arXiv subjects

James D. Meiss

Publications and source records attributed to James D. Meiss.

34 records · Page 2Linked to original sources

arXiv2015

Exploring the Topology of Dynamical Reconstructions

Computing the state-space topology of a dynamical system from scalar data requires accurate reconstruction of those dynamics and construction of an appropriate simplicial complex from the results. The reconstruction process involves a number of free parameters and the computation of homology for a large number of simplices can be expensive. This paper is a study of how to avoid a full (diffeomorphic) reconstruction and how to decrease the computational burden. Using trajectories from the classic Lorenz system, we reconstruct the dynamics using the method of delays, then build a parsimonious simplicial complex---the "witness complex"---to compute its homology. Surprisingly, we find that the witness complex correctly resolves the homology of the underlying invariant set from noisy samples of that set even if the reconstruction dimension is well below the thresholds specified in the embedding theorems for assuring topological conjugacy between the true and reconstructed dynamics. We conjecture that this is because the requirements for reconstructing homology, are less stringent than those in the embedding theorems. In particular, to faithfully reconstruct the homology, a homeomorphism is sufficient---as opposed to a diffeomorphism, as is necessary for the full dynamics. We provide preliminary evidence that a homeomorphism, in the form of a delay-coordinate reconstruction map, may manifest at a lower dimension than that required to achieve an embedding.

math.DS↗

arXiv2015

The Hamiltonian Mean Field model: effect of network structure on synchronization dynamics

The Hamiltonian Mean Field (HMF) model of coupled inertial, Hamiltonian rotors is a prototype for conservative dynamics in systems with long-range interactions. We consider the case where the interactions between the rotors are governed by a network described by a weighted adjacency matrix. By studying the linear stability of the incoherent state, we find that the transition to synchrony occurs at a coupling constant $K$ inversely proportional to the largest eigenvalue of the adjacency matrix. We derive a closed system of equations for a set of local order parameters and use these equations to study the effect of network heterogeneity on the synchronization of the rotors. We find that for values of $K$ just beyond the transition to synchronization the degree of synchronization is highly dependent on the network's heterogeneity, but that for large values of $K$ the degree of synchronization is robust to changes in the heterogeneity of the network's degree distribution. Our results are illustrated with numerical simulations on Erdös-Renyi networks and networks with power-law degree distributions.

cond-mat.stat-mech↗

arXiv2015

Simplicial Multivalued Maps and the Witness Complex for Dynamical Analysis of Time Series

Topology based analysis of time-series data from dynamical systems is powerful: it potentially allows for computer-based proofs of the existence of various classes of regular and chaotic invariant sets for high-dimensional dynamics. Standard methods are based on a cubical discretization of the dynamics and use the time series to construct an outer approximation of the underlying dynamical system. The resulting multivalued map can be used to compute the Conley index of isolated invariant sets of cubes. In this paper we introduce a discretization that uses instead a simplicial complex constructed from a witness-landmark relationship. The goal is to obtain a natural discretization that is more tightly connected with the invariant density of the time series itself. The time-ordering of the data also directly leads to a map on this simplicial complex that we call the witness map. We obtain conditions under which this witness map gives an outer approximation of the dynamics, and thus can be used to compute the Conley index of isolated invariant sets. The method is illustrated by a simple example using data from the classical Hénon map.

nlin.CD↗

arXiv2014

Statistics of the Island-Around-Island Hierarchy in Hamiltonian Phase Space

The phase space of a typical Hamiltonian system contains both chaotic and regular orbits, mixed in a complex, fractal pattern. One oft-studied phenomenon is the algebraic decay of correlations and recurrence time distributions. For area-preserving maps, this has been attributed to the stickiness of boundary circles, which separate chaotic and regular components. Though such dynamics has been extensively studied, a full understanding depends on many fine details that typically are beyond experimental and numerical resolution. This calls for a statistical approach, the subject of the present work. We calculate the statistics of the boundary circle winding numbers, contrasting the distribution of the elements of their continued fractions to that for uniformly selected irrationals. Since phase space transport is of great interest for dynamics, we compute the distributions of fluxes through island chains. Analytical fits show that the "level" and "class" distributions are distinct, and evidence for their universality is given.

nlin.CD↗

arXiv2014

Onset of Synchronization in the Disordered Hamiltonian Mean Field Model

We study the Hamiltonian Mean Field (HMF) model of coupled Hamiltonian rotors with a heterogeneous distribution of moments of inertia and coupling strengths. We show that when the parameters of the rotors are heterogeneous, finite size fluctuations can greatly modify the coupling strength at which the incoherent state loses stability by inducing correlations between the momenta and parameters of the rotors. When the distribution of initial frequencies of the oscillators is sufficiently narrow, an analytical expression for the modification in critical coupling strength is obtained that confirms numerical simulations. We find that heterogeneity in the moments of inertia tends to stabilize the incoherent state, while heterogeneity in the coupling strengths tends to destabilize the incoherent state. Numerical simulations show that these effects disappear for a wide, bimodal frequency distribution.

nlin.CD↗

arXiv2013

Critical Invariant Circles in Asymmetric and Multiharmonic Generalized Standard Maps

Invariant circles play an important role as barriers to transport in the dynamics of area-preserving maps. KAM theory guarantees the persistence of some circles for near-integrable maps, but far from the integrable case all circles can be destroyed. A standard method for determining the existence or nonexistence of a circle, Greene's residue criterion, requires the computation of long-period orbits, which can be difficult if the map has no reversing symmetry. We use de la Llave's quasi-Newton, Fourier-based scheme to numerically compute the conjugacy of a Diophantine circle conjugate to rigid rotation, and the singularity of a norm of a derivative of the conjugacy to predict criticality. We study near-critical conjugacies for families of rotational invariant circles in generalizations of Chirikov's standard map. A first goal is to obtain evidence to support the long-standing conjecture that when circles breakup they form cantori, as is known for twist maps by Aubry-Mather theory. The location of the largest gaps is compared to the maxima of the potential when anti-integrable theory applies. A second goal is to support the conjecture that locally most robust circles have noble rotation numbers, even when the map is not reversible. We show that relative robustness varies inversely with the discriminant for rotation numbers in quadratic algebraic fields. Finally, we observe that the rotation number of the globally most robust circle generically appears to be a piecewise-constant function in two-parameter families of maps.

nlin.CD↗

arXiv2012

Greene's Residue Criterion for the Breakup of Invariant Tori of Volume-Preserving Maps

Invariant tori play a fundamental role in the dynamics of symplectic and volume-preserving maps. Codimension-one tori are particularly important as they form barriers to transport. Such tori foliate the phase space of integrable, volume-preserving maps with one action and $d$ angles. For the area-preserving case, Greene's residue criterion is often used to predict the destruction of tori from the properties of nearby periodic orbits. Even though KAM theory applies to the three-dimensional case, the robustness of tori in such systems is still poorly understood. We study a three-dimensional, reversible, volume-preserving analogue of Chirikov's standard map with one action and two angles. We investigate the preservation and destruction of tori under perturbation by computing the "residue" of nearby periodic orbits. We find tori with Diophantine rotation vectors in the "spiral mean" cubic algebraic field. The residue is used to generate the critical function of the map and find a candidate for the most robust torus.

nlin.CD↗

arXiv2011

Iterated Function System Models in Data Analysis: Detection and Separation

We investigate the use of iterated function system (IFS) models for data analysis. An IFS is a discrete dynamical system in which each time step corresponds to the application of one of a finite collection of maps. The maps, which represent distinct dynamical regimes, may act in some pre-determined sequence or may be applied in random order. An algorithm is developed to detect the sequence of regime switches under the assumption of continuity. This method is tested on a simple IFS and applied to an experimental computer performance data set. This methodology has a wide range of potential uses: from change-point detection in time-series data to the field of digital communications.

math.DS↗

arXiv2009

Simultaneous Border-Collision and Period-Doubling Bifurcations

We unfold the codimension-two simultaneous occurrence of a border-collision bifurcation and a period-doubling bifurcation for a general piecewise-smooth, continuous map. We find that, with sufficient non-degeneracy conditions, a locus of period-doubling bifurcations emanates non-tangentially from a locus of border-collision bifurcations. The corresponding period-doubled solution undergoes a border-collision bifurcation along a curve emanating from the codimension-two point and tangent to the period-doubling locus here. In the case that the map is one-dimensional local dynamics are completely classified; in particular, we give conditions that ensure chaos.

math.DS↗

arXiv2008

Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations

We study the dynamics of the five-parameter quadratic family of volume-preserving diffeomorphisms of R^3. This family is the unfolded normal form for a bifurcation of a fixed point with a triple-one multiplier and also is the general form of a quadratic three-dimensional map with a quadratic inverse. Much of the nontrivial dynamics of this map occurs when its two fixed points are saddle-foci with intersecting two-dimensional stable and unstable manifolds that bound a spherical ``vortex-bubble''. We show that this occurs near a saddle-center-Neimark-Sacker (SCNS) bifurcation that also creates, at least in its normal form, an elliptic invariant circle. We develop a simple algorithm to accurately compute these elliptic invariant circles and their longitudinal and transverse rotation numbers and use it to study their bifurcations, classifying them by the resonances between the rotation numbers. In particular, rational values of the longitudinal rotation number are shown to give rise to a string of pearls that creates multiple copies of the original spherical structure for an iterate of the map.

nlin.CD↗

arXiv2008

Chaotic Advection and the Emergence of Tori in the Küppers-Lortz State

Motivated by the roll-switching behavior observed in rotating Rayleigh-Bénard convection, we define a Küppers-Lortz (K-L) state as a volume-preserving flow with periodic roll switching. For an individual roll state, the Lagrangian particle trajectories are periodic. In a system with roll-switching, the particles can exhibit three-dimensional, chaotic motion. We study a simple phenomenological map that models the Lagrangian dynamics in a K-L state. When the roll axes differ by $120^{\circ}$ in the plane of rotation, we show that the phase space is dominated by invariant tori if the ratio of switching time to roll turnover time is small. When this parameter approaches zero these tori limit onto the classical hexagonal convection patterns, and, as it gets large, the dynamics becomes fully chaotic and well-mixed. For intermediate values, there are interlinked toroidal and poloidal structures separated by chaotic regions. We also compute the exit time distributions and show that the unbounded chaotic orbits are normally diffusive. Although the map presumes instantaneous switching between roll states, we show that the qualitative features of the flow persist when the model has smooth, overlapping time-dependence for the roll amplitudes (the Busse-Heikes model).

nlin.CD↗

arXiv2005

Numerical Computation of the Stable and Unstable Manifolds of Invariant Tori

We develop an iterative technique for computing the unstable and stable eigenfunctions of the invariant tori of diffeomorphisms. Using the approach of Jorba, the linearized equations are rewritten as a generalized eigenvalue problem. Casting the system in this light allows us to take advantage of the speed of eigenvalue solvers and create an efficient method for finding the first order approximations to the invariant manifolds of the torus. We present a numerical scheme based on the power method that can be used to determine the behavior normal to such tori, and give some examples of the application of the method. We confirm the qualitative conclusions of the Melnikov calculations of Lomelí and Meiss (2003) for a volume-preserving mapping.

nlin.CD↗

arXiv2004

Symbolic Codes for Rotational Orbits

Symbolic codes for rotational orbits and "islands-around-islands" are constructed for the quadratic, area-preserving Henon map. The codes are based upon continuation from an anti-integrable limit, or alternatively from the horseshoe. Given any sequence of rotation numbers we obtain symbolic sequences for the corresponding elliptic and hyperbolic rotational orbits. These are shown to be consistent with numerical evidence. The resulting symbolic partition of the phase space consists of wedges constructed from images of the symmetry lines of the map.

nlin.CD↗

arXiv1999

Heteroclinic orbits and transport in a perturbed integrable Suris map

Explicit formulae are given for the saddle connection of an integrable family of standard maps studied by Y. Suris. When the map is perturbed this connection is destroyed, and we use a discrete version of Melnikov's method to give an explicit formula for the first order approximation of the area of the lobes of the resultant turnstile. These results are compared with computations of the lobe area.

chao-dyn↗

arXiv1997

Quadratic Volume Preserving Maps

We study quadratic, volume preserving diffeomorphisms whose inverse is also quadratic. Such maps generalize the Henon area preserving map and the family of symplectic quadratic maps studied by Moser. In particular, we investigate a family of quadratic volume preserving maps in three space for which we find a normal form and study invariant sets. We also give an alternative proof of a theorem by Moser classifying quadratic symplectic maps.

chao-dyn↗

arXiv1996

Average exit times in volume preserving maps

For a volume preserving map, the exit time, averaged over the incoming set of a region, is given by the ratio of the measure of the accessible subset of the region to that of the incoming set. This result is primarily of interest to show two things: first it gives a simple bound on the algebraic decay exponent of the survival probability. Second, it gives a tool which permits the computation of the measure of the accessible set. We use this to find the measure of the bounded orbits for Hénon's quadratic map.

math.DS↗