Aspects Of Global Dynamics Research Articles

Chaotic Behavior of a Generalized Sprott E Differential System

A chaotic system with only one equilibrium, a stable node-focus, was introduced by Wang and Chen [2012]. This system was found by adding a nonzero constant [Formula: see text] to the Sprott E system [Sprott, 1994]. The coexistence of three types of attractors in this autonomous system was also considered by Braga and Mello [2013]. Adding a second parameter to the Sprott E differential system, we get the autonomous system [Formula: see text] where [Formula: see text] are parameters and [Formula: see text]. In this paper, we consider theoretically some global dynamical aspects of this system called here the generalized Sprott E differential system. This polynomial differential system is relevant because it is the first polynomial differential system in [Formula: see text] with two parameters exhibiting, besides the point attractor and chaotic attractor, coexisting stable limit cycles, demonstrating that this system is truly complicated and interesting. More precisely, we show that for [Formula: see text] sufficiently small this system can exhibit two limit cycles emerging from the classical Hopf bifurcation at the equilibrium point [Formula: see text]. We also give a complete description of its dynamics on the Poincaré sphere at infinity by using the Poincaré compactification of a polynomial vector field in [Formula: see text], and we show that it has no first integrals in the class of Darboux functions.

Global dynamical aspects of a generalized Chen–Wang differential system

We study theoretically the global chaotic behavior of the generalized Chen–Wang differential system $$\begin{aligned} \dot{x} = y, \quad \dot{y} = z,\quad \dot{z} = -y - b x^2 - x z + 3 y^2 + a, \end{aligned}$$ where \(a,b \in {\mathbb {R}}\) are parameters and \(b\ne 0\). This polynomial differential system is relevant because is the first polynomial differential system in \({\mathbb {R}}^3\) with two parameters exhibiting chaotic motion without having equilibria. We first show that for \(a>0\) sufficiently small it can exhibit up to three small amplitude periodic solutions that bifurcate from a zero-Hopf equilibrium point located at the origin of coordinates when \(a=0\). We also show that the system exhibits two limit cycles emerging from two classical Hopf bifurcations at the equilibrium points \(({\pm }\sqrt{2a}, 0,0)\), for \(a>0, b=1/2\). We also give a complete description of its dynamics on the Poincare sphere at infinity by using the Poincare compactification of a polynomial vector field in \({\mathbb {R}}^3\), and we show that it has no first integrals neither in the class of analytic functions nor in the class of Darboux functions.

On global dynamics in a periodic differential equation with deviating argument

Several aspects of global dynamics and the existence of periodic solutions are studied for the scalar differential delay equation x′(t)=a(t)f(x([t-K])), where f(x) is a continuous negative feedback function, x·f(x)<0,x≠0,0<a(t) is continuous ω-periodic, [·] is the integer part function, and the integer K⩾0 is the delay. The case of integer period ω allows for a reduction to finite-dimensional difference equations. The dynamics of the latter are studied in terms of corresponding discrete maps, including the partial case of interval maps (K=0).

Global properties of tight Reeb flows with applications to Finsler geodesic flows on S2

AbstractWe show that if a Finsler metric on S2 with reversibility r has flag curvatures K satisfying (r/(r+1))2 &lt; K ≤ 1, then closed geodesics with specific contact-topological properties cannot exist, in particular there are no closed geodesics with precisely one transverse self-intersection point. This is a special case of a more general phenomenon, and other closed geodesics with many self-intersections are also excluded. We provide examples of Randers type, obtained by suitably modifying the metrics constructed by Katok [21], proving that this pinching condition is sharp. Our methods are borrowed from the theory of pseudo-holomorphic curves in symplectizations. Finally, we study global dynamical aspects of 3-dimensional energy levels C2-close to S3

Dynamics at infinity and other global dynamical aspects of Shimizu–Morioka equations

We present some global dynamical aspects of Shimizu–Morioka equations given by $$\dot{x}=y, \qquad\dot{y}=x-\lambda y-xz, \qquad\dot{z}=-\alpha z+x^2,$$ where (x,y,z)∈ℝ3 are the state variables and λ,α are real parameters. This system is a simplified model proposed for studying the dynamics of the well-known Lorenz system for large Rayleigh numbers. Using the Poincare compactification of a polynomial vector field in ℝ3, we give a complete description of the dynamics of Shimizu–Morioka equations at infinity. Then using analytical and numerical tools, we investigate for the case α=0 the existence of infinitely many singularly degenerate heteroclinic cycles, each one consisting of an invariant set formed by a line of equilibria together with a heteroclinic orbit connecting two of these equilibria. The dynamical consequences of the existence of these cycles are also investigated. The present study is part of an effort aiming to describe global properties of quadratic three-dimensional vector fields with chaotic dynamical behavior, as made for instance in (Dias et al. in Nonlinear Anal. Real World Appl. 11(5):3491–3500, 2010; Kokubu and Roussarie in J. Dyn. Differ. Equ. 16(2):513–557, 2004; Llibre and Messias in Physica D 238(3):241–252, 2009; Llibre et al. in J. Phys. A, Math. Theor. 41:275210, 2008; Llibre et al. in Int. J. Bifurc. Chaos Appl. Sci. Eng. 20(10):3137–3155, 2010; Lorenz in J. Atmos. Sci. 20:130–141, 1963; Lu et al. in Int. J. Bifurc. Chaos Appl. Sci. Eng. 14(5):1507–1537, 2004; Mello et al. in Chaos Solitons Fractals 37:1244–1255, 2008; Messias in J. Phys. A, Math. Theor. 42:115101, 2009; Messias et al. in TEMA Tend. Mat. Apl. Comput. 9(2):275–285, 2008).

Global dynamics of a differential equation with piecewise constant argument

Several aspects of global dynamics are studied for the scalar differential-difference equation ε x ̇ ( t ) + x ( t ) = f ( x ( [ t ] ) ) , 0 < ε ≪ 1 , where [ ⋅ ] is the integer part function. The equation is a particular case of the special discretization (discrete version) of the singularly perturbed differential delay equation ε x ̇ ( t ) + x ( t ) = f ( x ( t − 1 ) ) . Sufficient conditions for the invariance, global stability of equilibria, existence, stability/instability, and shape of periodic solutions, and the chaotic behavior are derived. The principal analysis is based on the reduction of its dynamics to the one-dimensional map F : x → f ( x ) + [ x − f ( x ) ] e − 1 / ε , many relevant properties of which follow from those of the interval map f .

Sensitivity to social contingency or stability of interaction? Modelling the dynamics of perceptual crossing

We introduce a series of evolutionary robotics simulations that address the behaviour of individuals in socially contingent interactions. The models are based on a recent study by Auvray, Lenay and Stewart [(2006). The attribution of intentionality in a simulated environment: The case of minimalist devices. In Tenth meeting of the association for the scientific study of consciousness, Oxford, UK, 23–26 June, 2006] on tactile perceptual crossing in a minimal virtual environment. In accordance, both the empirical experiments and our simulations point out the essential character of global embodied interaction dynamics for the sensitivity to contingency to arise. Rather than being individually perceived by any of the interactors, sensitivity to contingency arises from processes of circular causality that characterise the collective dynamics. Such global dynamical aspects are frequently neglected when studying social cognition. Furthermore, our synthetic studies point out interesting aspects of the task that are not immediately obvious in the empirical data. They, in addition, generate new hypotheses for further experiments. We conclude by promoting a minimal but tractable, dynamic and embodied account to social interaction, combining synthetic and empirical findings as well as concrete predictions regarding sensorimotor strategies, the role of time-delays and robustness to perturbations in interactive dynamics.