Mode analysis of the tree-like networks of nonlinear oscillators презентация

OUTLINE
Introduction
Tree-like networks in nature and mathematics
Tree-like networks in complex artificial systems
Theoretical analysis
Problem of normal modes
Quasi-Hamiltonian approach and truncated equations
Application of the Quasi-Hamiltonian approach
Complex network analysis. Synchronization
Topological properties of the complex network organization
Conclusions

TREE-LIKE NETWORKS IN NATURE
R. Rammal, et al. 1986.
T. Nakayama, et al. 1994
Tree roots
Egypt map
Vascular system

TREE-LIKE NETWORKS IN MATHEMATICS
rational numbers Q
Real numbers R
Completion using absolute value
Standard metric
Tree-like metric – ultrametric
p-adic metric
Ultrametric space M is a set of points with a distance function d:
1.d(x,y)≥0
2. d(x,y)=0 if x=y
3. d(x,y)=d(y,x)
4. d(x,z)≤max(d(x,y),d(y,z)).

ARTIFICIAL NEURAL NETWORKS
B. Benjamin, 2014
Silicon-based
neural networks
Spintronic-based
neural networks
J. Grollier, 2016

MOLECULAR MACHINES & PROTEINS
A molecular machine, or nanomachine, refers to any discrete number of molecular components that produce quasi-mechanical movements (output) in response to specific stimuli (input).

.
Synch-mode
COMPLEX NETWORKS
.
Non-synch mode
.
Many modes in a real network with self-oscillators!

MATHEMATICAL MODEL
Network of Landau-Stuart oscillators
Landau, 1944. Stuart, 1960.
Structure of normal modes is unknown
Different types of nonlinearity
?
- cubic nonlinear term
- given normal mode
Kuramoto model

QUASI-HAMILTONIAN APPROACH
complex amplitude of j-th oscillator,
Hamiltonian of all system
perturbation term (~ small parameter)
Equations of motion
General and simple way to write truncated equations (for slowly varying amplitudes and phases)
General structure of Hamiltonian

STRUCTURE OF HAMILTONIAN
adjacency matrix
Coefficients of the Hamiltonian
Normal modes coefficients
Normal modes equations

MODE STRUCTURE: SIMPLE NETWORKS
LINE
RING
GRID

MODE STRUCTURE: 2-ADIC NETWORKS

MODE STRUCTURES: 3-ADIC NETWORKS

MODE STRUCTURE: 4-ADIC NETWORK

MODE STRUCTURE: RANDOMIZATION

MODE STABILITY & SYNCHRONIZATION
Degenerate mode
(identical case)
Nondegenerate duplets
(nonidentical case)
Two frequencies born from degenerate mode, can synchronize if they are close enough, but they lose synchronization when they are separated to some extent.
Non-isochronous of oscillators increase the phase locking bandwidth between two modes in noidentical case
Phase locking area at the parameter plane becomes asymmetric

TOPOLOGICAL PROPERTIES OF COMPLEX NETWORKS

CONCLUSION
The structure of normal modes of tree-like (ultrametric) networks is fractal (“devil’s staircase”).
Increasing of p (number of branches) leads to the increasing of degenerate modes
We propose to apply the quasi-Hamiltonian approach to derive normal modes using priori knowledge of the network topology only.
We find that hierarchical networks are characterized by a smaller number of stable nontrivial modes than randomly organized.
Our analysis gives rise to an approach to specify topological transformations of networks that can enhance synchronization.
Randomization of the coupling frequencies leads to the modes non-degeneracy and difficulties with synchronization.