Related papers: Isochronous Centers of Lienard Type Equations and …

We first generalize a classical iteration formula for one variable holomorphic mappings to a formula for higher dimensional holomorphic mappings. Then, as an application, we give a short and intuitive proof of a classical theorem, due to H.…

We consider large random matrices $X$ with centered, independent entries but possibly different variances. We compute the normalized trace of $f(X) g(X^*)$ for $f,g$ functions analytic on the spectrum of $X$. We use these results to compute…

We consider the rational potentials of the one-dimensional mechanical systems, which have a family of periodic solutions with the same period (isochronous potentials). We prove that up to a shift and adding a constant all such potentials…

Assume $n\geq 2$. Consider the elementary symmetric polynomials $e_k(y_1,y_2,\ldots, y_n)$ and denote by $E_0,E_1,\ldots,E_{n-1}$ the elementary symmetric polynomials in reverse order \begin{align*}…

Let $t_{i}=\frac{i}{n}$ for $i=0,...,n$ be equally spaces knots in the unit interval $[0,1].$ Let $\mathcal{S}_{n}$ be the space of piecewise linear continuous functions on $[0,1]$ with knots $\pi_{n}=\{t_{i}:0\leq i\leq n\}.$ Then we have…

We consider a homoclinic orbit to a saddle fixed point of an arbitrary $C^\infty$ map $f$ on $\mathbb{R}^2$ and study the phenomenon that $f$ has an infinite family of asymptotically stable, single-round periodic solutions. From classical…

In this paper, we approach the problem of finding the zeros of the sum of a maximally monotone operator and a monotone and Lipschitz continuous one in a real Hilbert space via an implicit forward-backward-forward dynamical system with…

We present a general approach to the study of synchrony in networks of weakly nonlinear systems described by singularly perturbed equations of the type $x''+x+\epsilon f(x,x')=0$. By performing a perturbative calculation based on normal…

The aim of this paper is to prove the existence of periodic solutions to symmetric Newtonian systems in any neighborhood of an isolated orbit of equilibria. Applying equivariant bifurcation techniques we obtain a generalization of the…

Pleshkan proved in 1969 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in the family of cubic centers with homogeneous nonlinearities $\mathscr C_3.$ In this paper we prove that if we perturb…

This work presents a comprehensive study of the generalized Duffing oscillator, a fundamental model in nonlinear dynamics described by the system $$ \dot{x} = y, \quad \dot{y} = -\alpha y - \epsilon x^m - \sigma x, $$ where $\epsilon \neq…

We give an alternative proof for the celebrated Bertrand's theorem as a corollary of the isochronicity of a certain family of centers.

Isochronous islands are regular solutions related to different chains of elliptic points but with the same winding number. These isochronous islands emerge in phase space as a response to multiple resonant perturbations and can be simulated…

In principle, while coupled limit cycle oscillators can overcome mismatch in intrinsic rates and match their frequencies, but zero phase lag synchronization is just achievable in the limit of zero mismatch, i.e., with identical oscillators.…

We present a geometric characterization of the nonlinear smooth functions $V: R\to R$ for which the origin is a global isochronous center for the scalar equation $\ddot x=-V'(x)$. We revisit Stillinger and Dorignac isochronous potentials…

Let $f$ and $g$ be both continuous functions on $\left( 0,\infty \right) $ with $g\left( t\right) >0$ for $t\in \left( 0,\infty \right) $ and let $ F\left( x\right) =\mathcal{L}\left( f\right) $, $G\left( x\right) =\mathcal{L }\left(…

We study the classical planar two-center problem of a particle $m$ subjected to harmonic-like interactions with two fixed centers. For convenient values of the dimensionless parameter of this problem we use the averaging theory for showing…

We consider classical solutions to $-\Delta u = f(u)$ in half-spaces, under homogeneous Dirichlet boundary conditions. We prove that any positive solution is strictly monotone increasing in the direction orthogonal to the boundary, provided…

For a product of i.i.d. random maps or a memoryless stochastic flow on a compact space $X$, we find conditions under which the presence of locally asymptotically stable trajectories (e.g. as given by negative Lyapunov exponents) implies…

Let $X$ be a zero-dimensional locally compact Hausdorff space not necessarily metric and $G$ a compactly generated topological group not necessarily abelian or countable. We define recurrence at a point for any continuous action of $G$ on…