arXiv subjects

Malcolm Hillebrand

Publications and source records attributed to Malcolm Hillebrand.

13 recordsLinked to original sources

arXiv2026

On the efficient numerical computation of covariant Lyapunov vectors

Covariant Lyapunov vectors (CLVs) are useful in multiple applications, but the optimal time windows needed to accurately compute these vectors are yet unclear. To remedy this, we investigate two methods for determining when to safely terminate the forward and backward transient phases of the CLV computation algorithm by Ginelli et al.~\cite{GinelliEtAl2007} when applied to chaotic orbits of conservative Hamiltonian systems. We perform this investigation for two prototypical Hamiltonian systems, namely the well-known Hénon-Heiles system of two degrees of freedom and a system of three nonlinearly coupled harmonic oscillators having three degrees of freedom, finding very similar results for the two methods and thus recommending the more efficient one. We find that the accuracy of two-dimensional center subspace computations is significantly reduced when the backward evolution stages of the algorithm are performed over long time intervals. We explain this observation by examining the tangent dynamics of the center subspace wherein CLVs tend to align/anti-align, and we propose an adaptation of the algorithm that improves the accuracy of such computations over long times by preventing this alignment/anti-alignment of CLVs in the center subspace.

nlin.CD↗

arXiv2026

Discontinuous transition to active nematic turbulence

Active fluids exhibit chaotic flows at low Reynolds number known as active turbulence. Whereas the statistical properties of the chaotic flows are increasingly well understood, the nature of the transition from laminar to turbulent flows as activity increases remains unclear. Here, through simulations of a minimal model of unbounded and defect-free active nematics, we find that the transition to active turbulence is discontinuous. We show that the transition features a jump in the mean-squared velocity, as well as bistability and hysteresis between laminar and chaotic flows. From distributions of finite-time Lyapunov exponents, we identify the transition at a value $A^*\approx 4900$ of the dimensionless activity number. Below the transition to chaos, we find subcritical bifurcations that feature bistability of different laminar patterns. These bifurcations give rise to oscillations and to chaotic transients, which become very long close to the transition to turbulence. Overall, our findings contrast with the continuous transition to turbulence in channel confinement, where turbulent puffs emerge within a laminar background. We propose that, without confinement, the long-range hydrodynamic interactions of Stokes flow suppress the spatial coexistence of different flow states, and thus render the transition discontinuous.

cond-mat.soft↗

arXiv2025

Efficient detection of chaos through the computation of the Generalized Alignment Index (GALI) by the multi-particle method

We present a method for the computation of the Generalized Alignment Index (GALI), a fast and effective chaos indicator, using a multi-particle approach that avoids variational equations. We show that this approach is robust and accurate by deriving a leading-order error estimation for both the variational (VM) and the multi-particle (MPM) methods, which we validate by performing extensive numerical simulations on two prototypical models: the two degrees of freedom Hénon-Heiles system and the multidimensional $β$-Fermi-Pasta-Ulam-Tsingou chain of oscillators. The dependence of the accuracy of the GALI on control parameters such as the renormalization time, the integration time step and the deviation vector size is studied in detail. We test the MPM implemented with double precision accuracy ($\varepsilon \approx 10^{-16}$) and find that it performs reliably for deviation vector sizes $d_0\approx \varepsilon^{1/2}$, renormalization times $τ\lesssim 1$, and relative energy errors $E_r \lesssim \varepsilon^{1/2}$. These results hold for systems with many degrees of freedom and demonstrate that the MPM is a robust and efficient method for studying the chaotic dynamics of Hamiltonian systems. Our work makes it possible to explore chaotic dynamics with the GALI in a vast number of systems by eliminating the need for variational equations.

nlin.CD↗

arXiv2024

The evolution of the phase space structure along pitchfork and period-doubling bifurcations in a 3D galactic bar potential

We investigate how the phase space structure of a 3D autonomous Hamiltonian system evolves across a series of successive 2D and 3D pitchfork and period-doubling bifurcations, as the transition of the parent families of periodic orbits (POs) from stability to simple instability leads to the creation of new stable POs. Our research illustrates the consecutive alterations in the phase space structure near POs as the stability of the main family of POs changes. This process gives rise to new families of POs within the system, either maintaining the same or exhibiting higher multiplicity compared to their parent families. Tracking such a phase space transformation is challenging in a 3D system. By utilizing the color and rotation technique to visualize the 4D Poincar'e surfaces of section of the system, i.e. projecting them onto a 3D subspace and employing color to represent the fourth dimension, we can identify distinct structural patterns. Perturbations of parent and bifurcating stable POs result in the creation of tori characterized by a smooth color variation on their surface. Furthermore, perturbations of simple unstable parent POs beyond the bifurcation point which lead to the birth of new stable families of POs, result in the formation of figure-8 structures of smooth color variations. These figure-8 formations surround well-shaped tori around the bifurcated stable POs, losing their well-defined forms for energies further away from the bifurcation point. We also observe that even slight perturbations of highly unstable POs create a cloud of mixed color points, which rapidly move away from the location of the PO. Our study introduces, for the first time, a systematic visualization of 4D surfaces of section within the vicinity of higher multiplicity POs. It elucidates how, in these cases, the coexistence of regular and chaotic orbits contributes to shaping the phase space landscape.

nlin.CD↗

arXiv2023

Performance of chaos diagnostics based on Lagrangian descriptors. Application to the 4D standard map

We investigate the ability of simple diagnostics based on Lagrangian descriptor (LD) computations of initially nearby orbits to detect chaos in conservative dynamical systems with phase space dimensionality higher than two. In particular, we consider the recently introduced methods of the difference ($D_L^n$) and the ratio ($R_L^n$) of the LDs of neighboring orbits, as well as a quantity ($S_L^n$) related to the finite-difference second spatial derivative of the LDs, and use them to determine the chaotic or regular nature of ensembles of orbits of a prototypical area-preserving map model, the 4-dimensional (4D) symplectic standard map. Using the distributions of the indices' values we determine appropriate thresholds to discriminate between regular and chaotic orbits, and compare the obtained characterization against that achieved by the Smaller Alignment Index (SALI) method of chaos detection, by recording the percentage agreement $P_A$ between the two classifications. We study the influence of various factors on the performance of these indices, and show that the increase of the final number of orbit iterations T and the order n of the indices (i.e. the dimensionality of the space where the considered nearby orbits lie), as well as the decrease of the distance $σ$ of neighboring orbits, increase the $P_A$ values along with the required computational effort. Balancing between these two factors we find appropriate T, n and $σ$ values, which allow the efficient use of the $D_L^n$, $R_L^n$ and $S_L^n$ indices as short time and computationally cheap chaos diagnostics achieving $P_A \gtrsim 90 \%$, with $D_L^n$ and $S_L^n$ having larger $P_A$ values than $R_L^n$. Our results show that the three LDs-based indices perform better for systems with large percentages of chaotic orbits.

astro-ph.EP↗

arXiv2023

Navigating Phase Space Transport with the Origin-Fate Map

We introduce and demonstrate the usage of the origin-fate map (OFM) as a tool for the detailed investigation of phase space transport in reactant-product type systems. For these systems, which exhibit clearly defined start and end states, it is possible to build a comprehensive picture of the lobe dynamics by considering backward and forward integration of sets of initial conditions to index their origin and fate. We illustrate the method and its utility in the study of a two degrees of freedom caldera potential with four exits, demonstrating that the OFM not only recapitulates results from classical manifold theory, but even provides more detailed information about complex lobe structures. The OFM allows the detection of dynamically significant transitions caused by the creation of new lobes, and is also able to guide the prediction of the position of unstable periodic orbits (UPOs). Further, we compute the OFM on the periodic orbit dividing surface (PODS) associated with the transition state of a caldera entrance, which allows for a powerful analysis of reactive trajectories. The intersection of the manifolds corresponding to this UPO with other manifolds in the phase space results in the appearance of lobes on the PODS, which are directly classified by the OFM. This allows computations of branching ratios and the exploration of a fractal cascade of lobes as the caldera is stretched, which results in fluctuations in the branching ratio and chaotic selectivity. The OFM is found to be a simple and very useful tool with a vast range of descriptive and quantitative applications.

math.DS↗

arXiv2022

Quantifying chaos using Lagrangian descriptors

We present and validate simple and efficient methods to estimate the chaoticity of orbits in low dimensional dynamical systems from computations of Lagrangian descriptors (LDs) on short time scales. Two quantities are proposed for determining the chaotic or regular nature of orbits in a system's phase space, which are based on the values of the LDs of these orbits and of nearby ones: The difference (DNLD) and ratio (RNLD) of neighboring orbits' LDs. We find that, typically, these indicators are able to correctly characterize the chaotic or regular nature of orbits to better than 90 % agreement with results obtained by implementing the Smaller Alignment Index (SALI) method, which is a well established chaos detection technique. Further investigating the performance of the two introduced quantities we discuss the effects of the total integration time and of the spacing between the used neighboring orbits on the accuracy of the methods, finding that even typical short time, coarse grid LD computations are sufficient to provide a reliable quantification of a system's chaotic component, using less CPU time than the SALI. In addition to quantifying chaos, the introduced indicators have the ability to reveal details about the systems' local and global chaotic phase space structure. Our findings clearly suggest that LDs can also be used to quantify and investigate chaos in continuous and discrete low dimensional dynamical systems.

nlin.CD↗

arXiv2022

Bubble relaxation dynamics in homopolymer DNA sequences

Understanding the inherent timescales of large bubbles in DNA is critical to a thorough comprehension of its physicochemical characteristics, as well as their potential role on helix opening and biological function. In this work we employ the coarse-grained Peyrard-Bishop-Dauxois model of DNA to study relaxation dynamics of large bubbles in homopolymer DNA, using simulations up to the microsecond time scale. By studying energy autocorrelation functions of relatively large bubbles inserted into thermalised DNA molecules, we extract characteristic relaxation times from the equilibration process for both adenine-thymine (AT) and guanine-cytosine (GC) homopolymers. Bubbles of different amplitudes and widths are investigated through extensive statistics and appropriate fittings of their relaxation. Characteristic relaxation times increase with bubble height and width. We show that, within the model, relaxation times are two orders of magnitude longer in GC sequences than in AT sequences. Overall, our results confirm that large bubbles leave a lasting impact on the molecule's dynamics, for times between 0.4-200ns depending on the homopolymer type and bubble shape, thus clearly affecting long-time evolutions of the molecule.

physics.bio-ph↗

arXiv2021

Chaotic Dynamics of Polyatomic Systems with an Emphasis on DNA Models

We investigate the chaotic behaviour of multiparticle systems, in particular DNA and graphene models, by applying methods of nonlinear dynamics. Using symplectic integration techniques, we present an extensive analysis of chaos in the Peyrard-Bishop-Dauxois (PBD) model of DNA. The chaoticity is quantified by the maximum Lyapunov exponent (mLE) across a spectrum of temperatures, and the effect of base pair (BP) disorder on the dynamics is studied. In addition to heterogeneity due to the ratio of adenine-thymine (AT) and guanine-cytosine (GC) BPs, the distribution of BPs in the sequence is analysed by introducing the alternation index $α$. An exact probability distribution for BP arrangements and $α$ is derived using Pólya counting. The value of the mLE depends on the composition and arrangement of BPs in the strand, with a dependence on temperature. We probe regions of strong chaoticity using the deviation vector distribution, studying links between strongly nonlinear behaviour and the formation of bubbles. Randomly generated sequences and biological promoters are both studied. Further, properties of bubbles are analysed through molecular dynamics simulations. The distributions of bubble lifetimes and lengths are obtained, fitted with analytical expressions, and a physically justified threshold for considering a BP to be open is successfully implemented. In addition to DNA, we present analysis of the dynamical stability of a planar model of graphene, studying the mLE in bulk graphene as well as in graphene nanoribbons (GNRs). The stability of the material manifests in a very small mLE, with chaos being a slow process in graphene. For both armchair and zigzag edge GNRs, the mLE decreases with increasing width, asymptotically reaching the bulk behaviour. This dependence of the mLE on both energy density and ribbon width is fitted accurately with empirical expressions.

nlin.CD↗

arXiv2020

Distributions of Bubble Lifetimes and Bubble Lengths in DNA

We investigate the distribution of bubble lifetimes and bubble lengths in DNA at physiological temperature, by performing extensive molecular dynamics simulations with the Peyrard-Bishop-Dauxois (PBD) model, as well as an extended version (ePBD) having a sequence-dependent stacking interaction, emphasizing the effect of the sequences' guanine-cytosine (GC)/adenine-thymine (AT) content on these distributions. For both models we find that base pair-dependent (GC vs AT) thresholds for considering complementary nucleotides to be separated are able to reproduce the observed dependence of the melting temperature on the GC content of the DNA sequence. Using these thresholds for base pair openings, we obtain bubble lifetime distributions for bubbles of lengths up to ten base pairs as the GC content of the sequences is varied, which are accurately fitted with stretched exponential functions. We find that for both models the average bubble lifetime decreases with increasing either the bubble length or the GC content. In addition, the obtained bubble length distributions are also fitted by appropriate stretched exponential functions and our results show that short bubbles have similar likelihoods for any GC content, but longer ones are substantially more likely to occur in AT-rich sequences. We also show that the ePBD model permits more, longer-lived, bubbles than the PBD system.

physics.bio-ph↗

arXiv2020

Chaotic dynamics of graphene and graphene nanoribbons

We study the chaotic dynamics of graphene structures, considering both a periodic, defect free, graphene sheet and graphene nanoribbons (GNRs) of various widths. By numerically calculating the maximum Lyapunov exponent, we quantify the chaoticity for a spectrum of energies in both systems. We find that for all cases, the chaotic strength increases with the energy density, and that the onset of chaos in graphene is slow, becoming evident after more than $10^4$ natural oscillations of the system. For the GNRs, we also investigate the impact of the width and chirality (armchair or zigzag edges) on their chaotic behavior. Our results suggest that due to the free edges the chaoticity of GNRs is stronger than the periodic graphene sheet, and decreases by increasing width, tending asymptotically to the bulk value. In addition, the chaotic strength of armchair GNRs is higher than a zigzag ribbon of the same width. Further, we show that the composition of ${}^{12}C$ and ${}^{13}C$ carbon isotopes in graphene has a minor impact on its chaotic strength.

nlin.CD↗

arXiv2019

Heterogeneity and chaos in the Peyrard-Bishop-Dauxois DNA model

We discuss the effect of heterogeneity on the chaotic properties of the Peyrard-Bishop-Dauxois nonlinear model of DNA. Results are presented for the maximum Lyapunov exponent and the deviation vector distribution. Different compositions of adenine-thymine (AT) and guanine-cytosine (GC) base pairs are examined for various energies up to the melting point of the corresponding sequence. We also consider the effect of the alternation index, which measures the heterogeneity of the DNA chain through the number of alternations between different types (AT or GC) of base pairs, on the chaotic behavior of the system. Biological gene promoter sequences have been also investigated, showing no distinct behavior of the maximum Lyapunov exponent.

nlin.CD↗

arXiv2018

Distribution of Base Pair Alternations in a Periodic DNA Chain: Application of Polya Counting to a Physical System

In modeling DNA chains, the number of alternations between Adenine-Thymine (AT) and Guanine-Cytosine (GC) base pairs can be considered as a measure of the heterogeneity of the chain, which in turn could affect its dynamics. A probability distribution function of the number of these alternations is derived for circular or periodic DNA. Since there are several symmetries to account for in the periodic chain, necklace counting methods are used. In particular, Polya's Enumeration Theorem is extended for the case of a group action that preserves partitioned necklaces. This, along with the treatment of generating functions as formal power series, allows for the direct calculation of the number of possible necklaces with a given number of AT base pairs, GC base pairs and alternations. The theoretically obtained probability distribution functions of the number of alternations are accurately reproduced by Monte Carlo simulations and fitted by Gaussians. The effect of the number of base pairs on the characteristics of these distributions is also discussed, as well as the effect of the ratios of the numbers of AT and GC base pairs.

math.CO↗