Featured Researches

Dynamical Systems

A Central Limit Theorem for Rosen Continued Fractions

We prove a central limit theorem for Birkhoff sums of the Rosen continued fraction algorithm. A Lasota-Yorke bound is obtained for general one-dimensional continued fractions with the bounded variation space, which implies quasi-compactness of the transfer operator. The main result is a direct proof of the existence of a spectral gap, assuming a certain behavior of the transformation when iterated. This condition is explicitly proved for the Rosen system. We conclude via well-known results of A. Broise that the central limit theorem holds.

Read more
Dynamical Systems

A Computer-Assisted Study of Red Coral Population Dynamics

We consider a 13-dimensional age-structured discrete red coral population model varying with respect to a fitness parameter. Our numerical results give a bifurcation diagram of both equilibria and stable invariant curves of orbits. We observe that not only for low levels of fitness, but also for high levels of fitness, populations are extremely vulnerable, in that they spend long time periods near extinction. We then use computer-assisted proofs techniques to rigorously validate the set of regular and bifurcation fixed points that have been found numerically.

Read more
Dynamical Systems

A Delay Equation Model for the Atlantic Multidecadal Oscillation

A new technique to derive delay models from systems of partial differential equations, based on the Mori-Zwanzig formalism, is used to derive a delay difference equation model for the Atlantic Multidecadal Oscillation. The Mori-Zwanzig formalism gives a rewriting of the original system of equations which contains a memory term. This memory term can be related to a delay term in a resulting delay equation. Here the technique is applied to an idealized, but spatially extended, model of the Atlantic Multidecadal Oscillation. The resulting delay difference model is of a different type than the delay differential model which has been used to describe the El Niño- Southern Oscillation. In addition to this model, which can also be obtained by integration along characteristics, error terms for a smoothing approximation of the model have been derived from the Mori-Zwanzig formalism. Our new method of deriving delay models from spatially extended models has a large potential to use delay models to study a range of climate variability phenomena.

Read more
Dynamical Systems

A Dichotomy for the Weierstrass-type functions

For a real analytic periodic function ϕ:R→R , an integer b≥2 and λ∈(1/b,1) , we prove the following dichotomy for the Weierstrass-type function W(x)= ∑ n≥0 λ n ϕ( b n x) : Either W(x) is real analytic, or the Hausdorff dimension of its graph is equal to 2+ log b λ . Furthermore, given b and ϕ , the former alternative only happens for finitely many λ unless ϕ is constant.

Read more
Dynamical Systems

A Dynamized Power Flow Method based on Differential Transformation

This paper proposes a novel method for solving and tracing power flow solutions with changes of a loading parameter. Different from the conventional continuation power flow method, which repeatedly solves static AC power flow equations, the proposed method extends the power flow model into a fictitious dynamic system by adding a differential equation on the loading parameter. As a result, the original solution curve tracing problem is converted to solving the time domain trajectories of the reformulated dynamic system. A non-iterative algorithm based on differential transformation is proposed to analytically solve the aforementioned dynamized model in form of power series of time. This paper proves that the nonlinear power flow equations in the time domain are converted to formally linear equations in the domain of the power series order after the differential transformation, thus avoiding numerical iterations. Case studies on several test systems including a 2383-bus system show the merits of the proposed method.

Read more
Dynamical Systems

A Galoisian proof of Ritt theorem on the differential transcendence of Poincaré functions

Using Galois theory of functional equations, we give a new proof of the main result of the paper "Transcendental transcendency of certain functions of Poincaré" by J.F. Ritt, on the differential transcendence of the solutions of the functional equation R(y(t))=y(qt), where R is a rational function with complex coefficients which verifies R(0)=0, R'(0)=q, where q is a complex number with |q|>1. We also give a partial result in the case of an algebraic function R.

Read more
Dynamical Systems

A Markovian and Roe-algebraic approach to asymptotic expansion in measure

In this paper, we conduct further studies on geometric and analytic properties of asymptotic expansion in measure. More precisely, we develop a machinery of Markov expansion and obtain an associated structure theorem for asymptotically expanding actions. Based on this, we establish an analytic characterisation for asymptotic expansion in terms of the Druţu-Nowak projection and the Roe algebra of the associated warped cones. As an application, we provide new counterexamples to the coarse Baum-Connes conjecture.

Read more
Dynamical Systems

A Rauzy fractal unbounded in all directions of the plane

We construct an Arnoux-Rauzy word for which the set of all differences of two abelianized factors is equal to Z 3 . In particular, the imbalance of this word is infinite - and its Rauzy fractal is unbounded in all directions of the plane.

Read more
Dynamical Systems

A Regular Gonosomal Evolution Operator with uncountable set of fixed points

In this paper we study dynamical systems generated by a gonosomal evolution operator of a bisexual population. We find explicitly all (uncountable set) of fixed points of the operator. It is shown that each fixed point has eigenvalues less or equal to 1. Moreover, we show that each trajectory converges to a fixed point, i.e. the operator is reqular. There are uncountable family of invariant sets each of which consisting unique fixed point. Thus there is one-to-one correspondence between such invariant sets and the set of fixed points. Any trajectory started at a point of the invariant set converges to the corresponding fixed point.

Read more
Dynamical Systems

A Special Conic Associated with the Reuleaux Negative Pedal Curve

The Negative Pedal Curve of the Reuleaux Triangle w.r. to a point M on its boundary consists of two elliptic arcs and a point P 0 . Interestingly, the conic passing through the four arc endpoints and by P 0 has a remarkable property: one of its foci is M . We provide a synthetic proof based on Poncelet's polar duality and inversive techniques. Additional intriguing properties of Reuleaux negative pedal are proved using straightforward techniques.

Read more

Ready to get started?

Join us today