Space Charges Can Significantly Affect the Dynamics of Accelerator Maps

Space charge effects can be very important for the dynamics of intense particle beams, as they repeatedly pass through nonlinear focusing elements, aiming to maximize the beam's luminosity properties in the storage rings of a high energy accelerator. In the case of hadron beams, whose charge distribution can be considered as "frozen" within a cylindrical core of small radius compared to the beam's dynamical aperture, analytical formulas have been recently derived \cite{BenTurc} for the contribution of space charges within first order Hamiltonian perturbation theory. These formulas involve distribution functions which, in general, do not lead to expressions that can be evaluated in closed form. In this paper, we apply this theory to an example of a charge distribution, whose effect on the dynamics can be derived explicitly and in closed form, both in the case of 2--dimensional as well as 4--dimensional mapping models of hadron beams. We find that, even for very small values of the "perveance" (strength of the space charge effect) the long term stability of the dynamics changes considerably. In the flat beam case, the outer invariant "tori" surrounding the origin disappear, decreasing the size of the beam's dynamical aperture, while beyond a certain threshold the beam is almost entirely lost. Analogous results in mapping models of beams with 2-dimensional cross section demonstrate that in that case also, even for weak tune depressions, orbital diffusion is enhanced and many particles whose motion was bounded now escape to infinity, indicating that space charges can impose significant limitations on the beam's luminosity.Comment: 16 pages, 4 figures, to appear in Physics Letters

Complex statistics in Hamiltonian barred galaxy models

We use probability density functions (pdfs) of sums of orbit coordinates, over time intervals of the order of one Hubble time, to distinguish weakly from strongly chaotic orbits in a barred galaxy model. We find that, in the weakly chaotic case, quasi-stationary states arise, whose pdfs are well approximated by $q$-Gaussian functions (with $1<q<3$), while strong chaos is identified by pdfs which quickly tend to Gaussians ($q=1$). Typical examples of weakly chaotic orbits are those that "stick" to islands of ordered motion. Their presence in rotating galaxy models has been investigated thoroughly in recent years due of their ability to support galaxy structures for relatively long time scales. In this paper, we demonstrate, on specific orbits of 2 and 3 degree of freedom barred galaxy models, that the proposed statistical approach can distinguish weakly from strongly chaotic motion accurately and efficiently, especially in cases where Lyapunov exponents and other local dynamic indicators appear to be inconclusive.Comment: 14 pages, 9 figures, submitted for publicatio

Explicit Construction of First Integrals with Quasi-monomial Terms from the Painlev\'{e} Series

The Painlev\'{e} and weak Painlev\'{e} conjectures have been used widely to identify new integrable nonlinear dynamical systems. For a system which passes the Painlev\'{e} test, the calculation of the integrals relies on a variety of methods which are independent from Painlev\'{e} analysis. The present paper proposes an explicit algorithm to build first integrals of a dynamical system, expressed as `quasi-polynomial' functions, from the information provided solely by the Painlev\'{e} - Laurent series solutions of a system of ODEs. Restrictions on the number and form of quasi-monomial terms appearing in a quasi-polynomial integral are obtained by an application of a theorem by Yoshida (1983). The integrals are obtained by a proper balancing of the coefficients in a quasi-polynomial function selected as initial ansatz for the integral, so that all dependence on powers of the time $\tau=t-t_0$ is eliminated. Both right and left Painlev\'{e} series are useful in the method. Alternatively, the method can be used to show the non-existence of a quasi-polynomial first integral. Examples from specific dynamical systems are given.Comment: 16 pages, 0 figure

Spectral Signatures of Exceptional Points and Bifurcations in the Fundamental Active Photonic Dimer

The fundamental active photonic dimer consisting of two coupled quantum well lasers is investigated in the context of the rate equation model. Spectral transition properties and exceptional points are shown to occur under general conditions, not restricted by PT-symmetry as in coupled mode models, suggesting a paradigm shift in the field of non-Hermitian photonics. The optical spectral signatures of system bifurcations and exceptional points are manifested in terms of self-termination effects and observable drastic variations of the spectral line shape that can be controlled in terms of optical detuning and inhomogeneous pumping.Comment: 13 pages, 5 figure

Periodically Forced Nonlinear Oscillators With Hysteretic Damping

We perform a detailed study of the dynamics of a nonlinear, one-dimensional oscillator driven by a periodic force under hysteretic damping, whose linear version was originally proposed and analyzed by Bishop in [1]. We first add a small quadratic stiffness term in the constitutive equation and construct the periodic solution of the problem by a systematic perturbation method, neglecting transient terms as $t\rightarrow \infty$. We then repeat the analysis replacing the quadratic by a cubic term, which does not allow the solutions to escape to infinity. In both cases, we examine the dependence of the amplitude of the periodic solution on the different parameters of the model and discuss the differences with the linear model. We point out certain undesirable features of the solutions, which have also been alluded to in the literature for the linear Bishop's model, but persist in the nonlinear case as well. Finally, we discuss an alternative hysteretic damping oscillator model first proposed by Reid [2], which appears to be free from these difficulties and exhibits remarkably rich dynamical properties when extended in the nonlinear regime.Comment: Accepted for publication in the Journal of Computational and Nonlinear Dynamic

Homoclinic points of 2-D and 4-D maps via the Parametrization Method

An interesting problem in solid state physics is to compute discrete breather solutions in $\mathcal{N}$ coupled 1--dimensional Hamiltonian particle chains and investigate the richness of their interactions. One way to do this is to compute the homoclinic intersections of invariant manifolds of a saddle point located at the origin of a class of $2\mathcal{N}$--dimensional invertible maps. In this paper we apply the parametrization method to express these manifolds analytically as series expansions and compute their intersections numerically to high precision. We first carry out this procedure for a 2--dimensional (2--D) family of generalized Henon maps ($\mathcal{N}$=1), prove the existence of a hyperbolic set in the non-dissipative case and show that it is directly connected to the existence of a homoclinic orbit at the origin. Introducing dissipation we demonstrate that a homoclinic tangency occurs beyond which the homoclinic intersection disappears. Proceeding to $\mathcal{N}=2$, we use the same approach to determine the homoclinic intersections of the invariant manifolds of a saddle point at the origin of a 4--D map consisting of two coupled 2--D cubic H\'enon maps. In dependence of the coupling the homoclinic intersection is determined, which ceases to exist once a certain amount of dissipation is present. We discuss an application of our results to the study of discrete breathers in two linearly coupled 1--dimensional particle chains with nearest--neighbor interactions and a Klein--Gordon on site potential.Comment: 24 pages, 10 figures, videos can be found at https://comp-phys.tu-dresden.de/supp