A geometric approach to phase response curves and its numerical computation through the parameterization method

The final publication is available at link.springer.comThe phase response curve (PRC) is a tool used in neuroscience that measures the phase shift experienced by an oscillator due to a perturbation applied at different phases of the limit cycle. In this paper, we present a new approach to PRCs based on the parameterization method. The underlying idea relies on the construction of a periodic system whose corresponding stroboscopic map has an invariant curve. We demonstrate the relationship between the internal dynamics of this invariant curve and the PRC, which yields a method to numerically compute the PRCs. Moreover, we link the existence properties of this invariant curve as the amplitude of the perturbation is increased with changes in the PRC waveform and with the geometry of isochrons. The invariant curve and its dynamics will be computed by means of the parameterization method consisting of solving an invariance equation. We show that the method to compute the PRC can be extended beyond the breakdown of the curve by means of introducing a modified invariance equation. The method also computes the amplitude response functions (ARCs) which provide information on the displacement away from the oscillator due to the effects of the perturbation. Finally, we apply the method to several classical models in neuroscience to illustrate how the results herein extend the framework of computation and interpretation of the PRC and ARC for perturbations of large amplitude and not necessarily pulsatile.Peer ReviewedPostprint (author's final draft

Chaos in the hysteretic grazing-sliding codimension-one saddle-node bifurcation of piecewise dynamical systems

We present two ways of regularizing a parameter family of piecewise smooth dynamical systems undergoing a grazing- sliding bifurcation. We use the Sotomayor-Teixeira regularization and prove that the bifurcation is a saddle-node (see [ ? ]). Then we perform a hysteretic regularization. However, in spite that the two regularization will give the same dynamics in the sliding modes (see [ ? ]), when a tangency appears, so is in the case of grazing-sliding, the hysteretic process generate chaotic dynamics. Finally, we smooth the hysteresis by embedding the system in a higher dimension. Now the discontinuous control variable u is also a continuous time dependent variable although a fast-fast one. We then encounter loop feedback chaotic behaviourPostprint (author's final draft

Breakdown of heteroclinic orbits for some analytic unfoldings of the Hopf-zero singularity

In this paper we study the exponentially small splitting of a heteroclinic orbit in some unfoldings of the central singularity also called Hopf-zero singularity. The fields under consideration are of the form: dx dτ = −δxz − y (α + cδz) + δp+1f(δx, δy, δz, δ) dy dτ = −δyz + x (α + cδz) + δp+1g(δx, δy, δz, δ) dz dτ = δ ?−1 + b(x2 + y2) + z2? + δp+1h(δx, δy, δz, δ), where f, g and h are real analytic functions, α, b and c are constants and δ is a small parameter. When f = g = h = 0 the system has a heteroclinic orbit between the critical points (0, 0,±1) given by: {(x, y) = (0, 0) ;−1 < z < 1}. Let ds,u be the distance between the one dimensional stable and unstable manifold of the perturbed system measured at the plane z = 0. We prove that for any f, g such that ˆm(i α) ?= 0, where ˆm is the Borel transform of the function m(u) = u1+i c(f + i g)(0, 0, u, 0) |ds,u| = 2π ecπ/2 | ˆm(i α)|δp e−π|α|/(2δ)(1 + O(δp+2| log δ|)), p>−2

On the numerical computation of Diophantine rotation numbers of analytic circle maps

In this paper we present a numerical method to compute Diophantine rotation numbers of circle maps with high accuracy. We mainly focus on analytic circle diffeomorphisms, but the method also works in the case of (enough) finite differentiability. The keystone of the method is that, under these conditions, the map is conjugate to a rigid rotation of the circle. Moreover, albeit it is not fully justified by our construction, the method turns to be quite efficient for computing rational rotation numbers. We discuss the method through several numerical examples

Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom

The splitting of separatrices for Hamiltonians with $1{1\over 2}$ degrees of freedom $$h(x,t /\varepsilon ) = h^{0}(x) + \mu \varepsilon ^{p} h^{1}(x,t /\varepsilon )$$ is measured. We assume that $h^{0}(x)= h^{0}(x_{1},x_{2})= x_{2}^{2}/2+V(x_{1})$ has a separatrix $x^{0}(t)$, $h^{1}(x,\theta )$ is $2\pi$-periodic in $\theta$, $\mu$ and $\varepsilon >0$ are independent small parameters, and $p\ge 0$. Under suitable conditions of meromorphicity for $x_{2}^{0}( u )$ and the perturbation $h^{1}(x^{0}( u ),\theta )$, the order $\ell$ of the perturbation on the separatrix is introduced, and it is proved that, for $p \ge \ell$, the splitting is exponentially small in $\varepsilon$, and is given in first order by the Melnikov function

Asymptotic behaviour of the domain of analyticity of invariant curves of the standard map

In this paper we consider the standard map, and we study the invariant curve obtained by analytical continuation, with respect to the perturbative parameter E, of the invariant circle of rotation number the golden mean corresponding to the case E=0. We show that, if we consider the parameterization that conjugates the dynamics of this curve to an irrational rotation, the domain of definition of this conjugation has an asymptotic boundary of analyticity when E->0 (in the sense of the singular perturbation theory). This boundary is obtained studying the conjugation problem for the so-called semi-standard map. To prove this result we have used KAM-like methods adapted to the framework of singular perturbation theory, as well as matching techniques to join di erent pieces of the conjugation, obtained in different parts of its domain of analyticity

A unified approach to explain contrary effects of hysteresis and smoothing in nonsmooth systems

Piecewise smooth dynamical systems make use of discontinuities to model switching between regions of smooth evolution. This introduces an ambiguity in prescribing dynamics at the discontinuity: should the dynamics be given by a limiting value on one side or other of the discontinuity, or a member of some set containing those values? One way to remove the ambiguity is to regularize the discontinuity, the most common being either to smooth it out, or to introduce a hysteresis between switching in one direction or the other across it. Here we show that the two can in general lead to qualitatively different dynamical outcomes. We then define a higher dimensional model with both smoothing and hysteresis, and study the competing limits in which hysteretic or smoothing effects dominate the behaviour, only the former of which correspond to Filippov’s standard ‘sliding modes’.Peer ReviewedPostprint (author's final draft

Splitting of separatrices for rapid degenerate perturbations of the classical pendulum

In this work we study the splitting distance of a rapidly perturbed pendulum H(x, y, t) = 1 2 y 2 + (cos(x) - 1) + µ(cos(x) - 1)g t e with g(t ) = P |k|>1 g [k] e ikt a 2p-periodic function and µ, e 1. Systems of this kind undergo exponentially small splitting and, when µ 1, it is known that the Melnikov function actually gives an asymptotic expression for the splitting function provided g [±1] 6= 0. Our study focuses on the case g [±1] = 0 and it is motivated by two main reasons. On the one hand the general understanding of the splitting, as current results fail for a perturbation as simple as g(t ) = cos(5t ) + cos(4t ) + cos(3t ). On the other hand, a study of the splitting of invariant manifolds of tori of rational frequency p/q in Arnold’s original model for diffusion leads to the consideration of pendulum-like Hamiltonians with g(t ) = sin p · t e + cos q · t e , where, for most p, q ¿ Z the perturbation satisfies g [±1] 6= 0. As expected, the Melnikov function is not a correct approximation for the splitting in this case. To tackle the problem we use a splitting formula based on the solutions of the so-called inner equation and make use of the Hamilton-Jacobi formalism. The leading exponentially small term appears at order µ n , where n is an integer determined exclusively by the harmonics of the perturbation. We also provide an algorithm to compute it.Preprin

Generic bifurcations of low codimension of planar Filippov Systems

In this article some qualitative and geometric aspects of non-smooth dynamical systems theory are discussed. The main aim of this article is to develop a systematic method for studying local (and global) bifurcations in non-smooth dynamical systems. Our results deal with the classification and characterization of generic codimension-2 typical singularities of planar Filippov systems as well as the presentation of the bifurcation diagrams and some dynamical consequencesPreprin

Critical velocity in kink-defect interaction models: rigorous results

In this work we study a model of interaction of kinks of the sine-Gordon equation with a weak defect. We obtain rigorous results concerning the so-called critical velocity derived in [7] by a geometric approach. More specifically, we prove that a heteroclinic orbit in the energy level 0 of a 2-dof Hamiltonian is destroyed giving rise to heteroclinic connections between certain elements (at infinity) for exponentially small (in e) energy levels. In this setting Melnikov theory does not apply because there are exponentially small phenomena.Peer ReviewedPostprint (published version