The Turing bifurcation in network systems: Collective patterns and single differentiated nodes

  title={The Turing bifurcation in network systems: Collective patterns and single differentiated nodes},
  author={Matthias Wolfrum},
  journal={Physica D: Nonlinear Phenomena},
  • M. Wolfrum
  • Published 15 August 2012
  • Physics
  • Physica D: Nonlinear Phenomena

Figures from this paper

Predictable topological sensitivity of Turing patterns on graphs.
Reaction-diffusion systems implemented as dynamical processes on networks have recently renewed the interest in their self-organized collective patterns known as Turing patterns. We investigate the
Snaking on Networks: From Local Solutions to Turing Patterns
Numerical continuation reveals snaking bifurcations connecting different solutions, similar to those found in reaction diffusion systems on regular lattice network topologies, shedding light on the origin of the multistable “Turing' patterns reported previously.
Theory of Turing Patterns on Time Varying Networks.
The process of pattern formation for a multispecies model anchored on a time varying network is studied and a closed analytical prediction for the onset of the instability in the time dependent framework is derived.
Turing instability in reaction–diffusion models on complex networks
Feedback-induced stationary localized patterns in networks of diffusively coupled bistable elements
Effects of feedbacks on self-organization phenomena in networks of diffusively coupled bistable elements are investigated. For regular trees, an approximate analytical theory for localized stationary
Turing patterns mediated by network topology in homogeneous active systems.
It is demonstrated that networks with large degree fluctuations tend to have stable patterns over the space of initial perturbations, whereas patterns in more homogenous networks are purely stochastic, and the Turing instability can be induced in any network topology by tuning the diffusion of the competing species or by altering network connectivity.
Dispersal-induced destabilization of metapopulations and oscillatory Turing patterns in ecological networks
The original analysis by Turing to networks is extended and applied to ecological metapopulations with dispersal connections between habitats, finding such oscillatory instabilities for all possible food webs with three predator or prey species.
Pattern Formation on Networks: from Localised Activity to Turing Patterns
Through the application of a generalisation of dynamical systems analysis this work reveals a fundamental connection between small-scale modes of activity on networks and localised pattern formation seen throughout science, such as solitons, breathers andLocalised buckling.
A theory of pattern formation for reaction–diffusion systems on temporal networks
Networks have become ubiquitous in the modern scientific literature, with recent work directed at understanding ‘temporal networks’—those networks having structure or topology which evolves over
Pattern formation in multiplex networks
The theory demonstrates that the existence of such topology-driven instabilities is generic in multiplex networks, providing a new mechanism of pattern formation.


Diffusion-induced instability and chaos in random oscillator networks.
  • H. Nakao, A. Mikhailov
  • Physics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2009
It is argued that uniform oscillations can be linearly unstable with respect to spontaneous phase modulations due to diffusional coupling-the effect corresponding to the Benjamin-Feir instability in continuous media.
Instability and dynamic pattern in cellular networks.
Stable Stationary Solutions in Reaction-diffusion Systems Consisting of a 1-d Array of Bistable Cells
This paper shows the existence of stable spatially periodic (pattern) solutions that persist for large coupling constants in reaction–diffusion systems consisting of a 1-D array of bistable cells with a cubic nonlinearity and a cubic-like piecewise-linear non linearity.
Loss of coherence in dynamical networks: spatial chaos and chimera states.
A dynamical bifurcation scenario for the coherence-incoherence transition is uncovered which starts with the appearance of narrow layers of incoherence occupying eventually the whole space.
Complex networks: Structure and dynamics
Turing patterns in network-organized activator–inhibitor systems
A general framework now provides the tools for studying so-called Turing patterns in systems organized in complex networks, leading to the spontaneous emergence of periodic spatial patterns.
Reaction–diffusion processes and metapopulation models in heterogeneous networks
This work lays out a theoretical and computational microscopic framework for the study of a wide range of realistic metapopulation and agent-based models that include the complex features of real-world networks.
Laplacian spectra as a diagnostic tool for network structure and dynamics.
The effects of clustering, degree distribution, and a particular type of coupling asymmetry (input normalization), all of which are known to have effects on the synchronizability of oscillator networks, are studied.