arXiv subjects

Christian Bick

Publications and source records attributed to Christian Bick.

At least 19 recordsLinked to original sources

arXiv2026

Self-organizing Chimera States in Adaptive Networks

We propose a minimal adaptive network model with product-form coupling that reduces dimensionality while capturing key features of synaptic plasticity. The system spontaneously self-organizes into adaptive chimera states, where identical oscillators separate into synchronized and desynchronized groups through adaptive weight dynamics. These adaptive chimeras organize into branches with fixed coherent cluster fraction and exhibit transitions between stationary, breathing, and chaotic collective dynamics, revealing a collective bifurcation structure. Crucially, the resulting attractor landscape is highly multistable: repeated cluster reorganizations generate distinct dynamical pathways that coexist within the same parameter regime and depend sensitively on initial conditions.

nlin.AO↗

arXiv2025

Higher-order phase reduction for delay-coupled oscillators beyond the phase-shift approximation

Network interactions between dynamical units are often subject to time delay. We develop a phase reduction method for delay-coupled oscillator networks. The method is based on rewriting the delay-differential equation as an ordinary differential equation coupled with a transport equation, expanding in the coupling strength, and solving the resulting equations order-by-order. This approach yields an approximation of the finite-dimensional phase dynamics to arbitrary order. While in the first-order approximation the time delay acts as a phase shift as expected, the higher-order phase reduction generally displays a less trivial dependence on the delay. In particular, exploiting second-order phase reduction, we prove the existence of a region of bistability in the synchronization dynamics of two delay-coupled Stuart-Landau oscillators.

math.DS↗

arXiv2025

Collective dynamics on higher-order networks

Higher-order interactions that nonlinearly couple more than two nodes are important in many networked systems, and their effects on collective dynamics are increasingly being studied. Here we provide an overview of this rapidly growing field, and of the techniques that can be used to describe and analyze them. We focus in particular on new phenomena and challenges that emerge when nonpairwise interactions are considered. We conclude by discussing open questions and promising future directions on the collective dynamics of higher-order networks.

nlin.AO↗

arXiv2025

Multiple Timescale Dynamics of Network Adaptation with Constraints

Adaptive network dynamical systems describe the co-evolution of dynamical quantities on the nodes as well as dynamics of the network connections themselves. For dense networks of many nodes, the resulting dynamics are typically high-dimensional. Here we consider adaptive dynamical systems subject to constraints on network adaptation: Asymptotically, the adaptive dynamics of network connections evolve on a low-dimensional subset of possible connectivity. Such dimension reduction may be intrinsic to the adaptation rule or arise from an additional dynamical mechanism acting on a timescale distinct from that of network adaptation. We illustrate how network adaptation with various constraints influences the dynamics of Kuramoto oscillator networks and elucidate the role of multiple timescales in shaping the dynamics. Our results shed light on why one may expect effective low-dimensional adaptation dynamics in generally high-dimensional adaptive network dynamical systems.

nlin.AO↗

arXiv2025

Frequency Synchronization Induced by Frequency Detuning

It is widely held that identical systems tend to behave similarly under comparable conditions. Yet, for systems that interact through a network, symmetry breaking can lead to scenarios in which this expectation does not hold. Prominent examples are chimera states in multistable phase-oscillator networks. Here, we show that for a broad class of such networks, asynchronous states can be converted into frequency-synchronized states when identical oscillators are detuned to have different intrinsic frequencies. Importantly, we show that frequency synchronization is achieved over a range of intrinsic frequency detuning and is thus a robust effect. These results, which are supported by theory, simulations, and electrochemical oscillator experiments, reveal a counterintuitive opportunity to use parameter heterogeneity to promote synchronization.

nlin.PS↗

arXiv2024

Decoding how higher-order network interactions shape complex contagion dynamics

Complex contagion models that involve contagion along higher-order structures, such as simplicial complexes and hypergraphs, yield new classes of mean-field models. Interestingly, the differential equations arising from many such models often exhibit a similar form, resulting in qualitatively comparable global bifurcation patterns. Motivated by this observation, we investigate a generalized mean-field-type model that provides a unified framework for analysing a range of different models. In particular, we derive analytical conditions for the emergence of different bifurcation regimes exhibited by three models of increasing complexity -- ranging from three- and four-body interactions to two connected populations with both pairwise and three-body interactions. For the first two cases, we give a complete characterisation of all possible outcomes, along with the corresponding conditions on network and epidemic parameters. In the third case, we demonstrate that multistability is possible despite only three-body interactions. Our results reveal that single population models with three-body interactions can only exhibit simple transcritical transitions or bistability, whereas with four-body interactions multistability with two distinct endemic steady states is possible. Surprisingly, the two-population model exhibits multistability via symmetry breaking despite three-body interactions only. Our work sheds light on the relationship between equation structure and model behaviour and makes the first step towards elucidating mechanisms by which different system behaviours arise, and how network and dynamic properties facilitate or hinder outcomes.

physics.soc-ph↗

arXiv2024

How many points converge to a heteroclinic network in an aperiodic way?

Homoclinic and heteroclinic connections can form cycles and networks in phase space, which organize global phenomena in dynamical systems. On the one hand, stability notions for (omni)cycles give insight into how many initial conditions approach the network along a single given (omni)cycle. On the other hand, the term switching is used to describe situations where there are trajectories that follow any possible sequence of heteroclinic connections along the network. Here we give a notion of asymptotic stability for general sequences along a network of homoclinic and heteroclinic connections. We show that there cannot be uncountably many aperiodic sequences that attract a set with nontrivial measure. Finally, we discuss examples where one may or may not expect aperiodic convergence towards a network and conclude with some open questions.

math.DS↗

arXiv2024

A Random-Walk Concentration Principle for Occupancy Processes on Finite Graphs

This paper concerns discrete-time occupancy processes on a finite graph. Our results can be formulated in two theorems, which are stated for vertex processes, but also applied to edge process (e.g., dynamic random graphs). The first theorem shows that concentration of local state averages is controlled by a random walk on the graph. The second theorem concerns concentration of polynomials of the vertex states. For dynamic random graphs, this allows to estimate deviations of edge density, triangle density, and more general subgraph densities. Our results only require Lipschitz continuity and hold for both dense and sparse graphs.

math.PR↗

arXiv2024

Dynamics on invariant tori emerging through forced symmetry breaking in phase oscillator networks

We consider synchrony patterns in coupled phase oscillator networks that correspond to invariant tori. For specific nongeneric coupling, these tori are equilibria relative to a continuous symmetry action. We analyze how the invariant tori deform under forced symmetry breaking as more general network interaction terms are introduced. We first show in general that perturbed tori that are relative equilibria can be computed using a parametrization method; this yields an asymptotic expansion of an embedding of the perturbed torus, as well as the local dynamics on the torus. We then apply this result to a coupled oscillator network, and we numerically study the dynamics on the persisting tori in the network by looking for bifurcations of their periodic orbits in a boundary-value-problem setup. This way we find new bifurcating stable synchrony patterns that can be the building blocks of larger global structures such as heteroclinic cycles.

math.DS↗

arXiv2024

Global bifurcations organizing weak chimeras in three symmetrically coupled Kuramoto oscillators with inertia

Frequency desynchronized attractors cannot appear in identically coupled symmetric phase oscillators because "overtaking" of phases cannot occur. This restriction no longer applies for more general identically coupled oscillators. Hence, it is interesting to understand precisely how frequency synchrony is lost and how invariant sets such as attracting weak chimeras are generated at torus breakup, where the phase description breaks down. Maistrenko et al (2016) found numerical evidence of an organizing center for weak chimeras in a system of $N=3$ coupled identical Kuramoto oscillators with inertia. This paper identifies this organizing center and shows that it corresponds to a particular type of non-transverse heteroclinic bifurcation that is generic in the context of symmetry. At this codimension two bifurcation there is a splitting of connecting orbits between the in-phase (fully synchronized) state. This generates a wide variety of associated bifurcations to weak chimeras. We further highlight a second organizing center associated with a codimension two symmetry-breaking heteroclinic connection.

math.DS↗

arXiv2024

Higher-order phase reduction captures delay-dependent synchronization phenomena in physical oscillator networks

Coupled oscillators with time-delayed network interactions are critical to understand synchronization phenomena in many physical systems. Phase reductions to finite-dimensional phase oscillator networks yield explicit insights into their dynamics. However, first-order phase approximations - in which the time delay acts as a phase shift - fail to capture the delay-dependence of synchronization. We develop a systematic approach to derive phase reductions for delay-coupled oscillators to higher order. Beyond first order, already a second-order phase reduction captures delay-induced synchronization as demonstrated in coupled Stuart-Landau oscillators and experiments with delay-coupled electrochemical oscillators. Our results establish a general mechanism by which time delays reshape synchronization phenomena and reveal intrinsic limitations of widely used reduced phase models, with implications for a broad range of oscillator networks.

math.DS↗

arXiv2024

Strong coupling yields abrupt synchronization transitions in coupled oscillators

Coupled oscillator networks often display transitions between qualitatively different phase-locked solutions -- such as synchrony and rotating wave solutions -- following perturbation or parameter variation. In the limit of weak coupling, these transitions can be understood in terms of commonly studied phase approximations. As the coupling strength increases, however, predicting the location and criticality of transition, whether continuous or discontinuous, from the phase dynamics may depend on the order of the phase approximation -- or a phase description of the network dynamics that neglects amplitudes may become impossible altogether. Here we analyze synchronization transitions and their criticality systematically for varying coupling strength in theory and experiments with coupled electrochemical oscillators. First, we analyze bifurcations analysis of synchrony and splay states in an abstract phase model and discuss conditions under which synchronization transitions with different criticalities are possible. Second, we illustrate that transitions with different criticality indeed occur in experimental systems. Third, we highlight that the amplitude dynamics observed in the experiments can be captured in a numerical bifurcation analysis of delay-coupled oscillators. Our results showcase that reduced order phase models may miss important features that one would expect in the dynamics of the full system.

math.DS↗

arXiv2023

Hopf Bifurcations of Twisted States in Phase Oscillators Rings with Nonpairwise Higher-Order Interactions

Synchronization is an essential collective phenomenon in networks of interacting oscillators. Twisted states are rotating wave solutions in ring networks where the oscillator phases wrap around the circle in a linear fashion. Here, we analyze Hopf bifurcations of twisted states in ring networks of phase oscillators with nonpairwise higher-order interactions. Hopf bifurcations give rise to quasiperiodic solutions that move along the oscillator ring at nontrivial speed. Because of the higher-order interactions, these emerging solutions may be stable. Using the Ott--Antonsen approach, we continue the emergent solution branches which approach anti-phase type solutions (where oscillators form two clusters whose phase is $\pi$ apart) as well as twisted states with a different winding number.

math.DS↗

arXiv2023

Heteroclinic Dynamics in Network Dynamical Systems with Higher-Order Interactions

Heteroclinic structures organize global features of dynamical systems. We analyze whether heteroclinic structures can arise in network dynamics with higher-order interactions which describe the nonlinear interactions between three or more units. We find that while commonly analyzed model equations such as network dynamics on undirected hypergraphs may be useful to describe local dynamics such as cluster synchronization, they give rise to obstructions that allow to design heteroclinic structures in phase space. By contrast, directed hypergraphs break the homogeneity and lead to vector fields that support heteroclinic structures.

math.DS↗

arXiv2023

Higher-Order Network Interactions through Phase Reduction for Oscillators with Phase-Dependent Amplitude

Coupled oscillator networks provide mathematical models for interacting periodic processes. If the coupling is weak, phase reduction -- the reduction of the dynamics onto an invariant torus -- captures the emergence of collective dynamical phenomena, such as synchronization. While a first-order approximation of the dynamics on the torus may be appropriate in some situations, higher-order phase reductions become necessary, for example, when the coupling strength increases. However, these are generally hard to compute and thus they have only been derived in special cases: This includes globally coupled Stuart--Landau oscillators, where the limit cycle of the uncoupled nonlinear oscillator is circular as the amplitude is independent of the phase. We go beyond this restriction and derive second-order phase reductions for coupled oscillators for arbitrary networks of coupled nonlinear oscillators with phase-dependent amplitude, a scenario more reminiscent of real-world oscillations. We analyze how the deformation of the limit cycle affects the stability of important dynamical states, such as full synchrony and splay states. By identifying higher-order phase interaction terms with hyperedges of a hypergraph, we obtain natural classes of coupled phase oscillator dynamics on hypergraphs that adequately capture the dynamics of coupled limit cycle oscillators.

math.DS↗

arXiv2022

Phase Oscillator Networks with Nonlocal Higher-Order Interactions: Twisted States, Stability and Bifurcations

The Kuramoto model provides a prototypical framework to synchronization phenomena in interacting particle systems. Apart from full phase synchrony where all oscillators behave identically, identical Kuramoto oscillators with ring-like nonlocal coupling can exhibit more elaborate patterns such as uniformly twisted states. It was discovered by Wiley, Strogatz and Girvan in 2006 that the stability of these twisted states depends on the coupling range of each oscillator. In this paper, we analyze twisted states and their bifurcations in the infinite particle limit of ring-like nonlocal coupling. We not only consider traditional pairwise interactions as in the Kuramoto model but also demonstrate the effects of higher-order nonpairwise interactions, which arise naturally in phase reductions. We elucidate how pairwise and nonpairwise interactions affect the stability of the twisted states, compute bifurcating branches, and show that higher-order interactions can stabilize twisted states that are unstable if the coupling is only pairwise.

math.DS↗

arXiv2022

Symmetry breaking yields chimeras in two small populations of Kuramoto-type oscillators

Despite their simplicity, networks of coupled phase oscillators can give rise to intriguing collective dynamical phenomena. However, the symmetries of globally and identically coupled identical units do not allow solutions where distinct oscillators are frequency-unlocked -- a necessary condition for the emergence of chimeras. Thus, forced symmetry breaking is necessary to observe chimera-type solutions. Here, we consider the bifurcations that arise when full permutational symmetry is broken for the network to consist of coupled populations. We consider the smallest possible network composed of four phase oscillators and elucidate the phase space structure, (partial) integrability for some parameter values, and how the bifurcations away from full symmetry lead to frequency-unlocked weak chimera solutions. Since such solutions wind around a torus they must arise in a global bifurcation scenario. Moreover, periodic weak chimeras undergo a period doubling cascade leading to chaos. The resulting chaotic dynamics with distinct frequencies do not rely on amplitude variation and arise in the smallest networks that support chaos.

nlin.AO↗

arXiv2022

Network Dynamics with Higher-Order Interactions: Coupled Cell Hypernetworks for Identical Cells and Synchrony

Network interactions that are nonlinear in the state of more than two nodes - also known as higher-order interactions - can have a profound impact on the collective network dynamics. Here we develop a coupled cell hypernetwork formalism to elucidate the existence and stability of (cluster) synchronization patterns in network dynamical systems with higher-order interactions. More specifically, we define robust synchrony subspace for coupled cell hypernetworks whose coupling structure is determined by an underlying hypergraph and describe those spaces for general such hypernetworks. Since a hypergraph can be equivalently represented as a bipartite graph between its nodes and hyperedges, we relate the synchrony subspaces of a hypernetwork to balanced colorings of the corresponding incidence digraph.

math.DS↗