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Topological properties of physical systems can lead to robust behaviors that are insensitive to microscopic details. Such topologically robust phenomena are not limited to static systems but can also appear in driven quantum systems. In…
Higher-order topological insulators (HOTIs) are systems with topologically protected in-gap boundary states localized at their $(d-n)$-dimensional boundaries, with $d$ the system dimension and $n$ the order of the topology. This work…
Often, experiments, observations or simulations generate large numbers of snapshots of the configurations of complex many-particle systems. It is important to find methods of extracting useful information from these ensembles of snapshots…
In the past decade, synchronization on complex networks has attracted increasing attentions from various research disciplines. Most previous works, however, focus only on the dynamic behaviors of synchronization process in the stable…
Many biological and physical systems exhibit behaviour at multiple spatial, temporal or population scales. Multiscale processes provide challenges when they are to be simulated using numerical techniques. While coarser methods such as…
The evolution of a large class of biological, physical and engineering systems can be studied through both dynamical systems theory and Hamiltonian mechanics. The former theory, in particular its specialization to study systems with…
We introduce two numerical conjugacy invariants for dynamical systems -- the complexity and weak complexity indices -- which are well-suited for the study of "completely integrable" Hamiltonian systems. These invariants can be seen as "slow…
The term "hybrid system" refers to a continuous time dynamical system that undergoes Markovian perturbations at discrete time intervals. In this paper, we find that under the right formulation, a hybrid system can be treated as a dynamical…
We show that a two-dimensional system of flocking microswimmers interacting hydrodynamically can be expressed using a Hamiltonian formalism. The Hamiltonian depends strictly on the angles between the particles and their swimming…
To study discrete dynamical systems of different types --- deterministic, statistical and quantum --- we develop various approaches. We introduce the concept of a system of discrete relations on an abstract simplicial complex and develop…
We consider discrete dynamical systems and lattice models in statistical mechanics from the point of view of their symmetry groups. We describe a C program for symmetry analysis of discrete systems. Among other features, the program…
We consider an ensemble of coupled oscillators whose individual states, in addition to the phase, are characterized by an internal variable with autonomous evolution. The time scale of this evolution is different for each oscillator, so…
The phase-space of a simple synchronization model is thoroughly investigated. The model considers two-mode stochastic oscillators, coupled through a pulse-like interaction controlled by simple optimization rules. A complex phase space is…
Two global symmetries are holo-equivalent if their algebras of local symmetric operators are isomorphic. Holo-equivalent classes of global symmetries are classified by gappable-boundary topological orders (TO) in one higher dimension…
Piecewise-linear maps describe dynamical phenomena that switch between distinct states and readily generate complex bifurcation structures due to their strong nonlinearity. We show that two-dimensional continuous piecewise-linear maps near…
Invariant manifolds are the skeleton of the chaotic dynamics in Hamiltonian systems. In Celestial Mechanics, for instance, these geometrical structures are applied to a multitude of physical and practical problems, such as to the…
Symmetries impose structure on the Hilbert space of a quantum mechanical model. The mathematical units of this structure are the irreducible representations of symmetry groups and I consider how they function as conceptual units of…
The analysis performed as well as extensive numerical simulations have revealed the possibility of the generation of homoclinic orbits as a result of homoclinic bifurcation in a porous pellet. A method has been proposed for the development…
We consider a 2 d.o.f. Hamiltonian system with one degree of freedom corresponding to fast motion and the other corresponding to slow motion. The ratio of the time derivatives of slow and fast variables is of order $0<\eps \ll 1$. At frozen…
We proposed a data-driven approach to dissect multivariate time series in order to discover multiple phases underlying dynamics of complex systems. This computing approach is developed as a multiple-dimension version of Hierarchical Factor…