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Living systems exhibit complex yet organized behavior on multiple spatiotemporal scales. To investigate the nature of multiscale coordination in living systems, one needs a meaningful and systematic way to quantify the complex dynamics, a…

Coupled map lattices have been widely used as models in several fields of physics, such as chaotic strings, turbulence, and phase transitions, as well as in other disciplines, such as biology (ecology, evolution) and information processing.…

We introduce a systematic mathematical language for describing fixed point models and apply it to the study to topological phases of matter. The framework is reminiscent of state-sum models and lattice topological quantum field theories,…

We introduce a method to successively locate equilibria (steady states) of dynamical systems on Riemannian manifolds. The manifolds need not be characterized by an a priori known atlas or by the zeros of a smooth map. Instead, they can be…

This paper exposes the language of geometric contexts and elementary schemes, which is a functorial formalism to study categories of geometric objects such as schemes, topological manifolds, differential manifolds, analytic manifolds, etc.…

Quasi-periodic structures of quasicrystals yield novel effects in diverse systems. However, there is little investigation on employing quasi-periodic structures in the morphology control. Here, we show the use of quasi-periodic surface…

In this article, we review the analytical and numerical approaches for computing the phase space structures in two degrees-of-freedom Hamiltonian systems that arise in chemical reactions. In particular, these phase space structures are the…

This work deals with planar dynamical systems with and without noise. In the first part, we seek to gain a refined understanding of such systems by studying their differential-geometric transformation properties under an arbitrary smooth…

We classify the topological types of surfaces in the 3-dimensional unit sphere that contain both a great and a small circle through each point. In particular, these surfaces are homeomorphic to one of five normal forms and are either the…

Nonexpansive mappings play a central role in modern optimization and monotone operator theory because their fixed points can describe solutions to optimization or critical point problems. It is known that when the mappings are sufficiently…

Topological invariants have proved useful for analyzing emergent function as they characterize a property of the entire system, and are insensitive to local details, disorder, and noise. They support boundary states, which reduce the system…

In this paper we study topological entropy and recurrence properties of non-autonomous dynamical system generated by a family of continuous self maps on a compact space X. Specially, we introduce the pseudo-entropy and…

In this paper we give an elementary treatment of the dynamics of skew tent maps. We divide the two-parameter space into six regions. Two of these regions are further subdivided into infinitely many regions. All of the regions are given…

We consider metrizable ergodic topological dynamical systems over locally compact, $\sigma$-compact abelian groups. We study pure point spectrum via suitable notions of almost periodicity for the points of the dynamical system. More…

We get new results (and rederive some know ones) on smooth surfaces in $\mathbb{R}^n$ by unifying several view points into a coherent general view. Namely, we show and use new relations of the evolute (caustic) with the curvature ellipse,…

We determine which connected surfaces can be partitioned into topological circles. There are exactly seven such surfaces up to homeomorphism: those of finite type, of Euler characteristic zero, and with compact boundary components. As a…

In four-dimensional symplectic maps complex instability of periodic orbits is possible, which cannot occur in the two-dimensional case. We investigate the transition from stable to complex unstable dynamics of a fixed point under parameter…

The paper is devoted to the questions connected with the investigation of the S.P. Novikov problem of the description of the geometry of level lines of quasiperiodic functions on a plane with different numbers of quasiperiods. We consider…

We study in this paper the behavior of a periodically driven nonlinear mechanical system. Bifurcation diagrams are found which locate regions of quasiperiodic, periodic and chaotic behavior within the parameter space of the system. We also…

In the directed setting, the spaces of directed paths between fixed initial and terminal points are the defining feature for distinguishing different directed spaces. The simplest case is when the space of directed paths is homotopy…