Frequency Map Analysis Research Articles

Applying Zero-Crossing Method for Frequency Map Analysis of Dynamical Systems

Abstract We present a simple method for estimating fundamental frequencies from orbit’s time series of triaxial dynamical systems based on the zero-crossing technique. The simplicity and high performance of the method make it a viable alternative to more advanced, but rather complex and computationally expensive techniques based on Fourier spectral analysis and iterative torus-mapping with action-angle variables. More importantly, the method provides a simple and quick way to identify irregular orbits and quantify their level of chaoticity, allowing to detect even a subtle onset of chaos. To validate the proposed algorithm, we performed comparison tests with synthetic quasi-periodic signals and with other methods using the triaxial Dehnen potential. We found noticeable differences in frequency maps constructed with our method and methods based on Fourier spectral analysis that we attributed to the way we assign static properties to chaotic orbits. At the same time, our frequency maps turn out to be in very good agreement with the maps obtained using the action-based technique.

Energetic and structural insights behind calcium induced conformational transition in calmodulin.

Calmodulin (CaM) is a key signaling protein that triggers several cellular and physiological processes inside the cell. Upon binding with calcium ion, CaM undergoes large scale conformational transition from a closed state to an open state that facilitates its interaction with various target protein and regulates their activity. This work explores the origin of the energetic and structural variation of the wild type and mutated CaM and explores the molecular origin for the structural differences between them. We first calculated the sequential calcium binding energy to CaM using the PDLD/S-LRA/β approach. This study shows a very good correlation with experimental calcium binding energies. Next we calculated the calcium binding energies to the wild type CaM and several mutated CaM systems which were reported experimentally. On the structural aspect, it has been reported experimentally that certain mutation (Q41L-K75I) in calcium bound CaM leads to complete conformational transition from an open to a closed state. By using equilibrium molecular dynamics simulation, free energy calculation and contact frequency map analysis, we have shown that the formation of a cluster of long-range hydrophobic contacts, initiated by the Q41L-K75I CaM variant is the driving force behind its closing motion. This study unravels the energetics and structural aspects behind calcium ion induced conformational changes in wild type CaM and its variant.

Eviction-like resonances for satellite orbits

The motion of a satellite can experience secular resonances between the precession frequencies of its orbit and the mean motion of the host planet around the star. Some of these resonances can significantly modify the eccentricity (evection resonance) and the inclination (eviction resonance) of the satellite. In this paper, we study in detail the secular resonances that can disturb the orbit of a satellite, in particular the eviction-like ones. Although the inclination is always disturbed while crossing one eviction-like resonance, capture can only occur when the semi-major axis is decreasing. This is, for instance, the case of Phobos, the largest satellite of Mars, that will cross some of these resonances in the future because its orbit is shrinking owing to tidal effects. We estimate the impact of resonance crossing in the orbit of the satellite, including the capture probabilities, as a function of several parameters, such as the eccentricity and the inclination of the satellite, and the obliquity of the planet. Finally, we use the method of the frequency map analysis to study the resonant dynamics based on stability maps, and we show that some of the secular resonances may overlap, which leads to chaotic motion for the inclination of the satellite.

Long-term dynamical evolution of Pallene (Saturn XXXIII) and its diffuse, dusty ring

ABSTRACT The distinctive set of Saturnian small satellites, Aegaeon, Methone, Anthe, and Pallene, constitutes an excellent laboratory to understand the evolution of systems immersed in co-orbital dusty rings/arcs, subjected to perturbations from larger satellites and non-gravitational forces. In this work, we carried out a comprehensive numerical exploration of the long-term evolution of Pallene and its ring. Through frequency map analysis, we characterized the current dynamical state around Pallene. A simple tidal evolution model serves to set a time frame for the current orbital configuration of the system. With detailed short- and long-term N-body simulations we determine whether Pallene is currently in resonance with one or more of six of Saturn’s major moons. We analysed a myriad of resonant arguments extracted from the direct and indirect parts of the disturbing function, finding that Pallene is not in mean motion resonance from the present up to 5 Myr into the future; none the less, some resonant arguments exhibit intervals of libration and circulation at different time-scales and moon pairings. We studied the dynamical evolution of micrometric particles forming the ring, considering gravitational and non-gravitational forces. Non-gravitational forces are responsible for particles vertical excursions and outward migration. By estimating the satellite’s mass production rate, we find that Pallene could be responsible for keeping its ring in steady-state only if it is mainly composed of large micrometre-sized particles. If mainly composed of particles with a few micrometres for which Pallene is the only source, the ring will spread out, both radially and vertically, until it finally disappears.

Numerical optimization of dc wire parameters for mitigation of the long range beam-beam interactions in High Luminosity Large Hadron Collider

Several configurations of the High Luminosity Large Hadron Collider, whose performance at collision is mainly limited due to the strong beam-beam long-range interactions, are studied in the presence of dc wire compensators. This analysis is based on analytical and numerical calculations where the main observables are the dynamic aperture (correlated to the beam lifetime) and the frequency map analysis. It is demonstrated that, with a proper optimization of the wire compensator parameters (distance from the beam and wire current) and without violating the machine protection restrictions, these long range beam-beam interactions can be very well mitigated, making these scenarios viable and complementary with respect to the present HL-LHC baseline. The impact in the integrated luminosity and the operational flexibility gained are also presented.

Investigation of Electrical Tree Growth of XLPE Nano Composites using Time–Frequency Map and Clustering Analysis of PD Signals

Failure of underground cables due to electrical treeing phenomena, i.e. formation of electrical discharges in the imperfections of cable insulation, is a major problem faced by electrical utilities. Many research works on nanocomposites are being carried out to improve the electrical treeing resistance of the XLPE cable insulation material. Technological advancements in communication and data analytic systems have shown the way for implementing online continuous partial discharge (PD) monitoring systems for high voltage apparatus. Hence collection of PD database of XLPE nanocomposites in the laboratory during entire electrical tree growth process is important for implementing efficient condition monitoring systems and relatively little work has been published in this area. In this work, PD characteristics of XLPE nanocomposites with 1, 3, 5 and 10 wt% silica were investigated. Electrical tree growth and corresponding phase resolved PD (PRPD) pattern were analysed with respect to time. Cluster analysis of equivalent time–frequency mapping of PD signals was carried out with respect to tree growth time period. Statistical analysis was performed for the entire set of PD data. Results show that cluster analysis of T–F map of PD data is useful in estimating early failure of insulating material due to treeing. Addition of silica nano fillers in the range of 3–5 wt% concentration significantly improves the PD resistance and breakdown time of XLPE material.

Rapidly converging chaos indicator for studying dynamic aperture in a storage ring with space charge

The determination of dynamic aperture in storage rings and colliders is a numerically intensive procedure. When realistic space-charge forces come into consideration, the numerical load becomes even heavier. Furthermore, dynamic aperture estimation using chaos indicators like frequency map analysis (FMA) raises reliability issues when the dynamical system has a time-dependent perturbation like the space-charge force. In this article, we apply a rapidly converging chaos indicator called reversibility error method (REM) to study the space-charge contribution to the dynamic aperture of the integrable optics test accelerator (IOTA) storage ring at a small value of the space charge tune shift. The strength of REM is addressed through examples, including a particle-core model of halo formation. We also develop a toy model of the IOTA lattice to further reduce the computing time required to estimate the dynamic aperture, and we compare this model with a realistic space-charge simulation for verification.

Optimization and Measurements on the Double Mini-Betay Lattice in the TPS Storage Ring

The Taiwan Photon Source (TPS) is capable of operating in multi-bunch and single-bunch mode. To operate in a hybrid mode as requested by users, we developed a lattice with chromaticities of 0.8/2 (ξx/ξy) to provide higher single bunch currents. Beam dynamics simulations and lattice characterizations including dynamic aperture, frequency map analysis, tune shift with energy, tune shift with amplitude, and betatron coupling will be discussed in this report.

Optimization and Measurements on the Double Mini-Betay Lattice in the TPS Storage Ring

The Taiwan Photon Source (TPS) is capable of operat-ing in multi-bunch and single-bunch mode. To operate in a hybrid mode as requested by users, we developed a lattice with chromaticities of 0.8/2 (x/y) to provide higher single bunch currents. Beam dynamics simulations and lattice characterizations including dynamic aperture, frequency map analysis, tune shift with energy, tune shift with amplitude, and betatron coupling will be discussed in this report.

Characterizing and studying the nonlinear beam dynamics performance of Iranian Light Source Facility storage ring

In this paper, we will investigate the nonlinear beam dynamics issues (performance and optimization) of Iranian Light Source Facility storage ring. The strong sextupole fields together with the imperfections in a low emittance machine will result in the large detuning effects and driving the different order of the resonance lines. We have used most recently achievement for the nonlinear optimization of the storage ring. We have optimized the nonlinearity of the lattice by analyzing the resonance driving terms, direct dynamic aperture tracking and frequency map analysis calculations. We have evaluated the performance of the lattice against different imperfections such as magnet misalignments, multipole harmonics, fringe field of the quadrupoles and insertion devices. Furthermore, the effect of the lattice imperfections on the off-momentum particles dynamics has been studied through momentum acceptance and lifetime calculations.

Modeling of nonlinear effects due to head-on beam-beam interactions

The beam-beam interaction is one of the most severe limitations on the performance of circular colliders, as it is an unavoidable strong nonlinear effect. As one aspires for greater luminosity in future colliders, one will simultaneously achieve stronger beam-beam interactions. We study the limitations caused by strong incoherent head-on beam-beam interactions, using a new code (cabin) that calculates on a graphics processing unit (GPU), allowing for a detailed description of the long-term particle trajectories in 6D phase space. The evolution of the beam emittance and beam intensity has been monitored to study the impact quantitatively, while frequency map analysis has been performed to understand the impact qualitatively. Results from cabin have shown good quantitative agreement with dedicated experiments in the Large Hadron Collider (LHC). For large beam-beam tune shifts, alternatives to the LHC tunes have been found to improve the beam quality. Schemes devised to cancel beam-beam driven resonances, by use of specific intermediate phase advances between the interaction points, work very well with zero crossing angle, and the accuracy required is achievable. Due to lack of symmetry, these schemes have an almost negligible impact with a significant crossing angle. The hourglass effect has been found to reduce the detrimental effects caused by the chromaticity and vice versa. The optimal level of the hourglass effect has been achieved when ${\ensuremath{\beta}}^{*}=1.5{\ensuremath{\sigma}}_{s}$. The ultimate parameters of the Future Circular Hadron Collider (FCC-hh) seem within reach, in absence of residual odd resonances.

Single-particle dynamics in a nonlinear accelerator lattice: attaining a large tune spread with octupoles in IOTA

Fermilab is constructing the Integrable Optics Test Accelerator (IOTA) as the centerpiece of the Accelerator R&D Program towards high-intensity circular machines. One of the factors limiting the beam intensity in present circular accelerators is collective instabilities, which can be suppressed by a spread of betatron frequencies (tunes) through the Landau damping mechanism or by an external damper, if the instability is slow enough. The spread is usually created by octupole magnets, which introduce the tune dependence on the amplitude and, in some cases, by a chromatic spread (tune dependence on particle's momentum). The introduction of octupoles usually has both beneficial (improved Landau damping) and harmful properties, such as a resonant behavior and a reduction of the dynamic aperture. One of the research goals at the IOTA ring is to achieve a large betatron tune spread, while retaining a large dynamic aperture, using conventional octupole magnets in a special but realistic accelerator configuration. The configuration, although not integrable by design, approximates an autonomous 2D Hamiltonian system. In this paper, we present results of computer simulations of an electron beam in the IOTA by particle tracking and the Frequency Map Analysis. The results show that the ring's octupole magnets can be configured to provide a betatron tune shift of 0.08 (for particles at large amplitudes) with the dynamical aperture of over 20 beam sigma for a 150-MeV electron beam. The influence of the synchrotron motion, lattice errors, and magnet imperfections is insignificant for the parameters and levels of tolerances set by the design of the ring. The described octupole insert could be beneficial for enhancing Landau damping in high intensity machines.

Stability of multiplanet systems in binaries

When exploring the stability of multiplanet systems in binaries, two parameters are normally exploited: the critical semimajor axis ac computed by Holman and Wiegert (1999) within which planets are stable against the binary perturbations, and the Hill stability limit Delta determining the minimum separation beyond which two planets will avoid mutual close encounters. Our aim is to test whether these two parameters can be safely applied in multiplanet systems in binaries or if their predictions fail for particular binary orbital configurations. We have used the frequency map analysis (FMA) to measure the diffusion of orbits in the phase space as an indicator of chaotic behaviour. First we revisited the reliability of the empirical formula computing ac in the case of single planets in binaries and we find that, in some cases, it underestimates by 10-20% the real outer limit of stability. For two planet systems, the value of Delta is close to that computed for planets around single stars, but the level of chaoticity close to it substantially increases for smaller semimajor axes and higher eccentricities of the binary orbit. In these configurations ac also begins to be unreliable and non linear secular resonances with the stellar companion lead to chaotic behaviour well within ac, even for single planet systems. For two planet systems, the superposition of mean motion resonances, either mutual or with the binary companion, and non linear secular resonances may lead to chaotic behaviour in all cases. We have developed a parametric semiempirical formula determining the minimum value of the binary semimajor axis, for a given eccentricity of the binary orbit, below which stable two planet systems cannot exist.

Effects of an elliptically polarizing undulator on the beam dynamics in PLS-II

The non-linear effects caused by the intrinsic field transverse roll-off in an Elliptically Polarizing Undulator with 72-mm period and maximum peak field of 0.72 T (EPU72) on the dynamics aperture of the Pohang Light Source II (PLS-II) were investigated. A kick map and Frequency map analysis both demonstrated that EPU72 will not reduce the lifetime or cause injection problems in PLS-II.

Optimization of the multi-turn injection efficiency for a medical synchrotron

We present a method for optimization of the multi-turn injection effciency for medical synchrotron. We show that for given injection energy the injection efficiency can be greatly enhanced by choosing transverse tunes appropriately as well as optimizing the injection bump and the number of turns required for beam injection. We verify our study by applying the method to the Korea Heavy Ion Medical Accelerator (KHIMA) synchrotron which is currently built at the campus of Dongnam Institute of Radiological and Medical Sciences (DIRAMS) in Busan, Korea. First the frequency map analysis is performed with the help of ELEGANT and ACCSIM codes. The tunes which yield the good injection efficiency are then selected. With these tunes the injection bump and the number of turns required for injection are then optimized by tracking a number of particles up to one thousand turns after injection beyond which there is no further beam loss. Results for optimization of the injection efficiency for proton ion are presented.

Frequency analysis of the non-principal-axis rotation of uniaxial space debris in circular orbit subjected to gravity-gradient torque

The non-principal-axis rotational motion of uniaxial space debris can be decomposed into periodic motions associated with two frequencies: the polhode frequency of the space debris rotating around the symmetry axis, and the tumbling frequency of the symmetry axis rotating around the angular momentum. To determine from optical measurements the rotational motion of upper rocket stages in circular orbits subjected to gravity-gradient torque, the evolutions of these two frequencies need to be analyzed. Taking into account only the long-term changes in the long-period variables, the differential equations of the non-principal axis rotational motion of the uniaxial space debris are averaged and reduced, from which the evolutions of the polhode and tumbling frequencies are then obtained analytically. The theoretical results are verified by numerical simulations of the diffuse reflection model. The frequencies in the variation of the reflected light intensity in the simulation are analyzed using the frequency map analysis (FMA) method. Errors of these results are found to be less than 1%. Based on the theoretical expressions, the rotational state of the uniaxial space debris can be estimated in the simulation without any prior information except the orbital parameters. A series of state variables are estimated, including the ratio of the moments of inertia about the transverse axis and the symmetry axis, the instantaneous rotation velocity, the orientation of the angular momentum, and the precession cone of the symmetry axis.

Quasi-periodic motions in a special class of dynamical equations with dissipative effects: A pair of detection methods

We consider a particular class of equations of motion, generalizing to $n$ degrees of freedom the ``dissipative spin--orbit problem'', commonly studied in Celestial Mechanics. Those equations are formulated in a pseudo-Hamiltonian framework with action-angle coordinates; they contain a quasi-integrable conservative part and friction terms, assumed to be linear and isotropic with respect to the action variables. In such a context, we transfer two methods determining quasi-periodic solutions, which were originally designed to analyze purely Hamiltonian quasi-integrable problems. First, we show how the frequency map analysis can be adapted to this kind of dissipative models. Our approach is based on a key remark: the method can work as usual, by studying the behavior of the angular velocities of the motions as a function of the so called ``external frequencies'', instead of the actions. Moreover, we explicitly implement the Kolmogorov's normalization algorithm for the dissipative systems considered here. In a previous article, we proved a theoretical result: such a constructing procedure is convergent under the hypotheses usually assumed in KAM theory. In the present work, we show that it can be translated to a code making algebraic manipulations on a computer, so to calculate effectively quasi-periodic solutions on invariant tori (and the attracting dynamics in their neighborhoods). Both the methods are carefully tested, by checking that their predictions are in agreement, in the case of the so called ``dissipative forced pendulum''. Furthermore, the results obtained by applying our adaptation of the frequency analysis method to the dissipative standard map are compared with some existing ones in the literature.

Dynamical behaviour of multiplanet systems close to their stability limit

The dynamics of systems of two and three planets, initially placed on circular and nearly coplanar orbits, is explored in the proximity of their stability limit. The evolution of a large number of systems is numerically computed and their dynamical behaviour is investigated with the frequency map analysis as chaos indicator. Following the guidance of this analysis, it is found that for two-planet systems the dependence of the Hill limit on the planet mass, usually made explicit through the Hill's radius parametrization, does not appear to be fully adequate. In addition, frequent cases of stable chaos are found in the proximity of the Hill limit. For three-planet systems, the usual approach adopted in numerical explorations of their stability, where the planets are initially separated by multiples of the mutual Hill radius, appears too reducing. A detailed sampling of the parameter space reveals that systems with more packed inner planets are stable well within previous estimates of the stability limit. This suggests that a two-dimensional approach is needed to outline when three-planet systems are dynamically stable.

Detecting chaos in particle accelerators through the frequency map analysis method.

The motion of beams in particle accelerators is dominated by a plethora of non-linear effects, which can enhance chaotic motion and limit their performance. The application of advanced non-linear dynamics methods for detecting and correcting these effects and thereby increasing the region of beam stability plays an essential role during the accelerator design phase but also their operation. After describing the nature of non-linear effects and their impact on performance parameters of different particle accelerator categories, the theory of non-linear particle motion is outlined. The recent developments on the methods employed for the analysis of chaotic beam motion are detailed. In particular, the ability of the frequency map analysis method to detect chaotic motion and guide the correction of non-linear effects is demonstrated in particle tracking simulations but also experimental data.

The effect of insertion devices on the beam dynamics in PLS-II

The effects of an insertion device (ID) on the beam dynamics of the PLS-II storage ring have been investigated quantitatively. Although the linear perturbations caused by high field wigglers, such as tune shift and beta-beating, are corrected by using iterative Twiss-matching, particles may be unstable in the transverse planes due to nonlinear ID fields. Intrinsic non-linear fields and transverse magnetic field roll off cause ID field-perturbations making the beam unstable and possibly causing a reduction in the injection efficiency and the beam’s lifetime. Therefore, a frequency map analysis is used to analyze the transverse dynamics with high field IDs. In order to analyze the transverse dynamics by using frequency map analysis, we model a three dimensional magnetic field based on numerical data with a Taylor expansion to accurately represent the actual field. As a result, new operation tunes, 15.245 and 9.18, are suggested rather than the designed tunes of 15.28 and 9.18. The new tunes are used for Twiss-matching and require several iterations to reduce both the horizontal and the vertical beta beatings. In this paper, the effects of an insertion device and their compensations will be presented.