Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies
Amadeu Delshams Marina Gonchenko Sergey V. Gonchenko J. Tomás Lázaro
Discrete & Continuous Dynamical Systems - A 2018, 38(9): 4483-4507 doi: 10.3934/dcds.2018196

We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider one-parameter families of reversible maps unfolding the initial homoclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations related to the birth of asymptotically stable, unstable and elliptic periodic orbits.

keywords: Newhouse phenomenon homoclinic and heteroclinic tangencies reversible mixed dynamics
Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies
Amadeu Delshams Marina Gonchenko Pere Gutiérrez
Electronic Research Announcements 2014, 21(0): 41-61 doi: 10.3934/era.2014.21.41
We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a frequency vector $\omega=(1,\Omega)$, where $\Omega$ is a quadratic irrational number, or a 3-dimensional torus with a frequency vector $\omega=(1,\Omega,\Omega^2)$, where $\Omega$ is a cubic irrational number. Applying the Poincaré--Melnikov method, we find exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, and we show that such estimates depend strongly on the arithmetic properties of the frequencies. In the quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fulfilled in 24 cases, which allows us to provide asymptotic estimates in a simple way. In the cubic case, we focus our attention to the case in which $\Omega$ is the so-called cubic golden number (the real root of $x^3+x-1=0$), obtaining also asymptotic estimates. We point out the similitudes and differences between the results obtained for both the quadratic and cubic cases.
keywords: Melnikov integrals Splitting of separatrices quadratic and cubic frequencies.

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