Poincaré Maps and Aperiodic Oscillations in Leukemic Cell Proliferation Reveal Chaotic Dynamics

Biological systems are dynamic systems featuring two very common characteristics; Initial conditions and progression over time. Conceptualizing this on tumour models it can lead to important conclusions about disease progression, as well as the disease’s “starting point”. In the present study we tried to answer two questions: (a) which are the evolving properties of proliferating tumour cells that started from different initial conditions and (b) we have attempted to prove that cell proliferation follows chaotic orbits and it can be described by the use of Poincaré maps. As a model we have used the acute lymphoblastic leukemia cell line CCRF-CEM. Measurements of cell population were taken at certain time points every 24 h or 48 h. In addition to the population measurements flow cytometry studies have been conducted in order to examine the apoptotic and necrotic rate of the system and also the DNA content of the cells as they progress through. The cells exhibited a proliferation rate of nonlinear nature with aperiodic oscillatory behavior. In addition to that, the (positive) Lyapunov indices and the Poincaré representations in phase-space that we performed confirmed the presence of chaotic orbits. Several studies have dealt with the complex dynamic behaviour of animal populations, but few with cellular systems. This type of approach could prove useful towards the understanding of leukemia dynamics, with particular interest in the understanding of leukemia onset and progression.

Orbit Classification and Sensitivity Analysis in Dynamical Systems Using Surrogate Models

Dynamics of many classical physics systems are described in terms of Hamilton’s equations. Commonly, initial conditions are only imperfectly known. The associated volume in phase space is preserved over time due to the symplecticity of the Hamiltonian flow. Here we study the propagation of uncertain initial conditions through dynamical systems using symplectic surrogate models of Hamiltonian flow maps. This allows fast sensitivity analysis with respect to the distribution of initial conditions and an estimation of local Lyapunov exponents (LLE) that give insight into local predictability of a dynamical system. In Hamiltonian systems, LLEs permit a distinction between regular and chaotic orbits. Combined with Bayesian methods we provide a statistical analysis of local stability and sensitivity in phase space for Hamiltonian systems. The intended application is the early classification of regular and chaotic orbits of fusion alpha particles in stellarator reactors. The degree of stochastization during a given time period is used as an estimate for the probability that orbits of a specific region in phase space are lost at the plasma boundary. Thus, the approach offers a promising way to accelerate the computation of fusion alpha particle losses.

New forms of structure in ecosystems revealed with the Kuramoto model

Ecological systems, as is often noted, are complex. Equally notable is the generalization that complex systems tend to be oscillatory, whether Huygens' simple patterns of pendulum entrainment or the twisted chaotic orbits of Lorenz’ convection rolls. The analytics of oscillators may thus provide insight into the structure of ecological systems. One of the most popular analytical tools for such study is the Kuramoto model of coupled oscillators. We apply this model as a stylized vision of the dynamics of a well-studied system of pests and their enemies, to ask whether its actual natural history is reflected in the dynamics of the qualitatively instantiated Kuramoto model. Emerging from the model is a series of synchrony groups generally corresponding to subnetworks of the natural system, with an overlying chimeric structure, depending on the strength of the inter-oscillator coupling. We conclude that the Kuramoto model presents a novel window through which interesting questions about the structure of ecological systems may emerge.

Chaotic Cryptosystem for Selective Encryption of Faces in Photographs

In this article, a safe communication system is proposed that implements one or more portable devices denominated SBC (single-board computers), with which photographs are taken and that later utilizes the OpenCV Library for the detection and identification of the faces that appear in them. Subsequently, it consults the information in a stored database, whether locally in SBC or in a remote server, to verify that the faces should be coded, and it encrypts these, implementing a new cryptosystem that executes mathematical models to generate chaotic orbits, one of which is used for application on two occasions the technique of diffusion with the purpose of carrying out a small change in one of the pixels of the image, generating very different cryptograms. In addition, in order to make a safer system, it implements other chaotic orbits during the technique of confusion. With the purpose of verifying the robustness of the encryption algorithm, a statistical analysis is performed employing histograms, horizontal, vertical, and diagonal correlation diagrams, entropy, number of pixel change rate (NPCR), unified average change intensity (UACI), sensitivity of the key, encryption quality analysis, and the avalanche effect. The cryptosystem is very robust in that it generates highly disordered cryptograms, supports differential attacks, and in addition is highly sensitive to changes in the pixels as well as in the encrypted keys.

The Lyapunov exponents and the neighbourhood of periodic orbits

ABSTRACT We show that the Lyapunov exponents of a periodic orbit can be easily obtained from the eigenvalues of the monodromy matrix. It turns out that the Lyapunov exponents of simply stable periodic orbits are all zero, simply unstable periodic orbits have only one positive Lyapunov exponent, doubly unstable periodic orbits have two different positive Lyapunov exponents, and the two positive Lyapunov exponents of complex unstable periodic orbits are equal. We present a numerical example for periodic orbits in a realistic galactic potential. Moreover, the centre manifold theorem allowed us to show that stable, simply unstable, and doubly unstable periodic orbits are the mothers of families of, respectively, regular, partially, and fully chaotic orbits in their neighbourhood.

Computational Bifurcations Occurring on Red Fixed Components in the λ-Parameter Plane for a Family of Optimal Fourth-Order Multiple-Root Finders under the Möbius Conjugacy Map

Optimal fourth-order multiple-root finders with parameter λ were conjugated via the Möbius map applied to a simple polynomial function. The long-term dynamics of these conjugated maps in the λ -parameter plane was analyzed to discover some properties of periodic, bounded and chaotic orbits. The λ -parameters for periodic orbits in the parameter plane are painted in different colors depending on their periods, and the bounded or chaotic ones are colored black to illustrate λ -dependent connected components. When a red fixed component in the parameter plane branches into a q-periodic component, we encounter geometric bifurcation phenomena whose characteristics determine the desired boundary equation and bifurcation point. Computational results along with illustrated components support the bifurcation phenomena underlying this paper.

Controlling the Chaos of Logistic Map using Switching Strategy

In our very recent work (2019), we extended the stability performance of logistic map up to a higher value of r using SP orbit. In this article, we further extend this range of stability by adopting switching strategy (Parrondo’s Paradox) of controlling the chaos of dynamical systems. We observe that even the earlier chaotic orbits of four step feedback procedure can be converted into periodic orbits. Our approach can be used to solve a wider circle of engineering problems.

Effects of chaos on the detectability of stellar streams

ABSTRACT Observations show that stellar streams originating in satellite dwarf galaxies are frequent in the Universe. While such events are predicted by theory, it is not clear how many of the streams that are generated are washed out afterwards to the point in which it is impossible to detect them. Here, we study how these diffusion times are affected by the fact that typical gravitational potentials of the host galaxies can sustain chaotic orbits. We do this by comparing the behaviour of simulated stellar streams that reside in chaotic or non-chaotic regions of the phase space. We find that chaos does reduce the time interval in which streams can be detected. By analysing detectability criteria in configuration and velocity space, we find that the impact of these results on the observations depends on the quality of both the data and the underlying stellar halo model. For all the stellar streams, we obtain a similar upper limit to the detectable mass.