Topological synchronization of coupled nonlinear oscillators

Abstract

Synchronization of coupled oscillators is a ubiquitous phenomenon found throughout nature. Its robust realization is crucial to our understanding of various nonlinear systems, ranging from biological functions to electrical engineering. On another front, in condensed matter physics, topology is utilized to realize robust properties like topological edge modes, as demonstrated by celebrated topological insulators. Here, we integrate these two research avenues and propose a nonlinear topological phenomenon, namely topological synchronization, where only the edge oscillators synchronize while the bulk ones exhibit chaotic dynamics. We analyze concrete prototypical models to demonstrate the presence of positive Lyapunov exponents and Lyapunov vectors localized along the edge. As a unique characteristic of topology in nonlinear systems, we find that unconventional extra topological boundary modes appear at emerging effective boundaries. Furthermore, our proposal shows promise for spatially controlling synchronization, such as on-demand pattern designing and defect detection. The topological synchronization can ubiquitously appear in topological nonlinear oscillators and thus can provide a guiding principle to realize synchronization in a robust, geometrical, and flexible way.

arXiv comment24 pages, 26 figures

References in corpus (41)

• Topological Insulators
• Topological insulators and superconductors
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Edge states and topological invariants of non-Hermitian systems
• Chimera States for Coupled Oscillators
• Topological phases of non-Hermitian systems
• Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems
• Topological Band Theory for Non-Hermitian Hamiltonians
• Topolectrical circuit realization of topological corner modes
• Topological energy transfer in an optomechanical system with exceptional points
• Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors
• Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators
• Topolectrical circuits
• Chimera States in Mechanical Oscillator Networks
• Nonlinear topological photonics
• Coexistence of Coherence and Incoherence in Nonlocally Coupled Phase Oscillators
• Non-reciprocal phase transitions
• Time Reversal Invariant Topologically Insulating Circuits
• Edge Solitons in Nonlinear Photonic Topological Insulators
• Topological properties of linear circuit lattices
• Quantum synchronization of quantum van der Pol oscillators with trapped ions
• Quantum many-body dynamics in optomechanical arrays
• Chiral voltage propagation in a self-calibrated topolectrical Chern circuit
• Characterizing dynamics with covariant Lyapunov vectors
• Nonlinear conduction via solitons in a topological mechanical insulator
• Active topological photonics
• Non-Hermitian boundary and interface states in nonreciprocal higher-order topological metals and electrical circuits
• Observation of Edge Solitons in Photonic Graphene
• Controlling unstable chaos: Stabilizing chimera states by feedback
• Observation of quantum phase synchronization in spin-1 atoms
• Active topolectrical circuits
• Nonlinearity induced topological physics in momentum space and real space
• Covariant Lyapunov vectors
• Exceptional non-Hermitian topological edge mode and its application to active matter
• Controlling Chimera States - The influence of excitable units
• $\mathcal{P}\mathcal{T}$-Symmetric Topological Edge-Gain Effect
• Lyapunov analysis captures the collective dynamics of large chaotic systems
• Non-Abelian braiding of Majorana-like edge states and topological quantum computations in electric circuits
• Critical Switching in Globally Attractive Chimeras
• Dissipative nonequilibrium synchronization of topological edge states via self-oscillation
• Lyapunov Analysis of Chimera States in Globally Coupled Stuart-Landau oscillators

Cited by in corpus (13)

• Topological synchronization of quantum van der Pol oscillators
• Transition from topological to chaos in the nonlinear Su-Schrieffer-Heeger model
• Dirac synchronization is rhythmic and explosive
• Bulk-edge correspondence for nonlinear eigenvalue problems
• Nonlinear Topological Mechanics in Elliptically Geared Isostatic Metamaterials
• Percolation-induced PT symmetry breaking
• Chiral edge soliton in nonlinear Chern systems
• Protected chaos in a topological lattice
• Hermitian and non-Hermitian topology in active matter
• Fractional Thouless pumping of solitons: a unique manifestation of bulk-edge correspondence of nonlinear eigenvalue problems
• Many topological regions on the Bloch sphere of the spin-1/2 double kicked top
• Topological Phases in Non-Hermitian Nonlinear-Eigenvalue Systems
• Nonlinear dynamical topological phases in Cooper-pair box array