Exploring synchronized oscillations in Kuramoto-type oscillators with nonlinear feedback

EXPLORING SYNCHRONIZED OSCILLATIONS IN KURAMOTO-TYPE OSCILLATORS WITH NONLINEAR FEEDBACK EXPLORING SYNCHRONIZED OSCILLATIONS IN KURAMOTO-TYPE OSCILLATORS WITH NONLINEAR FEEDBACK Arpan Dey Under the supervision of: Prof. Sitabhra Sinha Institute of Mathematical Sciences, Chennai June 2025 Acknowledgements I would like to thank my supervisor Prof Sitabhra Sinha for introducing me to this topic, and for his guidance and critical insights. I would also like to extend my gratitude to the entire IMSc community for supporting me in every way possible during my stay here. Ordered complexity is a fortunate product of random processes... How can order arise from randomness? The Sierpinski triangle Can be generated stochastically (chaos game) Rules and outcome are deterministic, but randomness encoded (outcome after rolling a die remains random at every step, but averages out statistically) Can also be generated deterministically Source: Wikipedia Emergent properties (greater than the sum) - knowledge of parts insufficient Weak emergence or strong emergence? Downward causation (6-fold symmetry in snowflakes) - can the whole affect the parts? Why study complex systems? Source: sciencenotes.org Interesting stuff happens when two opposing forces interact! Complex systems: energy vs entropy Music: predictability vs unpredictability Universe: gravity vs expansion Humans: autonomy vs interaction Sierpinski triangle: rules vs randomness Competing effects The study of complex, nonlinear systems Differential equations or difference equations Differential equations → Continuous → Nonlinear coupled oscillators Difference equations → Discrete → Logistic map Non-linear dynamics Population dynamics Self-similarity Logistic Map Source: Wolfram MathWorld Internal frequency Phase (between 0 and 2π) A group of oscillators are synchronized → same frequency → constant phase difference (not constant phase) → phase locking (if the oscillators don’t have a common frequency, they cannot maintain a constant phase difference) Synchronization is observed in cardiac muscle cells that aid in coordinated contraction, firing of neurons, flashing patterns in some species of fireflies etc. Oscillators and Synchronization Source: Wikipedia Visualizing phase trajectories and fixed points For ω<A, if oscillator is beyond the unstable fixed point threshold, it takes the longer path and settles at the stable fixed point. Otherwise, it directly settles at the stable point from the other direction The two fixed points are diametrically opposite only if ω=0 Visualizing phase trajectories and fixed points For ω=A, there is only one fixed point, and all the oscillators approach and settle at this point Visualizing phase trajectories and fixed points For ω>A, there are no fixed points, and the oscillators keep drifting continuously. However, they slow down near the regions where the fixed points would be if ω<A Networks Coupled oscillators → connected network (node → oscillator, edges → coupling) Fully-connected network → Quicker synchronization Loosely-connected network → Slower synchronization (information flow between nodes takes time) Source: Savarimuthu, BTR et al. DOI: 10.1007/978-3-540-79003-7_15 Networks Aᵢⱼ=1 if oscillators i and j are connected, and 0 if not Undirected networks → Aᵢⱼ=Aⱼᵢ Directed networks → Aᵢⱼ≠Aⱼᵢ (better synchronization under controlled conditions like strong and global connectivity, uneven phase pulling → cluster synchronization is more common) Networks Generating networks → specify number of nodes (n) and number of connections between the nodes (k) If the connections are generated randomly (not fully-connected network) → ensure all the nodes are part of the network In both cases, n=5 and k=4 But in the left network, one node is not part of the network, n=4 effectively Lattice Place the oscillators on a lattice Lattice is not in real 3D space The lattice points are fixed → oscillators remain in the points → oscillations described only by the phase The purpose of the lattice is only to define connectivity among the oscillators Lattice Toroid, Flat 2D space → topologically similar Toroid lattice → closed boundary (no edge effects) Flat lattice → open boundaries (edge effects) A torus can be constructed from a flat 2D surface by joining its edges Source: Sophia Potoczak Bragdon, ResearchGate Lattice Hexagonal lattice → often favored in Nature → efficient packing, better contact with neighbors → better synchronized oscillations? Square lattice → 4 nearest neighbors Hexagonal lattice → 6 nearest neighbors (more robust synchronization) Rough analogy: Dilemma (both options are equally appealing → homogeneous/symmetry) We toss a coin → inhomogeneity or asymmetry We settle to a stable configuration faster → we make a choice Can inhomogeneity sometimes cause faster synchronization? Random ω, Random Phases, Loosely-Connected Network - Unwrapped Time Series p ~ 0.1 (network connection probability) Poorer synchronization Identical ω, Random Phases, Fully-Connected Network - Unwrapped Time Series Quick and complete synchronization, as expected (same ω) Identical ω, Random Phases, Loosely-Connected Network - Unwrapped Time Series p ~ 0.1 (network connection probability) Relatively poorer synchronization Identical ω, Random Phases, Fully-Connected Network - Wrapped Time Series Wrapped time series using modulo 2π N = 10 K = 1 A = 0.5 ω = 1 (same ω for all oscillators → faster synchronization) Identical ω, Identical Phases, Fully-Connected Network - Wrapped Time Series Already fully synchronized (identical ω, identical phases) Random ω, Identical Phases, Fully-Connected Network - Wrapped Time Series Random ω → mean = 1, standard deviation = 0.2 Good synchronization maintained, slight divergence from common initial phase due to random ω Over a long period of time... Random ω, Random Phases, Fully-Connected Network - Wrapped Time Series Random initial phases and frequencies → random initial behavior Synchronizes significantly over time (fully connected network) Identical ω, Identical Phases, Loosely-Connected Network - Wrapped Time Series Since identical initial frequencies and phases, already synchronized (connections don’t matter) Random ω, Random Phases, Loosely-Connected Network - Wrapped Time Series Very poor synchronization → random ω and phases, looselyconnected network Identical ω, Random Phases, ω very slightly greater than A, Fully-Connected Network - Unwrapped Time Series ω = 0.51, A = 0.5 When ω is very slightly greater than A, we get extremely slow drift → the oscillators slow down for long periods of time before phase slips The competition between ω and A is crucial! Identical ω, Random Phases, ω<A, Fully-Connected Network - Wrapped Time Series No sustained oscillation as ω<A → all oscillators collapse to the stable fixed point Oscillators that directly converge → originally very near to stable fixed point Oscillators that take the longer path → originally beyond unstable fixed point → phase slip Both positive and negative ω → oscillators sliding clockwise and anticlockwise We will now look at systems where oscillators slide both ways on the phase circle This means ω can take both positive and negative values → mean value of ω should be less than standard deviation (mean could be zero to avoid bias) Random ω (both positive and negative), Random Phases, Fully-Connected Network - Unwrapped Time Series N = 30, K = 1, A = 0.2, Mean ω = 0, S.D. = 0.5 Since the mean is zero and standard deviation is 0.5, we have half of the oscillators moving clockwise and the other half counterclockwise through the phase circle The damping factor A is 0.2, it affects all oscillators in the same way (whether they are moving clockwise or counterclockwise) → it always slows down the oscillators Oscillators can move in both directions, upward and downward (since ω is drawn from a distribution containing both positive and negative values) → in wrapped phase plots, this appears as both positive and negative slopes, corresponding to increasing or decreasing phase with time Random ω (both positive and negative), Random Phases, Fully-Connected Network - Unwrapped Time Series Oscillators drift both upward and downward (in both directions) in the unwrapped time series Over a long time... The three converging phases differ by multiples of 2π → very close points on the phase circle in wrapped time series (not clusters) They all converge to the same or very close phases, maybe arriving after different number of revolutions around the phase circle Same system, same initial conditions and parameter values Almost all oscillators synchronize and settle to a fixed point, but oscillators with ω sufficiently large to overcome damping can drift indefinitely Same system, same initial conditions and parameter values Due to the inherent randomness in the system, oscillators with a large magnitude of ω can oscillate in either direction Cluster synchronization How can we get the oscillators to form stable clusters? By clusters, we mean subgroups of oscillators that either converge to the same (or nearby) fixed point(s) or remain phaselocked while oscillating collectively Cluster synchronization Bimodal ω distribution → ω drawn randomly but equiprobably from two normal distributions with mean +0.9 and -0.2 (we do not choose symmetric means, for example, +0.9 and -0.9), standard deviation = 0.05 Damping constant A drawn from a normal distribution with mean +0.7, standard deviation = 0.02 100 oscillators, K = 1, random initial phases, randomly-connected network with connection probability 0.1 Random ω, Random Phases, Toroidal Lattice - Wrapped Time Series 20X20 square lattice → periodic boundary conditions → 4 nearest neighbors Good overall synchronization (except for one oscillator, the red one) Random ω, Random Phases, Flat Square Lattice - Wrapped Time Series 20X20 flat square lattice → open boundary conditions → 4 nearest neighbors Overall slightly poorer synchronization Random ω, Random Phases, Hexagonal Lattice - Wrapped Time Series Hexagonal lattice → open boundary conditions → 6 nearest neighbors Much better synchronization (over the same period of time) Applications Modeling real-world systems with synchronization and feedback Better understanding of circadian rhythms Studying contraction dynamics of human heart and cardiac arrhythmias “The beauty of physics lies in the extent to which seemingly complex and unrelated phenomena can be explained and correlated through a high level of abstraction by a set of laws which are amazing in their simplicity.” - Melvin Schwartz “ T h e bea uty of physi c s li e s in th e e xt e nt to whi c h s ee mingly c ompl e x a n d unr e l a t ed ph e nom e n a ca n be e xpl a in ed a n d c orr e l a t ed through a high l e v e l of ab str ac tion b y a s e t of l a ws whi c h a r e a m a zing in th e ir simpli c ity .” - M e lvin S c hw a rtz THANK YOU THANK YOU