Hyperbolic Equilibrium Points Research Articles

Notes on geometric aspects of effectively hyperbolic critical points when the characteristic roots are real on one side of time

Notes on geometric aspects of effectively hyperbolic critical points when the characteristic roots are real on one side of time

Some Classical Methods in the Analysis of an Aedes aegypti Model


 
 
 
 The Taylor series approximation is often used to convert non-linear dynamical systems to linear systems, while the Hartman-Großman theorem analyzes the local qualitative behavior of the non-linear system around a hyperbolic equilibrium point. The global stability of an equilibrium point in the Lyapunov sense is based on the principle that if the equilibrium point is disturbed and the flow of the system is dissipative, then the system must be stable. This article applies these methods to an ecological Aedes aegypti model, whose local and global stability are characterized by a population growth threshold. In conclusion, the classical theory of dynamical systems, validated computationally, yields theoretical results in favor of controlling the local population of Aedes aegypti. It becomes usable if the proposed model is reinforced with the estimation of the parameters that describe the relationships between stages (aquatic and aerial) of the mosquito population and the inclusion of vector control strategies to protect people from the viruses transmitted by Aedes aegypti.
 
 
 

The set of hyperbolic equilibria and of invertible zeros on the unit ball is computable

In this note, we construct an algorithm that, on input of a description of a structurally stable planar dynamical flow f defined on the closed unit disk, outputs the exact number of the (hyperbolic) equilibrium points and their locations with arbitrary accuracy. By arbitrary accuracy it is meant that any accuracy required by the input can be achieved. The algorithm can be further extended to a root-finding algorithm that computes the exact number of zeros as well the location of each zero of a continuously differentiable function f defined on the closed unit ball of Rd, provided that the Jacobian of f is invertible at each zero of f; moreover, the computation is uniform in f.

Estabilidad en sistemas cuadráticos del tipo Kolmogorov describiendo interacciones entre dos especies. Una breve revisión

Population dynamics is a relevant topic in Biomathematics, being the study of the long-term behavior of interaction models between species, one of its central problems. A large part of these relationships are described by ordinary differential equations (ODE), having as main objectives the study of the stability of their solutions. In this document we mainly describe the dynamic behavior of the Volterra predation model. In addition, we make a review of some derived predation models and a brief review of the dynamical properties of models describing other interactions between species such as: competition, mutualism, amensalism, and commensalism; also described by nonlinear ODE systems of the second order of Kolmogorov-type. For each of these models, the non-existence of limit cycles can be demonstrated and in most of them, there is a globally stable equilibrium point. In one of them, there are conditions in the parameters for which the only positive equilibrium point is a center, as in the original Lotka-Volterra model. The methodology used is the usual one for the analysis of models with hyperbolic equilibrium points, but it can guide the analysis of other more complicated models.

Dynamical complexity of the teleparallel gravity cosmology

The exploration of teleparallel gravity has been done from a dynamical systems point of view in order to be tested against the cosmological evolution currently observed. So far, the proposed autonomous systems have been restrictive over a constant dynamical variable, which contains information related to the dynamics on the $H_0$ value. It is therefore that in this paper we consider a generalization of the dynamical system by imposing a nonconstant degree of freedom over it which allows us to rewrite a generic autonomous dynamical analysis. We describe the treatment of our nonlinear autonomous system by studying the hyperbolic critical points and discuss an interesting phenomenological feature in regards to $H_0$: the possibility to obtain a best-fit value for this parameter in a cosmologically viable $f(T,B)$ model, a mixed power law. This result allows us to present a generic scenario in which it is possible to fix constraints to solve the $H_0$ tension at late times where its linearized solutions are considered.

Linear and center manifold analysis of FRW cosmological model with variable equation of state in Lyra geometry

The present work deals with the dynamical evolution of flat FRW cosmological model with variable equation of state in Lyra geometry. We investigate the phase-plane analysis of our constructed model. The evolution of cosmological solutions is studied by using dynamical system techniques. Stability and viability of the model are discussed using linear Jacobian stability analysis for hyperbolic equilibrium points and center manifold theory for non-hyperbolic equilibrium points.

Suppression of chaos in nonlinear oscillators using a linear vibration absorber

This paper proposes a nonfeedback control to suppress the chaotic response of nonlinear oscillators. A linear vibration absorber is used as nonfeedback method, whose key idea is to find conditions under which the nonlinear oscillator response converges to an equilibrium point or stay oscillating around it. Theoretical results show that if the ratio between the natural frequencies of the primary system $$(\omega _1)$$ and the undamped absorber $$(\omega _2)$$ , that is, $$\omega _r=\frac{\omega _2}{\omega _1}$$ is tuned to be equal to the excitation frequency $$(\Omega )$$ , then the chaotic behavior of a nonlinear oscillator is driven to a stable hyperbolic equilibrium point. Numerical results are presented for the Duffing oscillator shown that if $$\omega _r$$ is tuned close to excitation frequency, then the chaotic response of the Duffing oscillator is driven to periodic orbits. In the case of damped absorber, $$\omega _r$$ to be tuned close to the excitation frequency does not guarantee that the response of the primary system is periodic, it is also necessary to take into account the amplitude of the excitation.

Local Adiabatic Invariants Near a Homoclinic Set of a Slow–Fast Hamiltonian System

In slow–fast systems, fast variables change at a rate of the order of one, and slow variables, at a rate of the order of $$\varepsilon\ll 1$$ . The system obtained for $$\varepsilon=0$$ is said to be frozen. If the frozen (fast) system has one degree of freedom, then in the region where the level curves of the frozen Hamiltonian are closed there exists an adiabatic invariant. A. Neishtadt showed that near a separatrix of the frozen system the adiabatic invariant exhibits quasirandom jumps of order $$\varepsilon$$ . In this paper we partially extend Neishtadt’s result to the multidimensional case. We show that if the frozen system has a hyperbolic critical point possessing several transverse homoclinics, then for small $$\varepsilon$$ there exist trajectories shadowing homoclinic chains. The slow variables evolve in a quasirandom way, shadowing trajectories of systems with Hamiltonians similar to adiabatic invariants. This paper extends the work of V. Gelfreich and D. Turaev, who considered similar phenomena away from critical points of the frozen Hamiltonian.

Stability analysis for cosmological models in f(T,\xa0B) gravity

In this paper we study cosmological solutions of the f(T, B) gravity using dynamical system analyses. For this purpose, we consider cosmological viable functions of f(T, B) that are capable of reproducing the dynamics of the Universe. We present three specific models of f(T, B) gravity which have a general form of their respective solutions by writing the equations of motion as an autonomous system. Finally, we study its hyperbolic critical points and general trajectories in the phase space of the resulting dynamical variables which turn out to be compatible with the current late-time observations.

Chazy-Type Asymptotics and Hyperbolic Scattering for the n-Body Problem

We study solutions of the Newtonian $n$-body problem which tend to infinity hyperbolically, that is, all mutual distances tend to infinity with nonzero speed as $t \rightarrow +\infty$ or as $t \rightarrow -\infty$. In suitable coordinates, such solutions form the stable or unstable manifolds of normally hyperbolic equilibrium points in a boundary manifold "at infinity". We show that the flow near these manifolds can be analytically linearized and use this to give a new proof of Chazy's classical asymptotic formulas. We also address the scattering problem, namely, for solutions which are hyperbolic in both forward and backward time, how are the limiting equilibrium points related? After proving some basic theorems about this scattering relation, we use perturbations of our manifold at infinity to study scattering "near infinity", that is, when the bodies stay far apart and interact only weakly.

Identification of trapping in a peristaltic flow: A new approach using dynamical system theory

In this paper, we propose a new approach for the identification of characteristic peristaltic flow features such as “bolus” and “trapping.” Using dynamical system analysis, we relate the occurrence of a bolus to the existence of a center (an elliptic equilibrium point). Trapping occurs when centers exist under the wave crests along with a pair of saddles (hyperbolic equilibrium points) lying on the central line. For an augmented flow, centers form under the wave crests, whereas saddles lie above (below) the central line. The proposed approach works much better than the presently adopted approach in two ways: (1) it does not require random testing and (2) it characterizes the qualitative flow behavior for the complete range of parameter values. The literature is somewhat inconsistent with regard to the terminologies used for describing characteristic flow behaviors. We have addressed this issue by explicitly defining quantities such as “bolus,” “backward flow,” “trapping,” and “augmented flow.”

A novel 4D chaotic system with nonhyperbolic hyperbolic shape equilibrium points: Analysis, circuit implementation and color image encryption

This paper presents a new 4D chaotic system by extending a 3D system to investigate chaotic systems2 with hidden attractors further and explore their engineering applications in color image encryption. The system has hyperbolic shape equilibriums and the equilibriums are nonhyperbolic which is interesting and rarely mentioned before. The system has been investigated deeply by phase plot, time series, the largest Lyapunov exponent, bifurcation diagram, Poincaré map and 0–1 test. A coexisting hidden attractor has been found in this new system and the circuit simulation model of the system is built by Power Simulation (PSIM). Circuit simulation results are consistent with results of numerical simulations which verify its dynamic behavior further. Then, by combining the generalized Arnold transform with the new chaotic system, a new color image encryption algorithm has been proposed which has large key space and can encrypt either square images or nonsquare images with different size. Finally, the results of encryption experiment and security analysis prove the effectiveness of the algorithm.

A non-canonical scalar field cosmological model: Stability and bifurcation analysis

In this work, cosmological implications of a non-canonical scalar field model have been studied in the background of homogeneous and isotropic flat FLRW spacetime by using dynamical system analysis. By suitable change of variables, the field equations are transformed to an autonomous system and the stability of the system of critical points are examined. For hyperbolic critical points, the analysis of the system is done by using Hartman–Grobman theorem, while the center manifold theory is used to study non-hyperbolic critical points. It is found that the parameters in the autonomous system play a crucial role for phase transition of the Universe. Finally, possible bifurcation scenarios are discussed to identify the point of phase transition.

Dynamical system analysis of Einstein–Skyrme model in a Kantowski–Sachs spacetime

The present work deals with anisotropic (but zero heat flux) Skyrme fluid with a constant radial profile in locally rotational Kantowski–Sachs spacetime in the background of Einstein gravity. By suitably change of variables the field equations are transformed to an autonomous system. To examine the stability of the system critical points are determined. For hyperbolic critical points the analysis of the system is done using Hartman–Grobman theorem, while center manifold theory is used to analyze non-hyperbolic critical points. Also stability at infinity is analyzed to visualize the global evolution of the Universe. As a result the 3D-phase space is identified with the Poincaré 3-sphere embedded in R4. It is found that the parameters in the autonomous system play a crucial role for phase transition of the Universe. Finally possible bifurcation scenarios are discussed to identify the point of phase transition.

Dynamical system analysis of a three fluid cosmological model: an invariant manifold approach

The present paper considers a three-fluid cosmological model consisting of noninteracting dark matter, dark energy and baryonic matter in the background of the Friedman–Robertson–Walker–Lemaître flat spacetime. It has been assumed that the dark matter takes the form of dust whereas the dark energy is a quintessence (real) scalar field with exponential potential. It has been further assumed that the baryonic matter is a perfect fluid with barotropic equation of states. The field equations for this model takes the form of an autonomous dynamical system after some suitable changes of variables. Then a complete stability analysis is done considering all possible parameter (the adiabatic index of the baryonic matter and the parameter arising from the dark energy potential) values and for both the cases of hyperbolic and non-hyperbolic critical points. For non-hyperbolic critical points, the invariant manifold theory (center manifold approach) is applied. Finally various topologically different phase planes and vector field diagrams are produced and the cosmological interpretation of this model is presented.

Ejection-Collision orbits in the symmetric collinear four–body problem

In this paper, we consider the collinear symmetric four-body problem, where four masses m3=α,m1=1,m2=1, and m4=α,α > 0, are aligned in this order and move symmetrically about their center of mass. We introduce regularized variables to deal with binary collisions as well as McGehee coordinates to study the quadruple collision manifold for a negative value of the energy. The paper is mainly focused on orbits that eject from (or collide to) quadruple collision. The problem has two hyperbolic equilibrium points, located in the quadruple collision manifold. We use high order parametrizations of their stable/unstable manifolds to devise a numerical procedure to compute ejection-collision orbits, for any value of α. Some results from the explorations done for α=1 are presented. Furthermore, we prove the existence of ejection-direct escape orbits, which perform a unique type of binary collisions.

Dynamical system analysis of quintom dark energy model

The present work deals with dynamical system analysis of a quintom model of dark energy. By suitable transformation of variables the Einstein field equations are converted to an autonomous system. The critical points are determined and stability of hyperbolic critical points are determined by Hartman–Grobman theorem. To analyze non-hyperbolic critical points different tools (notably center manifold theory) are used. Possible bifurcation scenarios have also been explained.

Approximations of stable manifolds in the vicinity of hyperbolic equilibrium points for fractional differential equations

This paper is devoted to the numerical analysis of the abstract semilinear fractional problem $$D^\alpha u(t) = Au(t) + f(u(t)), u(0)=u^0,$$ in a Banach space E. We are developing a general approach to establish a semidiscrete approximation of stable manifolds. The phase space in the neighborhood of the hyperbolic equilibrium can be split in such a way that the original initial value problem is reduced to systems of initial value problems in the invariant subspaces corresponding to positive and negative real parts of the spectrum. We show that such a decomposition of the equation keeps the same structure under general approximation schemes. The main assumption of our results are naturally satisfied, in particular, for operators with compact resolvents and can be verified for finite element as well as finite difference methods.

An extension of the concept of exponential dichotomy in Fréchet spaces which is stable under perturbation

In this paper we prove versions, in Fréchet spaces, of the classical theorems related to exponential dichotomy for a sequence of continuous linear operators on Banach spaces. To be more specific, here we define a kind of exponential dichotomy in Fréchet spaces, which extends the former one in Banach spaces, establish necessary conditions for its existence and provide sufficient conditions for its stability under perturbation.We apply the conclusions by providing an example of a semigroup of bounded linear operators, on a Fréchet space, which has this new exponential dichotomy but does not in Banach spaces, namely, $\{e^{mΔ}:\; m∈ \mathbb{N}\}$, where $Δ$ is the Laplace operator on the unbounded domain $\mathbb{R}^{n}\setminus \{0\}$.Also, we show how these new concepts allow us to study a hyperbolic equilibrium point of a backwards heat equation with nonlinearity involving convolution products, which cannot be obtained from the knowledge of exponential dichotomy in Banach spaces.

On the viability of Planck scale cosmology with quartessence

Growing evidence as the observations of the CMB (cosmic microwave background), galaxy clustering and high-redshift supernovae address a stable dynamically universe dominated by the dark components. In this paper, using a qualitative theory of dynamical systems, we study the stability of a unified dark matter-dark energy framework known as quartessence Chaplygin model (QCM) with three different equation-of-states within ultraviolet (UV) deformed Friedmann–Robertson–Walker (FRW) cosmologies without Big-Bang singularity. The UV deformation is inspired by the non-commutative (NC) Snyder spacetime approach in which by keeping the transformation groups and rotational symmetry there is a dimensionless, Planck scale characteristic parameter mu _0 with dual implications dependent on its sign that addresses the required invariant cutoffs for length and momentum in nature, in a separate manner. Our stability analysis is done in the (H,rho ) phase space at a finite domain concerning the hyperbolic critical points. According to our analysis, due to constraints imposed on the signs of mu _0 from the phenomenological parameters involved in quartessence models (Omega _m^*, c_s^2, rho _*), for an expanding and accelerating late universe, all three QCMs can be stable in the vicinity of the critical points. The requirement of stability for these quartessence models in case of admission of a minimum invariant length, can yield a flat as well as non-flat expanding and accelerating universe in which Big-Bang singularity is absent. This feedback also phenomenologically credits to braneworld-like framework versus loop quantum cosmology-like one as two possible scenarios which can be NC Snyder spacetime generators (correspond to mu _0<0 and mu _0>0, respectively). As a result, our analysis show that between quartessence models with Chaplygin gas equation-of-states and accelerating FRW backgrounds occupied by a minimum invariant length, there is a possibility of viability.