TB point acts as a unifying organizing center for the onset of pattern formation and the transitions from static to traveling waves. A TB normal form with unfolding parameters linked to the NRSH parameters shows that the sign of $\delta$ selects between two universal unfolding types, with a fixed ratio $D/M=1/3$ restricting the bifurcation topology to two scenarios. The study also identifies five codimension-two bifurcations that shape nonlinear routes to traveling states, including tricritical Turing, saddle-loop, decoupled transition, and double SNIPer, offering a comprehensive framework for interpreting experiments and guiding extensions to stochastic or nonlocal variants. Overall, the results provide a universal, multi-level understanding of nonreciprocal phase transitions in systems with a dominant wavelength.

Abstract

Focusing on a two-field Swift-Hohenberg model with linear nonreciprocal interactions, this study investigates how emerging higher-codimension points act as organizing centers for the nonequilibrium phase diagram that features various steady and dynamic phases. Complementing the numerical analysis of the field equations with time simulations and path continuation techniques, we derive a reduced dynamical system corresponding to a one-mode approximation for the critical-wavenumber modes. Furthermore, we derive the normal form equations that are valid in the vicinity of the Takens-Bogdanov bifurcation with O(2)-symmetry, which allows us to draw on corresponding literature results. Comparing results obtained on the different levels of description, we discuss the bifurcation structure relating trivial uniform and inhomogeneous steady states as well as traveling, standing and modulated waves. We also contextualize the relevance of recently highlighted features of the linear mode structure, i.e., of the dispersion relations, termed "critical exceptional points" for the transitions between the nonequilibrium phases.

Figures (7)

• Figure 1: Simulation results of the (top row) one-dimensional NRSH model \ref{['eq:NRSH']} as spacetime plots of the order parameter field $\phi(x,t)$ and (bottom row) as corresponding trajectories in the complex plane of the amplitudes $\phi_1(t)$ and $\psi_1(t)$ , both at $(\varepsilon, \chi, \delta)=(1, 1, 0)$. (a) Steady-State (SS) at $\alpha = 1.27$, (b) Standing-Wave (SW) at $\alpha = 1.28$, (c) Traveling-Wave (TW) at $\alpha = 1.35$, and (d) Modulated-Wave (MW) at $\alpha = 1.29$.
• Figure 2: Phase diagram for (a) $\delta = -0.5$, (b) $\delta = 0$, and (c) $\delta = 0.5$ obtained by numerical calculation of the NRSH \ref{['eq:NRSH']} at $\chi = 1$. The numerical scheme and criteria for phase classification, including the distinction between in-phase (IP-SS) and anti-phase (AP-SS) steady states, are described in Appendix \ref{['app:methods']}. The theoretical phase boundaries corresponding to the local or global bifurcation curves obtained with the one-mode approximation \ref{['eq:nr_AE']} are also overlaid on the phase diagram. Black diamonds mark the location of the codimension-two Takens-Bogdanov bifurcation. The derivation of the theoretical curves is described in Appendices \ref{['app:Turing-wave']} and \ref{['app:local-bif-derivation']}.
• Figure 3: Three different bifurcation scenarios in the NRSH model \ref{['eq:NRSH']} with nonreciprocal coupling strength $\alpha$ as continuation parameter for varying values of $\varepsilon$ for the transition from steady states (SS) to traveling waves (TW) in the symmetric case $\delta = 0$. Panels (a) to (c) show bifurcation diagrams calculated in the one-mode approximation \ref{['eq:nr_AE']} with the root-mean-square amplitude $\|(\phi_1,\psi_1)\|_2$ in Eq. \ref{['eq:norm_one_mode']} as a solution measure. Stable [unstable] states are indicated by thick [thin] lines. Gray lines indicate the same bifurcation diagrams obtained with the full NRSH model with spatiotemporal $L^2$-norm \ref{['eq:norm_NRSH']}. Panels (d) to (f) show the growth rates $\mathrm{Re}(\lambda)$ of the relevant eigenvalues for the SS (blue) and TW state (green), for the latter calculated in the co-moving frame. Solid [dotted] lines indicate real [complex] eigenvalues. The SS and TW states spontaneously break the continuous O(2)-symmetry of Eq. \ref{['eq:NRSH']}, resulting in a permanently present Goldstone zero mode.
• Figure 4: The two realized bifurcation scenarios in the vicinity of the O(2)-Takens-Bogdanov bifurcation. Panels (a) and (c) show numerical bifurcation diagrams for $\varepsilon = 0.05$ at $\delta = -0.5$, and $\delta = 0.5$, respectively, as obtained with the one-mode approximation \ref{['eq:nr_AE']}. Stable [unstable] states are indicated as thick [thin] lines. Panels (b) and (d) show the corresponding conceptual bifurcation diagrams $\text{III}_-$ ($A > 0$) and $\text{II}_-$ ($A < 0$) occurring in the universal unfolding in Ref. DaKn1987ptrslsapes, with plus and minus signs indicating the number of stable and unstable eigenvalues of the respective branches (used with permission of the Royal Society (U.K.), from Ref. DaKn1987ptrslsapes; permission conveyed through Copyright Clearance Center, Inc.). The red shading indicates the bifurcations that are accessible with the nonreciprocal coupling strength $\alpha$ as a bifurcation parameter (the direction of panels (b) and (d) corresponds to $\alpha$ increasing from right to left, as indicated by the red arrow). Panel (e) shows a magnification of the successive saddle-node, gluing, and Hopf bifurcations that terminate the standing wave branch. Subpanels 1 and 2 [3 and 4] of (f) show the asymmetric [symmetric] SW limit cycles in the amplitude representation of the phase space \ref{['eq:phase_difference']}. The hollow circles represent the unstable fixed points corresponding to N and SS states.
• Figure 5: Quantitative evaluation of the predictions of the normal form of the O(2)-symmetric Takens-Bogdanov bifurcation. Panel (a) for $\delta = -0.5$ compares the loci of bifurcations in the $(\varepsilon, \alpha)$-plane computed with the normal form \ref{['eq:TB-normal']} (dashed lines) and with the one-mode approximation \ref{['eq:nr_AE']} (solid lines). Panel (b) shows the slope $(\alpha - \alpha_{\mathrm{DP}}) / \varepsilon$ in the vicinity of the TB point, proving that the corresponding lines of codimension-one bifurcations determined on the two levels of description enter the TB point tangentially to each other.