Cherry Flows Research Articles

A Chromatic Enzymatic Time-Temperature Integrator Device Based on the Degradation of Phenolic Compounds for the Real-Time Prediction of the Quality and Shelf Life of Cherries

The particular characteristics of cherries, such as color, firmness, and palate increase their demand, as does, among other things, their antioxidant properties that benefit human health. However, their high perishability leads to a reduced shelf life and consequently generates undesirable changes in the cherry flow chain. To ensure food quality and safety and prevent food waste, a smart device prototype is proposed. The concepts related to the formulation and design of the enzymatic-type chromatic time-temperature integrator (TTI) device used to monitor the real-time quality of cherries are described. The kinetic parameters for thermal inactivation of cultivar Santina cherries were determined based on the degradation of phenolic compounds that are substrates of the polyphenol oxidase enzyme, whose hydroxylation reaction of a monophenol to o-diphenol leads to the oxidation in o-quinone. The proposed device concept aims to help retailers and consumers decide upon selling and buying according to the remaining shelf life, thus promoting sustainability related to food processes.

Hyperbolicity of renormalization for dissipative gap mappings

Abstract A gap mapping is a discontinuous interval mapping with two strictly increasing branches that have a gap between their ranges. They are one-dimensional dynamical systems, which arise in the study of certain higher dimensional flows, for example the Lorenz flow and the Cherry flow. In this paper, we prove hyperbolicity of renormalization acting on $C^3$ dissipative gap mappings, and show that the topological conjugacy classes of infinitely renormalizable gap mappings are $C^1$ manifolds.

Creation of discontinuities in circle maps.

Circle maps frequently arise in mathematical models of physical or biological systems. Motivated by Cherry flows and ‘threshold’ systems such as integrate and fire neuronal models, models of cardiac arrhythmias, and models of sleep/wake regulation, we consider how structural transitions in circle maps occur. In particular, we describe how maps evolve near the creation of a discontinuity. We show that the natural way to create discontinuities in the maps associated with both threshold systems and Cherry flows results in a singularity in the derivative of the map as the discontinuity is approached from either one or both sides. For the threshold systems, the associated maps have square root singularities and we analyse the generic properties of such maps with gaps, showing how border collisions and saddle-node bifurcations are interspersed. This highlights how the Arnold tongue picture for tongues bordered by saddle-node bifurcations is amended once gaps are present. We also show that a loss of injectivity naturally results in the creation of multiple gaps giving rise to a novel codimension two bifurcation.

Cherry flows with non-trivial attractors

We provide an example of a Cherry flow (i.e. a C-infinity flow on the 2-dimensional torus with a sink and a saddle) having a quasi-minimal set which is an attractor. The first return map for such a ...

Cherry flow: physical measures and perturbation theory

In this article we consider Cherry flows on the torus which have two singularities, a source and a saddle, and no periodic orbits. We show that every Cherry flow admits a unique physical measure, whose basin has full volume. This proves a conjecture given by Saghin and Vargas [Invariant measures for Cherry flows.Comm. Math. Phys.317(1) (2013), 55–67]. We also show that the perturbation of Cherry flows depends on the divergence at the saddle: when the divergence is negative, this flow admits a neighborhood, such that any flow in this neighborhood belongs to one of the following three cases: it has a saddle connection; it is a Cherry flow; it is a Morse–Smale flow whose non-wandering set consists of two singularities and one periodic sink. In contrast, when the divergence is non-negative, this flow can be approximated by a non-hyperbolic flow with an arbitrarily large number of periodic sinks.

On physical measures for Cherry flows

Studies of the physical measures for Cherry flows were initiated by R. Saghin and E. Vargas in Invariant measures for Cherry flows. While the non-positive divergence case was resolved, the positive divergence one still lacked the complete description. Some conjectures were put forward. In this paper we contribute in this direction. Namely, under mild technical assumptions we solve conjectures stated by R. Saghin and E. Vargas by providing a description of the physical measures for Cherry flows in the positive divergence case.

Unbounded regime for circle maps with a flat interval

We study $\mathcal{C}^2$ weakly order preserving circle maps with a flat interval. In particular we are interested in the geometry of the mapping near to the singularities at the boundary of the flat interval. Without any assumption on the rotation number we show that the geometry is degenerate when the degree of the singularities is less than or equal to two and becomes bounded when the degree goes to three. As an example of application, the result is applied to study Cherry flows.

Invariant Measures for Cherry Flows

We investigate the invariant probability measures for Cherry flows, i.e. flows on the two-torus which have a saddle, a source, and no other fixed points, closed orbits or homoclinic orbits. In the case when the saddle is dissipative or conservative we show that the only invariant probability measures are the Dirac measures at the two fixed points, and the Dirac measure at the saddle is the physical measure. In the other case we prove that there exists also an invariant probability measure supported on the quasi-minimal set, we discuss some situations when this other invariant measure is the physical measure, and conjecture that this is always the case. The main techniques used are the study of the integrability of the return time with respect to the invariant measure of the return map to a closed transversal to the flow, and the study of the close returns near the saddle.

The Lamination of Infinitely Renormalizable Dissipative Gap Maps: Analyticity, Holonomies and Conjugacies

Motivated by return maps near saddles for three-dimensional flows and also by return maps in the torus associated to Cherry flows, we study gap maps with derivative positive and smaller than one outside the discontinuity point. We prove that the lamination of infinitely renormalizable maps (or else maps with irrational rotation numbers) has analytic leaves in a natural subset of a Banach space of analytic maps of this kind. With maps having Holder continuous derivative and derivative bounded away from zero, we also prove Holder continuity of holonomies of the lamination and also of conjugacies between maps having the same combinatorics.

CHAOTIC ATTRACTOR IN THE KURAMOTO MODEL

The Kuramoto model of globally coupled phase oscillators is an essentially nonlinear dynamical system with a rich dynamics including synchronization and chaos. We study the Kuramoto model from the standpoint of bifurcation and chaos theory of low-dimensional dynamical systems. We find a chaotic attractor in the four-dimensional Kuramoto model and study its origin. The torus destruction scenario is one of the major mechanisms by which chaos arises. L. P. Shilnikov has made decisive contributions to its discovery. We show also that in the Kuramoto model the transition to chaos is in accordance with the torus destruction scenario. We present the general bifurcation diagram containing phase chaos, Cherry flow as well as periodic and quasiperiodic dynamics.

Mechanism of desynchronization in the finite-dimensional Kuramoto model.

We study how a decrease of the coupling strength causes a desynchronization in the Kuramoto model of N globally coupled phase oscillators. We show that, if the natural frequencies are distributed uniformly or close to that, the synchronized state can robustly split into any number of phase clusters with different average frequencies, even culminating in complete desynchronization. In the simplest case of N=3 phase oscillators, the course of the splitting is controlled by a Cherry flow. The general N-dimensional desynchronization mechanism is numerically illustrated for N=5.

Fractal Fourier spectra of Cherry flows

Flows on the 2-torus with points of equilibrium are considered. Numerical observations suggest that, for the case of trajectories with irrational rotation numbers, power spectra are singular continuous. Dynamics is adequately represented by a symbolic sequence built from four symbols with the help of a substitution rule. This rule effectively takes into account the slowing down of the motion during passages near the saddle point and the superexponential scaling of returns onto a secant line.

On the cyclicity of recurrent flows on surfaces

We search for a bound for the number of isolated closed orbits of small perturbations of vector fields on surfaces. We prove that such a bound exists for Cherry flows on the torus.

Metric properties of Cherry flows

In a Cherry flow without sources, with positive divergence on the saddles, almost all points goes to a Sink.

On Cherry flows

AbstractThe purpose of this research is to describe all smooth vector fields on the torus T2 with a finite number of singularities, no periodic orbits and no saddleconnections. In this paper we are able to complete the description within the class of vector fields which are area contracting near all singularities. In particular we give a large class of analytic vector fields on the torus T2 which have non-trivial recurrence and also sinks.

Polynomial vector fields on the torus

In this paper it is shown that the structurally stable polynomial vector fields on the torusT2, with singularities, are open and dense in the set of such vector fields. Many kinds of distinct dynamical phenomena are also presented by a list of examples including the Cherry flows. The above result works for analytical vector fields onT2 with the same proof.

Topological structure of Cherry flows on the torus

Topological structure of Cherry flows on the torus