(k1,k2)-Mode Turing-Hopf Bifurcation

• The study defines (k1,k2)-mode Turing-Hopf bifurcation as a codimension-two interaction where a steady (Turing) mode and an oscillatory (Hopf) mode coexist.
• It employs spectral decomposition, center-manifold reduction, and normal forms to rigorously analyze the interplay of diffusion-driven pattern formation with temporal oscillations.
• The work distinguishes the (k1,k2) framework from traditional Turing-Hopf scenarios, highlighting its impact on predicting mixed spatial and temporal dynamics.

A $(k_1,k_2)$-mode Turing-Hopf bifurcation is a codimension-two Hopf-steady-state singularity in which a simple zero eigenvalue is associated with one spatial mode $k_1$, while a simple conjugate pair $\pm i\omega_0$ is associated with another spatial mode $k_2$. In the formulation that explicitly uses this terminology, the bifurcation is called a $(k_1,k_2)$-mode Turing-Hopf bifurcation when the steady branch is spatially inhomogeneous, i.e. $k_1\neq 0$ (Jiang et al., 2018). The resulting local dynamics organize the interaction of diffusion-driven pattern formation and temporal oscillation, and may generate patterned steady states, homogeneous or inhomogeneous periodic orbits, and mixed spatiotemporal states. A terminological complication is that some continuation literature uses “Turing-Hopf” or “wave” for a Hopf crossing at a single nonzero spatial mode, whereas the $(k_1,k_2)$-mode usage refers to simultaneous steady and oscillatory criticality on possibly different spatial modes (Uecker, 2016).

1. Definition and terminological scope

In the explicit normal-form framework for delayed reaction-diffusion systems, the underlying codimension-two hypothesis is that for mode $k_1$ the characteristic equation has a simple real eigenvalue $\gamma(\alpha)$ with $\gamma(0)=0$, while for mode $k_1$0 it has a simple pair $k_1$1 with $k_1$2 and $k_1$3; all other spectral branches stay off the imaginary axis (Jiang et al., 2018). Under homogeneous Neumann boundary conditions, $k_1$4 denotes the homogeneous spatial mode, while $k_1$5 denotes a spatially inhomogeneous Laplacian mode. In that language, $k_1$6 is the steady or Turing mode, and $k_1$7 is the oscillatory or Hopf mode (Jiang et al., 2018).

The phrase is not used uniformly across the literature. In the pde2path study of the extended Brusselator, a Hopf-type crossing at nonzero spatial mode is called “Turing-Hopf (aka wave),” so the relevant object is a single oscillatory finite-$k_1$8 instability rather than a Hopf-zero codimension-two interaction (Uecker, 2016). This distinction matters because the $k_1$9-mode formulation is specifically a two-mode codimension-two concept.

A second recurrent misconception is that the Hopf component must be spatially homogeneous. The general $\pm i\omega_0$0-mode framework does not require that: $\pm i\omega_0$1 may be $\pm i\omega_0$2 or nonzero, and the corresponding critical eigenfunctions may both be spatially inhomogeneous (Jiang et al., 2018). The same point appears in the non-instantaneous Kerr cavity, where the Hopf instability may occur at $\pm i\omega_0$3 or at $\pm i\omega_0$4, while the Turing instability is selected by a finite critical wave number $\pm i\omega_0$5 (Ouali et al., 2016).

2. Spectral formulation and modal decomposition

The general delayed reaction-diffusion formulation used for $\pm i\omega_0$6-mode analysis is

$\pm i\omega_0$7

posed on a bounded domain with homogeneous Neumann or Dirichlet boundary conditions (Jiang et al., 2018). After separation in Laplacian eigenfunctions $\pm i\omega_0$8, the characteristic equation becomes modewise: $\pm i\omega_0$9 The codimension-two point is therefore located by solving one steady condition at mode $k_2$0 and one Hopf condition at mode $k_2$1 (Jiang et al., 2018).

In one-dimensional Neumann problems, the spatial basis is typically $k_2$2, so the mode number has a direct wavelength interpretation. In the delayed Schnakenberg system, the $k_2$3-th modal characteristic equation is

$k_2$4

and a $k_2$5-mode Turing bifurcation means that $k_2$6 has a simple zero root, whereas a $k_2$7-mode Hopf bifurcation means that $k_2$8 has a simple pair of purely imaginary roots (Jiang et al., 2018). The paper then states that the system undergoes a $k_2$9-mode Turing-Hopf bifurcation at

$(k_1,k_2)$0

with $(k_1,k_2)$1 the Turing index and $(k_1,k_2)$2 the Hopf index (Jiang et al., 2018).

A closely related linear-algebraic formulation appears in the diffusive epidemic model. There the Neumann decomposition on $(k_1,k_2)$3 yields modal Jacobians

$(k_1,k_2)$4

with modal characteristic polynomial

$(k_1,k_2)$5

The Turing condition is $(k_1,k_2)$6, and the Hopf condition is $(k_1,k_2)$7 with $(k_1,k_2)$8. The paper’s theorem then locates a $(k_1,k_2)$9-mode Turing-Hopf point at

$k_1\neq 0$0

subject to $k_1\neq 0$1 and condition $k_1\neq 0$2 (Khateeb et al., 10 Sep 2025).

3. Center-manifold reduction and normal forms

At a $k_1\neq 0$3-mode Turing-Hopf point, the center space is three-dimensional over $k_1\neq 0$4: one real amplitude for the steady mode and one complex amplitude for the Hopf mode. In the general PFDE theory this yields

$k_1\neq 0$5

and the reduced normal form

$k_1\neq 0$6

or, more explicitly,

$k_1\neq 0$7

with quadratic and cubic coupling terms such as $k_1\neq 0$8, $k_1\neq 0$9, $(k_1,k_2)$0, and $(k_1,k_2)$1 (Jiang et al., 2018).

The mode dependence is not merely bookkeeping. The coefficients depend on Fréchet derivatives of the original PDE or PFDE and on overlap integrals such as

$(k_1,k_2)$2

which determine whether quadratic couplings vanish. The theory distinguishes five mode configurations, including $(k_1,k_2)$3 and $(k_1,k_2)$4, and therefore accommodates both homogeneous and inhomogeneous Hopf components (Jiang et al., 2018).

Two reduced amplitude systems recur. In the Hopf-transcritical case, the planar reduction is written in $(k_1,k_2)$5-coordinates with $(k_1,k_2)$6 the Hopf amplitude and $(k_1,k_2)$7 the steady amplitude (Jiang et al., 2018). In the Hopf-pitchfork case, which is especially common when the Turing branch is symmetry-related, the reduction takes the form

$(k_1,k_2)$8

and its equilibria correspond to the base equilibrium, a pure Hopf branch, a pure Turing branch, and a mixed branch with both amplitudes nonzero (Jiang et al., 2018). In the delayed Schnakenberg system the same planar form is obtained for $(k_1,k_2)$9, and explicit examples are worked out for $k_1$0 and $k_1$1 (Jiang et al., 2018).

The attractor interpretation is standard across these normal forms. Pure $k_1$2-branches correspond to spatially inhomogeneous steady states; pure $k_1$3-branches correspond to periodic solutions; mixed equilibria correspond to spatially inhomogeneous periodic orbits. In the delayed reaction-diffusion framework of the two-component Turing-Hopf analysis, periodic orbits of the planar amplitude system can further correspond to spatially inhomogeneous quasi-periodic solutions of the original PDE (An et al., 2017).

4. Pattern classes: steady, periodic, localized, and front-pinned states

The local codimension-two reduction organizes several distinct pattern classes. In the non-instantaneous Kerr cavity, the linear problem yields a finite-wavelength Turing instability together with a Hopf instability, and the numerics show a transition from a stationary Turing branch to a stable mixed-mode Turing-Hopf branch. The same model also supports localized mixed-mode solutions: stationary localized structures begin to pulse in time, producing time-periodic states that remain localized in space (Ouali et al., 2016).

Two-dimensional effects introduce additional structure that is not captured by one-dimensional front interaction alone. In the study of forced oscillatory media near a Hopf-Turing point, the authors derive a two-dimensional amplitude description with one Hopf mode and two Turing modes $k_1$4 and $k_1$5, associated with wave vectors along $k_1$6 and $k_1$7, respectively. This produces comb-like localized Turing states embedded in oscillatory Hopf backgrounds, including localized stripes inside $k_1$8-shifted Hopf oscillations and in spiral cores (Castillero et al., 2017). The key mechanism is a local dynamics pinning of Hopf fronts, which allows the Turing mode perpendicular to the front to become locally selected and pinned; the resulting states occur outside the one-dimensional flip-flop or homoclinic-snaking picture and extend frequency locking beyond the classical $k_1$9 resonance region (Castillero et al., 2017).

The same literature therefore shows that the $\gamma(\alpha)$0-mode viewpoint is not restricted to spatially uniform oscillatory backgrounds or to one-dimensional pattern branches. A plausible implication is that the codimension-two mechanism is best understood as a mode-selection problem with geometry-dependent coupling coefficients rather than as a single universal front scenario.

5. Symmetry, geometry, and multimode generalizations

On a disk, the scalar wave number $\gamma(\alpha)$1 is replaced by a pair of mode labels $\gamma(\alpha)$2, where $\gamma(\alpha)$3 is the azimuthal index and $\gamma(\alpha)$4 the radial index. Nonradial modes with $\gamma(\alpha)$5 are symmetry-degenerate, so a Turing-Hopf point may involve equivariant Turing or equivariant Hopf branches rather than simple scalar modes. The disk analysis distinguishes three types: ET-H, T-EH, and ET-EH. Their center manifolds have dimensions $\gamma(\alpha)$6, $\gamma(\alpha)$7, and $\gamma(\alpha)$8, respectively, and the associated patterns include breathing, standing wave-like, and rotating wave-like states (Chen et al., 2023). A resonance condition

$\gamma(\alpha)$9

changes the normal form by introducing additional phase-coupling terms, so the disk problem provides a symmetry-based analogue of wave-number resonance in standard mode-interaction theory (Chen et al., 2023).

A different extension appears in the codimension-three Hopf-Turing-Turing framework for two-component reaction-diffusion systems on an interval. There a $\gamma(0)=0$0-mode Hopf instability interacts simultaneously with two adjacent Turing modes $\gamma(0)=0$1 and $\gamma(0)=0$2, giving a $\gamma(0)=0$3 organizing center. For the explicitly derived $\gamma(0)=0$4 case the reduced normal form contains resonance terms of the form $\gamma(0)=0$5 and $\gamma(0)=0$6; for $\gamma(0)=0$7 these resonance terms drop out in the stated $\gamma(0)=0$8 normal form (Izuhara et al., 2023). The dynamics of this extension include mixed spatiotemporal oscillations, invariant $\gamma(0)=0$9-tori, possible invariant $k_1$00-tori, heteroclinic structures, and numerically suggested chaos (Izuhara et al., 2023). Although this is not a codimension-two $k_1$01-mode Turing-Hopf bifurcation in the strict sense, it shows how nearby Turing modes can enrich the standard two-mode picture.

6. Computation, continuation, and representative models

The computational side of $k_1$02-mode problems is developed most explicitly in pde2path. After spatial discretization of

$k_1$03

Hopf points are detected from the generalized eigenvalue problem

$k_1$04

localized by bisection, and continued as periodic orbits with phase and continuation conditions; Floquet multipliers are computed either by direct monodromy products or by periodic Schur decomposition (Uecker, 2016). In the two-dimensional Brusselator example on [ \Omega=(-l_x,l_x)\times(-l_y,l_y),\qquad l_x=\pi/2,\quad l_y=\pi/