Traveling Chimera States in Oscillator Networks

• Traveling chimera states are spatiotemporal patterns in coupled oscillators, featuring a drifting synchronized (coherent) region amid desynchronized (incoherent) background.
• Their dynamics are quantified using a local order parameter, with kernel asymmetry and gradient-based feedback controlling the drift speed and direction.
• These states enable practical applications such as binary encoding, logic operations in chimera computers, and insights into neurobiological memory formation.

Traveling chimera states are spatiotemporal patterns in networks of coupled nonlinear dynamical units, characterized by a coherent (synchronized) domain whose spatial location is not stationary but systematically shifts or drifts over time across a network, embedded within an incoherent (desynchronized) background. Unlike pinned or statically localized chimeras, traveling chimeras exhibit persistent, organized motion of the coherent region, offering interpretation as spatial information carriers and opening possibilities for feedback control, information encoding, and functional applications in artificial and biological systems.

1. Definition and Mathematical Characterization

A traveling chimera state is defined by the coexistence of two domains within a spatially extended system of coupled oscillators:

• A coherent domain: oscillators in this region are phase-synchronized.
• An incoherent domain: oscillators are desynchronized, typically displaying uncorrelated phase behaviors.

In the paradigm of non-locally coupled phase or limit-cycle oscillators distributed along a ring, the spatial position $y(t)$ of the coherent (synchronized) region evolves nontrivially in time. The dynamics of synchrony can be quantified via a local order parameter: $$Z(x, \varphi) = \int_{0}^{1} h(d(x, y)) e^{i \varphi(y)}\, dy$$ and its absolute value

$R(x, \varphi) = |Z(x, \varphi)|$

peaks where local synchrony is strongest. In traveling chimeras, the maximum of $R(x, \varphi(t))$ drifts in $x$ as a function of time.

Chimera drift is induced by breaking spatial symmetry (typically via an asymmetry parameter $a$ in the coupling kernel), yielding a position dynamics

$$\dot{y}(t) = -\gamma \partial_x R(x, \varphi(t))|_{x = x^*}$$

where $x^*$ is the control target and $\gamma$ is a control gain. In the uncontrolled case, traveling arises from inherent system symmetry breaking or parameter choices without external control.

2. Mechanisms Enabling Traveling Chimeras

Symmetry Breaking via Kernel Asymmetry

A traveling chimera can be produced by introducing a family of asymmetric coupling kernels, e.g.,

$$h_a(x) = \begin{cases} \exp[-\kappa(1-a)|x|] & x < 0 \ \exp[-\kappa(1+a)|x|] & x \geq 0 \end{cases}$$

The asymmetry parameter $a$ determines the drift speed $\nu(a)$ of the coherent region. By modulating $a$, one can directly control the velocity and direction of the traveling domain.

Observable-Guided Gradient Dynamics

Gradient-based feedback control exploits the spatial gradient of the order parameter to steer the coherent region: $$a(t) = \nu^{-1}\left( -\gamma \partial_x R(x, \varphi(t))|_{x = x^*} \right)$$ When implemented in finite systems, the discrete gradient approximation is used: $$\Delta_{x^*}^{(\delta)} R(\varphi(t)) = \frac{R(x^* + \delta, \varphi(t)) - R(x^* - \delta, \varphi(t))}{2\delta}$$ with bounded control via a sigmoidal function to prevent overshoot. This approach enables targeted spatial manipulation while suppressing random walks or Brownian drift of the chimera.

Spontaneous Drift in Dynamical Regimes

In certain neuron, oscillator, and excitable media models, traveling chimeras can arise spontaneously through intrinsic bifurcations or multistability, not necessarily requiring an explicit asymmetry. Such states may appear in regimes of high coupling strength, near bifurcation thresholds, or due to the interplay of local and nonlocal coupling.

3. Implementation in Finite-Dimensional Systems

For discrete rings of $n$ oscillators with phases $\{\varphi_k\}$, the dynamics take the form: $$\dot{\varphi}_k = \omega_k - \frac{1}{n} \sum_j h(d(\iota(k), \iota(j)))\, \sin(\varphi_k - \varphi_j + \alpha)$$ Here, $\iota(k) = k/n$ maps index to spatial position, and $h$ incorporates the kernel asymmetry. The discrete local order parameter: $$R(x, \varphi) = \left| \frac{1}{n} \sum_j h(d(x, \iota(j)))\, e^{i\varphi_j} \right|$$ is used both for characterizing local synchrony and for constructing gradient-based controls.

When implementing feedback control, the control law can be written as

$$a(t) = a_{\text{max}} \cdot \lambda\left( K \cdot \Delta_{x^*}^{(\delta)} R(\varphi(t)) \right)$$

where $\lambda(x) = 2/(1 + e^{-x}) - 1$, $K$ is a gain, and $a_{\text{max}}$ sets the control range. This law has been verified through numerical simulations (e.g., $n = 256$ oscillators), where it successfully moves the chimera to any prescribed $x^*$ and suppresses pseudo-random drift, even with as few as $n=30$ oscillators.

4. Functional Applications: Encoding and Computation

The ability to robustly control the position of the coherent region in a traveling chimera facilitates the encoding of information:

• Binary encoding: Assigning logical values (e.g., $0$ and $1$) to two antipodal positions on the ring, writing a bit as the presence of the chimera at either location.
• Logic operations: By dynamically linking rings (e.g., setting the target of ring B to the position of the maximum in ring A), operations like copying and NOT can be performed: $x^*_B(t) = \arg\max_x R(x, \varphi^A(t))$ This capacity suggests the feasibility of a “chimera computer,” where spatial synchrony localizations encode and process information.

Such paradigms parallel mechanisms hypothesized in neurobiology, where localized synchronization (bump states) underlies memory formation and spatial representation.

5. Generalization to Higher Dimensions and Other Systems

The control framework generalizes when the spatial order parameter gradient is available. For systems on a $d$-dimensional domain, control proceeds by

$$\dot{\mathbf{y}} = -\gamma\, \nabla_x R(x, \varphi(t))$$

with feedback mapped onto a physically accessible parameter that induces drift (e.g., anisotropy of coupling). This applies to spiral wave chimeras in excitable media or engineered nonlocally coupled arrays.

The key requirements are:

• Existence of a differentiable local synchrony observable.
• Presence of a tunable parameter (such as kernel asymmetry or anisotropy) that enables drift.

6. Practical Considerations and Limitations

System Size and Inhomogeneity

Performance is robust in moderate system sizes, but stochastic fluctuations and heterogeneity (intrinsic frequency disorder $\omega_k$) can introduce random walks or pinning. The control law is designed to suppress such effects by noninvasive feedback (the signal vanishes at the target).

Computational Overhead

Implementing gradient-based feedback requires continuous calculation of spatial derivatives of the order parameter and real-time adjustment of the kernel parameter. The computational burden increases with system dimensionality and size but remains tractable for standard oscillator counts.

Limitations

Traveling chimeras are sensitive to spatial inhomogeneities; introducing small spatial defects or heterogeneities can pin or destroy the motion of the coherent domain [see also (Xie et al., 2015)]. Strong disorder invalidates drift-based control.

7. Impact and Perspective

The integration of gradient-based feedback and symmetry-breaking control positions traveling chimera states as functionally relevant objects that can be manipulated, harnessed for unconventional computation, and realized in physical systems where control over localized synchronization is required. This mechanism is flexible, permitting generalization beyond 1D rings to arbitrary spatial networks, provided a controllable drift mechanism exists. The results provide foundational perspectives for both fundamental studies of spatiotemporal complexity and the engineering of dynamical systems with programmable spatial information processing capabilities.