Open Set Of Parameter Values Research Articles

Bright-Dark and Multi Solitons Solutions of (3 + 1)-Dimensional Cubic-Quintic Complex Ginzburg-Landau Dynamical Equation with Applications and Stability.

In this paper, bright-dark, multi solitons, and other solutions of a (3 + 1)-dimensional cubic-quintic complex Ginzburg–Landau (CQCGL) dynamical equation are constructed via employing three proposed mathematical techniques. The propagation of ultrashort optical solitons in optical fiber is modeled by this equation. The complex Ginzburg–Landau equation with broken phase symmetry has strict positive space–time entropy for an open set of parameter values. The exact wave results in the forms of dark-bright solitons, breather-type solitons, multi solitons interaction, kink and anti-kink waves, solitary waves, periodic and trigonometric function solutions are achieved. These exact solutions have key applications in engineering and applied physics. The wave solutions that are constructed from existing techniques and novel structures of solitons can be obtained by giving the special values to parameters involved in these methods. The stability of this model is examined by employing the modulation instability analysis which confirms that the model is stable. The movements of some results are depicted graphically, which are constructive to researchers for understanding the complex phenomena of this model.

Computer assisted proof of the existence of the Lorenz attractor in the Shimizu–Morioka system

We prove that the Shimizu–Morioka system has a Lorenz attractor for an open set of parameter values. For the proof we employ a criterion proposed by Shilnikov, which allows to conclude the existence of the attractor by examination of the behaviour of only one orbit. The needed properties of the orbit are established by using computer assisted numerics. Our result is also applied to the study of local bifurcations of triply degenerate periodic points of three-dimensional maps. It provides a formal proof of the birth of discrete Lorenz attractors at various global bifurcations.

Coupling of the cell cycle and metabolism in yeast cell-cycle-related oscillations via resource criticality and checkpoint gating

We investigate the possibility that slow metabolic, cell-cycle-related oscillations in yeast and associated temporal clustering of cells within the cell cycle could be due to an interplay between near-critical metabolism and cell cycle checkpoints. We construct a dynamical model of the cell cycles of a large culture of cells that incorporates checkpoint gating and metabolic mode switching that are triggered by resource thresholds. We investigate the model analytically and prove that there exist open sets of parameter values for which the model possesses stable periodic solutions that exhibit metabolic oscillations with cell cycle clustering. Simulations of the model give evidence that such solutions exist for large sets of parameter values. This demonstrates that checkpoint gating coupled with critical resources can be a robust mechanism for producing the phenomena observed in experiments.

Analytic proof of the existence of the Lorenz attractor in the extended Lorenz model

We give an analytic (free of computer assistance) proof of the existence of a classical Lorenz attractor for an open set of parameter values of the Lorenz model in the form of Yudovich–Morioka–Shimizu. The proof is based on detection of a homoclinic butterfly with a zero saddle value and rigorous verification of one of the Shilnikov criteria for the birth of the Lorenz attractor; we also supply a proof for this criterion. The results are applied in order to give an analytic proof for the existence of a robust, pseudohyperbolic strange attractor (the so-called discrete Lorenz attractor) for an open set of parameter values in a 4-parameter family of 3D Henon-like diffeomorphisms.

Analytical proof of space-time chaos in Ginzburg-Landau equations

We prove that the attractor of the 1D quintic complex Ginzburg-Landau equation with a broken phase symmetry has strictly positive space-time entropy for an open set of parameter values. The result is obtained by studying chaotic oscillations in grids of weakly interacting solitons in a class of Ginzburg-Landau type equations. We provide an analytic proof for the existence of two-soliton configurations with chaotic temporal behavior, and construct solutions which are closed to a grid of such chaotic soliton pairs, with every pair in the grid well spatially separated from the neighboring ones for all time. The temporal evolution of the well-separated multi-soliton structures is described by a weakly coupled lattice dynamical system (LDS) for the coordinates and phases of the solitons. We develop a version of normal hyperbolicity theory for the weakly coupled LDSs with continuous time and establish for them the existence of space-time chaotic patterns similar to the Sinai-Bunimovich chaos in discrete-time LDSs. While the LDS part of the theory may be of independent interest, the main difficulty addressed in the paper concerns with lifting the space-time chaotic solutions of the LDS back to the initial PDE. The equations we consider here are space-time autonomous, i.e. we impose no spatial or temporal modulation which could prevent the individual solitons in the grid from drifting towards each other and destroying the well-separated grid structure in a finite time. We however manage to show that the set of space-time chaotic solutions for which the random soliton drift is arrested is large enough, so the corresponding space-time entropy is strictly positive.

On “Acquisition of Information in Financial Markets”

In their paper Information Acquisition in Financial Markets (this journal, 2000), Barlevy and Veronesi present a model of a one-period financial market, and claim that for an open set of parameter values, the value of information increases with the mass of informed agents. That claim is shown here to be false. The property of strategic substitution is robust in their model.

Heteroclinic cycles between unstable attractors

We consider networks of pulse coupled linear oscillators with non-zero delay where the coupling between the oscillators is given by the Mirollo–Strogatz function. We prove the existence of heteroclinic cycles between unstable attractors for a network of four oscillators and for an open set of parameter values.

Dynamics of two-dimensional Blaschke products

AbstractIn this paper we study the dynamics on $\mathbb {T}^2$ and $\mathbb {C}^2$ of a two-dimensional Blaschke product. We prove that in the case when the Blaschke product is a diffeomorphism of $\mathbb {T}^2$ with all periodic points hyperbolic then the dynamics is hyperbolic. If a two-dimensional Blaschke product diffeomorphism of $\mathbb {T}^2$ is embedded in a two-dimensional family given by composition with translations of $\mathbb {T}^2$, then we show that there is a non-empty open set of parameter values for which the dynamics is Anosov or has an expanding attractor with a unique SRB measure.

Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells

A mathematical model that describes HIV infection of CD4+ T cells is analyzed. Global dynamics of the model is rigorously established. We prove that, if the basic reproduction number R0⩽1, the HIV infection is cleared from the T-cell population; if R0>1, the HIV infection persists. For an open set of parameter values, the chronic-infection equilibrium P∗ can be unstable and periodic solutions may exist. We establish parameter regions for which P∗ is globally stable.

Extreme Sensitivity to Detuning for Globally Coupled Phase Oscillators

We discuss the sensitivity of a population of coupled oscillators to differences in their natural frequencies, i.e., to detuning. We argue that for three or more oscillators, one can get great sensitivity even if the coupling is strong. For N globally coupled phase oscillators we find there can be bifurcation to extreme sensitivity, where frequency locking can be destroyed by arbitrarily small detuning. This extreme sensitivity is absent for N = 2, appears at isolated parameter values for N = 3 and N = 4, and can appear robustly for open sets of parameter values for N > or = 5 oscillators.

Unstable attractors: existence and robustness in networks of oscillators with delayed pulse coupling

We consider unstable attractors: Milnor attractors A such that, for some neighbourhood U of A, almost all initial conditions leave U. Previous research strongly suggests that unstable attractors exist and even occur robustly (i.e. for open sets of parameter values) in a system modelling biological phenomena, namely in globally coupled oscillators with delayed pulse interactions.In the first part of this paper we give a rigorous definition of unstable attractors for general dynamical systems. We classify unstable attractors into two types, depending on whether or not there is a neighbourhood of the attractor that intersects the basin in a set of positive measure. We give examples of both types of unstable attractor; these examples have non-invertible dynamics that collapse certain open sets onto stable manifolds of saddle orbits.In the second part we give the first rigorous demonstration of existence and robust occurrence of unstable attractors in a network of oscillators with delayed pulse coupling. Although such systems are technically hybrid systems of delay differential equations with discontinuous ‘firing’ events, we show that their dynamics reduces to a finite dimensional hybrid system after a finite time and, hence, we can discuss Milnor attractors for this reduced finite dimensional system. We prove that for an open set of phase resetting functions there are saddle periodic orbits that are unstable attractors.

Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor

First, the free frictionless dynamics of the triple linkage of Thurston and Weeks is proved to be Anosov on each positive energy level for a non-empty open set of parameter values. Second, in the presence of weak friction and any smooth potential, an open set of physically implementable feedback control laws are found for which the system possesses a uniformly hyperbolic attracting submanifold.

An equivariant, inclination-flip, heteroclinic bifurcation

We examine a heteroclinic bifurcation occurring in families of equivariant vector fields. Within these families, the flows contain structurally stable heteroclinic cycles. The flow can twist around the cycle to produce what is the equivalent of an `inclination-flip' homoclinic orbit. An unfolding of this bifurcation in a generic one-parameter equivariant family shows that Smale horseshoes are embedded in the flow for an open set of parameter values.

Isolating neighborhoods and chaos

We show that the map part of the discrete Conley index carries information which can be used to detect the existence of connections in the repeller-attractor decomposition of an isolated invariant set of a homeomorphism. We use this information to provide a characterization of invariant sets which admit a semi-conjugacy onto the space of sequences on K symbols with dynamics given by a subshift. These ideas are applied to the Henon map to prove the existence of chaotic dynamics on an open set of parameter values.

Secondary bifurcations in plane Poiseuille flow with three interacting modes

A degenerate secondary bifurcation point in a plane channel flow is investigated. In the absence of any symmetry assumption between the two channel halves the analysis of the corresponding three-dimensional normal form reveals the presence of mixed-mode equilibria with more complicated spatial structures. It is further shown that the dynamics of the normal form contains trajectories between unstable equilibria for open sets of parameter values.

Apparently chaotic amplitude maps of nonchaotic flows.

Amplitude maps that are constructed from a scalar time-series measurement along a trajectory of a dynamical system can be chaotic. This occurs if the underlying flow that generated the time series is chaotic. We explain how amplitude maps of toral flows might give the impression that a transition to chaos has occurred, in that the resulting amplitude map appears complicated and has nontrivial sensitivity to initial conditions. A two-parameter family of aperiodic toral flows is studied and apparently chaotic amplitude maps are found for an open set of parameter values. Such maps are shown to differ from genuinely chaotic maps in that the rate of divergence of orbits is linear, rather than exponential. Three methods of distinguishing them are provided, one analytical and two suitable for use with experimental data.

Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry

We analyze unfoldings of a codimension two, steady-state/steady-state modal interaction possessing O(2) symmetry. At the degenerate bifurcation point there are two zero eigenvalues, each of multiplicity two. The spatial wavenumbers of the critical modes k i are assumed to satisfy k 2 = 2 k 1. We base our analysis on a detailed study of the third order truncation of the resulting equivariant normal form, which is a four-dimensional vector field. We find that heteroclinic cycles and modulated travelling waves exist for open sets of parameter values near the codimension two bifurcation point. We provide conditions on parameters which guarantee existence and uniqueness of such solutions and we investigate their stability types. We argue that such motions will be prevalent in continuum systems having the symmetry of translation and reflection with respect to one (or more) spatial directions.

The Existence of One Dimensional Steady Detonation Waves in a Simple Model Problem

We study a system of three nonlinear ordinary differential equations arising from one dimensional steady combustion problems. The four singular points in the phase space correspond to equilibrium solutions before and after burning has taken place, and the desired detonation wave is a trajectory connecting two of these points (the cold and hot boundary points). In this paper we show that the problem of finding steady detonations is equivalent to proving the existence and uniqueness of a transversal heteroclinic orbit. The presence of a large parameter, the activation energy, permits us to show analytically that such an orbit exists for an open set of parameter values. In doing so, we are able to give a geometric interpretation of the cold boundary difficulty. We conclude with some numerical results for lower activation energies.