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The comparison between time-varying hybrid trajectories is crucial for tracking, observer design and synchronisation problems for hybrid systems with state-triggered jumps. In this paper, a systematic way of designing an appropriate distance function is proposed that can be used for this purpose. The so-called "peaking phenomenon", which occurs when using the Euclidean distance to compare two hybrid trajectories, is circumvented by taking the hybrid nature of the system explicitly into account in the design of the distance function. Based on the proposed distance function, we define the stability of a trajectory of a hybrid system with state-triggered jumps and present sufficient Lyapunov-type conditions for stability of a hybrid trajectory. A constructive design method for the distance function is presented for hybrid systems with affine flow and jump maps and a jump set that is a hyperplane. For this case, the mentioned Lyapunov-type stability conditions can be verified using linear matrix conditions. Finally, for this class of systems, we present a tracking controller that asymptotically stabilises a given hybrid reference trajectory, and we illustrate our results with examples.

We consider semigroup actions on the unit interval generated by strictly increasing $C^r$-maps. We assume that one of the generators has a pair of fixed points, one attracting and one repelling, and a heteroclinic orbit that connects the repeller and attractor, and the other generators form a robust blender, which can bring the points from a small neighborhood of the attractor to an arbitrarily small neighborhood of the repeller. This is a model setting for partially hyperbolic systems with one central direction. We show that, under additional conditions on the non-linearity and the Schwarzian derivative, the above semigroups exhibit, $C^r$-generically for any r, arbitrarily fast growth of the number of periodic points as a function of the period. We also show that a $C^r$-generic semigroup from the class under consideration supports an ultimately complicated behavior called universal dynamics., 36 pages, 1 figure

We investigate the effects of structural perturbations of both, undirected and directed diffusive networks on their ability to synchronize. We establish a classification of directed links according to their impact on synchronizability. We focus on adding directed links in weakly connected networks having a strongly connected component acting as driver. When the connectivity of the driver is not stronger than the connectivity of the slave component, we can always make the network strongly connected while hindering synchronization. On the other hand, we prove the existence of a perturbation which makes the network strongly connected while increasing the synchronizability. Under additional conditions, there is a node in the driving component such that adding a single link starting at an arbitrary node of the driven component and ending at this node increases the synchronizability.

In this paper we consider families of holomorphic maps defined on subsets of the complex plane, and show that the technique developed in \cite{LSvS1} to treat unfolding of critical relations can also be used to deal with cases where the critical orbit converges to a hyperbolic attracting or a parabolic periodic orbit. As before this result applies to rather general families of maps, such as polynomial-like mappings, provided some lifting property holds. Our Main Theorem states that either the multiplier of a hyperbolic attracting periodic orbit depends univalently on the parameter and bifurcations at parabolic periodic points are generic, or one has persistency of periodic orbits with a fixed multiplier.

International audience; The response of a low-frequency mode of climate variability, El Niño-Southern Oscillation, to stochastic forcing is studied in a high-dimensional model of intermediate complexity, the fully-coupled Cane-Zebiak model [ZC87], from the spectral analysis of Markov operators governing the decay of correlations and resonances in the power spectrum. Noise-induced oscillations excited before a supercritical Hopf bifurcation are examined by means of complex resonances , the reduced Ruelle-Pollicott (RP) resonances, via a numerical application of the reduction approach of the first part of this contribution [CTND19] to model simulations. The oscillations manifest themselves as peaks in the power spectrum which are associated with RP resonances organized along parabolas, as the bifurcation is neared. These resonances and the associated eigenvectors are furthermore well described by the small-noise expansion formulas obtained by [Gas02] and made explicit in the second part of this contribution [TCND19]. Beyond the bifurcation, the spectral gap between the imaginary axis and the real part of the leading resonances quantifies the diffusion of phase of the noise-induced oscillations and can be computed from the linearization of the model and from the diffusion matrix of the noise. In this model, the phase diffusion coefficient thus gives a measure of the predictability of oscillatory events representing ENSO. ENSO events being known to be locked to the seasonal cycle, these results should be extended to the non-autonomous case. More generally, the reduction approach theorized in [CTND19], complemented by our understanding of the spectral properties of reference systems such as the stochastic Hopf bifurcation, provides a promising methodology for the analysis of low-frequency variability in high-dimensional stochastic systems.

Two common types of dynamics, chaotic and monotone, are compared. It is shown that monotone maps in strongly ordered spaces do not have chaotic attractors.

International audience; A theory of Ruelle-Pollicott (RP) resonances for stochastic differential systems is presented. These resonances are defined as the eigenvalues of the generator (Kolmogorov operator) of a given stochastic system. By relying on the theory of Markov semigroups, decomposition formulas of correlation functions and power spectral densities (PSDs) in terms of RP resonances are then derived. These formulas describe, for a broad class of stochastic differential equations (SDEs), how the RP resonances characterize the decay of correlations as well as the signal’s oscillatory components manifested by peaks in the PSD.It is then shown that a notion reduced RP resonances can be rigorously defined, as soon as the dynamics is partially observed within a reduced state space V . These reduced resonances are obtained from the spectral elements of reduced Markov operators acting on functions of the state space V , and can be estimated from series. They inform us about the spectral elements of some coarse-grained version of the SDE generator. When the time-lag at which the transitions are collected from partial observations in V , is either sufficiently small or large, it is shown that the reduced RP resonances approximate the (weak) RP resonances of the generator of the conditional expectation in V , i.e. the optimal reduced system in V obtained by averaging out the contribution of the unobserved variables. The approach is illustrated on a stochastic slow-fast system for which it is shown that the reduced RP resonances allow for a good reconstruction of the correlation functions and PSDs, even when the time-scale separation is weak.The companions articles, Part II and Part III, deal with further practical aspects of the theory presented in this contribution. One important byproduct consists of the diagnosis usefulness of stochastic dynamics that RP resonances provide. This is illustrated in the case of a stochastic Hopf bifurcation in Part II. There, it is shown that such a bifurcation has a clear manifestation in terms of a geometric organization of the RP resonances along discrete parabolas in the left half plane. Such geometric features formed by (reduced) RP resonances are extractable from time series and allow thus for providing an unambiguous “signature” of nonlinear oscillations embedded within a stochastic background. By relying then on the theory of reduced RP resonances presented in this contribution, Part III addresses the question of detection and characterization of such oscillations in a high-dimensional stochastic system, namely the Cane-Zebiak model of El Niño-Southern Oscillation subject to noise modeling fast atmospheric fluctuations.

Tobasco et al. [Physics Letters A, 382:382-386, 2018; see https://doi.org/10.1016/j.physleta.2017.12.023] recently suggested that trajectories of ODE systems that optimize the infinite-time average of a certain observable can be localized using sublevel sets of a function that arise when bounding such averages using so-called auxiliary functions. In this paper we demonstrate that this idea is viable and allows for the computation of extremal unstable periodic orbits (UPOs) for polynomial ODE systems. First, we prove that polynomial optimization is guaranteed to produce auxiliary functions that yield near-sharp bounds on time averages, which is required in order to localize the extremal orbit accurately. Second, we show that points inside the relevant sublevel sets can be computed efficiently through direct nonlinear optimization. Such points provide good initial conditions for UPO computations. As a proof of concept, we then combine these methods with a single-shooting Newton-Raphson algorithm to study extremal UPOs for a nine-dimensional model of sinusoidally forced shear flow. We discover three previously unknown families of UPOs, one of which simultaneously minimizes the mean energy dissipation rate and maximizes the mean perturbation energy relative to the laminar state for Reynolds numbers approximately between 81.24 and 125., Comment: 22 pages, 7 figures. v3: update discussion after Theorem 1, fixed typos

The multiple futile cycle is a phosphorylation system in which a molecular substrate might be phosphorylated sequentially n times by means of an enzymatic mechanism. The system has been studied mathematically using reaction network theory and ordinary differential equations. It is known that the system might have at least as many as 2[n/2]+1 steady states (where [x] is the integer part of x) for particular choices of parameters. Furthermore, for the simple and dual futile cycles (n=1,2) the stability of the steady states has been determined in the sense that the only steady state of the simple futile cycle is globally stable, while there exist parameter values for which the dual futile cycle admits two asymptotically stable and one unstable steady state. For general n, evidence that the possible number of asymptotically stable steady states increases with $n$ has been given, which has led to the conjecture that parameter values can be chosen such that [n/2]+1 out of 2[n/2]+1 steady states are asymptotically stable and the remaining steady states are unstable. We prove this conjecture here by first reducing the system to a smaller one, for which we find a choice of parameter values that give rise to a unique steady state with multiplicity 2[n/2]+1. Using arguments from geometric singular perturbation theory, and a detailed analysis of the centre manifold of this steady state, we achieve the desired result. The multiple futile cycle is a phosphorylation system in which a molecular substrate might be phosphorylated sequentially n times by means of an enzymatic mechanism. The system has been studied mathematically using reaction network theory and ordinary differential equations. It is known that the system might have at least as many as $2\lfloor \frac{n}{2}\rfloor +1$ steady states (where lfloorxrfloor is the integer part of x) for particular choices of parameters. Furthermore, for the simple and dual futile cycles (n = 1, 2) the stability of the steady states has been determined in the sense that the only steady state of the simple futile cycle is globally stable, while there exist parameter values for which the dual futile cycle admits two asymptotically stable and one unstable steady state. For general n, evidence that the possible number of asymptotically stable steady states increases with n has been given, which has led to the conjecture that parameter values can be chosen such that $\lfloor \frac{n}{2}\rfloor +1$ out of $2\lfloor \frac{n}{2}\rfloor +1$ steady states are asymptotically stable and the remaining steady states are unstable. We prove this conjecture here by first reducing the system to a smaller one, for which we find a choice of parameter values that give rise to a unique steady state with multiplicity $2\lfloor \frac{n}{2}\rfloor +1$. Using arguments from geometric singular perturbation theory, and a detailed analysis of the centre manifold of this steady state, we achieve the desired result.

When an experimentalist measures a time series of qubits, the outcomes generate a classical stochastic process. We show that measurement induces high complexity in these processes in two specific senses: they are inherently unpredictable (positive Shannon entropy rate) and they require an infinite number of features for optimal prediction (divergent statistical complexity). We identify nonunifilarity as the mechanism underlying the resulting complexities and examine the influence that measurement choice has on the randomness and structure of measured qubit processes. We introduce new quantitative measures of this complexity and provide efficient algorithms for their estimation., Comment: 6 pages, 3 figures; supplemental material, 5 pages, 1 figure; http://csc.ucdavis.edu/~cmg/compmech/pubs/qdic_prl.htm

We look at the preservation of various notions of shadowing in discrete dynamical systems under inverse limits, products, factor maps and the induced maps for symmetric products and hyperspaces. The shadowing properties we consider are the following: shadowing, h-shadowing, eventual shadowing, orbital shadowing, strong orbital shadowing, the first and second weak shadowing properties, limit shadowing, s-limit shadowing, orbital limit shadowing and inverse shadowing., Comment: 39 pages

Biological systems with intertwined feedback loops pose a challenge to mathematical modeling efforts. Moreover, rare events, such as mutation and extinction, complicate system dynamics. Stochastic simulation algorithms are useful in generating time-evolution trajectories for these systems because they can adequately capture the influence of random fluctuations and quantify rare events. We present a simple and flexible package, BioSimulator.jl, for implementing the Gillespie algorithm, $\tau$-leaping, and related stochastic simulation algorithms. The objective of this work is to provide scientists across domains with fast, user-friendly simulation tools. We used the high-performance programming language Julia because of its emphasis on scientific computing. Our software package implements a suite of stochastic simulation algorithms based on Markov chain theory. We provide the ability to (a) diagram Petri Nets describing interactions, (b) plot average trajectories and attached standard deviations of each participating species over time, and (c) generate frequency distributions of each species at a specified time. BioSimulator.jl's interface allows users to build models programmatically within Julia. A model is then passed to the simulate routine to generate simulation data. The built-in tools allow one to visualize results and compute summary statistics. Our examples highlight the broad applicability of our software to systems of varying complexity from ecology, systems biology, chemistry, and genetics. The user-friendly nature of BioSimulator.jl encourages the use of stochastic simulation, minimizes tedious programming efforts, and reduces errors during model specification., Comment: 27 pages, 5 figures, 3 tables

We review some of the latest approaches to analysing cardiac electrophysiology data using machine learning and predictive modelling. Cardiac arrhythmias, particularly atrial fibrillation, are a major global healthcare challenge. Treatment is often through catheter ablation, which involves the targeted localized destruction of regions of the myocardium responsible for initiating or perpetuating the arrhythmia. Ablation targets are either anatomically defined, or identified based on their functional properties as determined through the analysis of contact intracardiac electrograms acquired with increasing spatial density by modern electroanatomic mapping systems. While numerous quantitative approaches have been investigated over the past decades for identifying these critical curative sites, few have provided a reliable and reproducible advance in success rates. Machine learning techniques, including recent deep-learning approaches, offer a potential route to gaining new insight from this wealth of highly complex spatio-temporal information that existing methods struggle to analyse. Coupled with predictive modelling, these techniques offer exciting opportunities to advance the field and produce more accurate diagnoses and robust personalised treatment. We outline some of these methods and illustrate their use in making predictions from the contact electrogram and augmenting predictive modelling tools, both by more rapidly predicting future states of the system and by inferring the parameters of these models from experimental observations.

A new diffusion mechanism from the neighborhood of elliptic equilibria for Hamiltonian flows in three or more degrees of freedom is introduced. We thus obtain explicit real entire Hamiltonians on $\R^{2d}$, $d\geq 4$, that have a Lyapunov unstable elliptic equilibrium with an arbitrary chosen frequency vector whose coordinates are not all of the same sign. For non-resonant frequency vectors, our examples all have divergent Birkhoff normal form at the equilibrium. On $\R^4$, we give explicit examples of real entire Hamiltonians having an equilibrium with an arbitrary chosen non-resonant frequency vector and a divergent Birkhoff normal form., This new version revises and supersedes the original submissions. It shows that all the examples constructed have divergent Birkhoff normal form at the origin. Moreover, it gives in all degrees of freedom larger or equal to 2 explicit examples of real entire Hamiltonians having an equilibrium with an arbitrary chosen non-resonant frequency vector and a divergent Birkhoff normal form

We prove that the Shimizu–Morioka system has a Lorenz attractor for an open set of parameter values. For the proof we employ a criterion proposed by Shilnikov, which allows to conclude the existence of the attractor by examination of the behaviour of only one orbit. The needed properties of the orbit are established by using computer assisted numerics. Our result is also applied to the study of local bifurcations of triply degenerate periodic points of three-dimensional maps. It provides a formal proof of the birth of discrete Lorenz attractors at various global bifurcations.

In this paper, tools to study forward invariance properties with robustness to dis- turbances, referred to as robust forward invariance, are proposed for hybrid dynamical systems modeled as hybrid inclusions. Hybrid inclusions are given in terms of dif- ferential and difference inclusions with state and disturbance constraints, for whose definition only four objects are required. The proposed robust forward invariance notions allow for the diverse type of solutions to such systems (with and without dis- turbances), including solutions that have persistent flows and jumps, that are Zeno, and that stop to exist after finite amount of (hybrid) time. Sufficient conditions for sets to enjoy such properties are presented. These conditions are given in terms of the objects defining the hybrid inclusions and the set to be rendered robust forward invariant. In addition, as special cases, these conditions are exploited to state results on nominal forward invariance for hybrid systems without disturbances. Furthermore, results that provide conditions to render the sublevel sets of Lyapunov-like functions forward invariant are established. Analysis of a controlled inverter system is presented as an application of our results. Academic examples are given throughout the paper to illustrate the main ideas., 39 pages, 7 figures, accepted to TAC

We prove existence of flips, special termination, the base point free theorem and, in the case of log general type, the existence of minimal models for F-dlt foliated log pairs of co-rank one on a projective threefold. As applications, we show the existence of F-dlt modifications and F-terminalisations for foliated log pairs and we show that foliations with canonical or F-dlt singularities admit non-dicritical singularities. Finally, we show abundance in the case of numerically trivial foliated log pairs.

We present a study of a delay differential equation (DDE) model for the Mid-Pleistocene Transition (MPT). We investigate the behavior of the model when subjected to periodic forcing. The unforced model has a bistable region consisting of a stable equilibrium along with a large amplitude stable periodic orbit. We study how forcing affects solutions in this region. Forcing based on astronomical data causes a sudden transition in time and under increase of the forcing amplitude, moving the model response from a non-MPT regime to an MPT regime. Similar transition behavior is found for periodic forcing. A bifurcation analysis shows that the transition is not due to a bifurcation but instead to a shifting basin of attraction. While determining the basin boundary we demonstrate how one can accurately compute the intersection of a stable manifold of a saddle with a slow manifold in a DDE by embedding the algorithm for planar maps proposed by England et al. (SIADS 2004(3)) into the equation-free framework by Kevrekidis et al. (Rev. Phys. Chem. 2009 (60))., Comment: 18 pages, supplementary material available on https://doi.org/10.6084/m9.figshare.7048292.v1

We present a study of a delay differential equation (DDE) model for the Mid-Pleistocene Transition. We investigate the behavior of the model when subjected to periodic forcing. The unforced model has a bistable region consisting of a stable equilibrium along with a large amplitude stable periodic orbit. We are interested in how forcing affects solutions in this region. The results here are compared to what is found when the model is forced with the quasiperiodic insolation. The quasiperiodic forcing displays a threshold behavior when the forcing amplitude is increased - moving the model from a non-transitioning regime to a transitioning regime. Similar threshold behavior is found when the periodic forcing amplitude is increased. A bifurcation analysis shows that the threshold is not due to a bifurcation but instead to a shifting basin of attraction.

Memcomputing is a novel computing paradigm that employs time non-local dynamical systems to compute with and in memory. The digital version of these machines [digital memcomputing machines or (DMMs)] is scalable , and is particularly suited to solve combinatorial optimization problems. One of its possible realizations is by means of standard electronic circuits, with and without memory. Since these elements are non-quantum, they can be described by ordinary differential equations. Therefore, the circuit representation of DMMs can also be simulated efficiently on our traditional computers. We have indeed previously shown that these simulations only require time and memory resources that scale linearly with the problem size when applied to finding a good approximation to the optimum of hard instances of the maximum-satisfiability problem. The state-of-the-art algorithms, instead, require exponential resources for the same instances. However, in that work, we did not push the simulations to the limit of the processor used. Since linear scalability at smaller problem sizes cannot guarantee linear scalability at much larger sizes, we have extended these results in a stress-test up to $64\times 10^{6}$ variables (corresponding to about 1 billion literals), namely the largest case that we could fit on a single core of an Intel Xeon E5-2860 with 128 GB of dynamic random-access memory (DRAM). For this test, we have employed a commercial simulator, Falcon of MemComputing, Inc. We find that the simulations of DMMs still scale linearly in both time and memory up to these very large problem sizes versus the exponential requirements of the state-of-the-art solvers. These results further reinforce the advantages of the physics-based memcomputing approach compared with traditional ones.

Coherent structures form spontaneously in nonlinear spatiotemporal systems and are found at all spatial scales in natural phenomena from laboratory hydrodynamic flows and chemical reactions to ocean, atmosphere, and planetary climate dynamics. Phenomenologically, they appear as key components that organize the macroscopic behaviors in such systems. Despite a century of effort, they have eluded rigorous analysis and empirical prediction, with progress being made only recently. As a step in this, we present a formal theory of coherent structures in fully-discrete dynamical field theories. It builds on the notion of structure introduced by computational mechanics, generalizing it to a local spatiotemporal setting. The analysis' main tool employs the \localstates, which are used to uncover a system's hidden spatiotemporal symmetries and which identify coherent structures as spatially-localized deviations from those symmetries. The approach is behavior-driven in the sense that it does not rely on directly analyzing spatiotemporal equations of motion, rather it considers only the spatiotemporal fields a system generates. As such, it offers an unsupervised approach to discover and describe coherent structures. We illustrate the approach by analyzing coherent structures generated by elementary cellular automata, comparing the results with an earlier, dynamic-invariant-set approach that decomposes fields into domains, particles, and particle interactions., 27 pages, 10 figures; http://csc.ucdavis.edu/~cmg/compmech/pubs/dcs.htm

We describe the statistical properties of the dynamics of the quadratic polynomials P_a(z):=e^{2\pi a i} z+z^2 on the complex plane, with a of high return times. In particular, we show that these maps are uniquely ergodic on their measure theoretic attractors, and the unique invariant probability is a physical measure describing the statistical behavior of typical orbits in the Julia set. This confirms a conjecture of Perez-Marco on the unique ergodicity of hedgehog dynamics, in this class of maps., Comment: 58 pages; pre-publication version

Let X be a connected open set in n-dimensional Euclidean space, partially ordered by a closed convex cone K with nonempty interior: y > x if and only if y-x is nonzero and in K. Theorem: If F is a monotone local flow in X whose periodic points are dense in X, then F is globally periodic., Incorrect proof of main theorem

In this paper we will give a short and elementary proof that critical relations unfold transversally in the space of rational maps., Comment: Published version (open access)

This paper deals with the existence, monotonicity, uniqueness, and asymptotic behaviour of travelling wavefronts for a class of temporally delayed, spatially nonlocal diffusion equations.

Let X be a subset of R^n whose interior is connected and dense in X, ordered by a polyhedral cone in R^n with nonempty interior. Let T be a monotone homeomorphism of X whose periodic points are dense. Then T is periodic., Presented at AIDS Conference on Dynamical Systems and Differential Equations, July 2016

We consider integral geometry inverse problems for unitary connections and skew-Hermitian Higgs fields on manifolds with negative sectional curvature. The results apply to manifolds in any dimension, with or without boundary, and also in the presence of trapped geodesics. In the boundary case, we show injectivity of the attenuated ray transform on tensor fields with values in a Hermitian bundle (i.e. vector valued case). We also show that a connection and Higgs field on a Hermitian bundle are determined up to gauge by the knowledge of the parallel transport between boundary points along all possible geodesics. The main tools are an energy identity, the Pestov identity with a unitary connection, which is presented in a general form, and a precise analysis of the singularities of solutions of transport equations when there are trapped geodesics. In the case of closed manifolds, we obtain similar results modulo the obstruction given by twisted conformal Killing tensors, and we also study this obstruction., 48 pages, 1 figure, final version to appear in CMP

Computational mechanics quantifies structure in a stochastic process via its causal states, leading to the process's minimal, optimal predictor---the $\epsilon$-machine. We extend computational mechanics to communication channels between two processes, obtaining an analogous optimal model---the $\epsilon$-transducer---of the stochastic mapping between them. Here, we lay the foundation of a structural analysis of communication channels, treating joint processes and processes with input. The result is a principled structural analysis of mechanisms that support information flow between processes. It is the first in a series on the structural information theory of memoryful channels, channel composition, and allied conditional information measures., Comment: 30 pages, 19 figures; http://csc.ucdavis.edu/~cmg/compmech/pubs/et1.htm; Updated to conform to published version plus additional corrections and updates

We derive some new conditions for integrability of dynamically defined C^1 invariant splittings in arbitrary dimension and co-dimension. In particular we prove that every 2-dimensional C^1 invariant decomposition on a 3-dimensional manifold satisfying a volume domination condition is uniquely integrable. In the special case of volume preserving diffeomorphisms we show that standard dynamical domination is already sufficient to guarantee unique integrability., Comment: 12 pages

Given an n-dimensional C r -diffeomorphism g, its renormalized iteration is an iteration of g, restricted to a certain n-dimensional ball and taken in some C r -coordinates in which the ball acquires radius 1. We show that for any r ≥ 1 the renormalized iterations of C r -close to identity maps of an n-dimensional unit ball B n (n ≥ 2) form a residual set among all orientation-preserving C r -diffeomorphisms B n → R n . In other words, any generic n-dimensional dynamical phenomenon can be obtained by iterations of C r -close to identity maps, with the same dimension of the phase space. As an application, we show that any C r -generic two-dimensional map that belongs to the Newhouse domain (i.e., it has a so-called wild hyperbolic set, so it is not uniformly-hyperbolic, nor uniformly partially-hyperbolic) and that neither contracts, nor expands areas, is C r -universal in the sense that its iterations, after an appropriate coordinate transformation, C r -approximate every orientation-preserving two-dimensional diffeomorphism arbitrarily well. In particular, every such universal map has an infinite set of coexisting hyperbolic attractors and repellers.

The construction of effective and informative landscapes for stochastic dynamical systems has proven a long-standing and complex problem. In many situations, the dynamics may be described by a Langevin equation while constructing a landscape comes down to obtaining the quasi-potential, a scalar function that quantifies the likelihood of reaching each point in the state-space. In this work we provide a novel method for constructing such landscapes by extending a tool from control theory: the Sum-of-Squares method for generating Lyapunov functions. Applicable to any system described by polynomials, this method provides an analytical polynomial expression for the potential landscape, in which the coefficients of the polynomial are obtained via a convex optimization problem. The resulting landscapes are based upon a decomposition of the deterministic dynamics of the original system, formed in terms of the gradient of the potential and a remaining "curl" component. By satisfying the condition that the inner product of the gradient of the potential and the remaining dynamics is everywhere negative, our derived landscapes provide both upper and lower bounds on the true quasi-potential; these bounds becoming tight if the decomposition is orthogonal. The method is demonstrated to correctly compute the quasi-potential for high-dimensional linear systems and also for a number of nonlinear examples., Comment: 12 pages, 4 figures

For a compact set $K\subset \mathbb{R}^1$ and a family $\{C_\lambda\}_{\lambda\in J}$ of dynamically defined Cantor sets sufficiently close to affine with $\text{dim}_H\, K+\text{dim}_H\, C_\lambda>1$ for all $\lambda\in J$, under natural technical conditions we prove that the sum $K+C_\lambda$ has positive Lebesgue measure for almost all values of the parameter $\lambda$. As a corollary, we show that generically the sum of two affine Cantor sets has positive Lebesgue measure provided the sum of their Hausdorff dimensions is greater than one., Comment: 15 pages, 1 figure

In this paper we analyze a two degree of freedom Hamiltonian system constructed from two planar Morse potentials. The resulting potential energy surface has two potential wells surrounded by an unbounded flat region containing no critical points. In addition, the model has an index one saddle between the potential wells. We study the dynamical mechanisms underlying transport between the two potential wells, with emphasis on the role of the flat region surrounding the wells. The model allows us to probe many of the features of the roaming mechanism whose reaction dynamics are of current interest in the chemistry community., 22 pages, 31 figures

This paper is devoted to the study of a problem of Cassels in multiplicative Diophantine approximation which involves minimising values of a product of affine linear forms computed at integral points. It was previously known that values of this product become arbitrary close to zero, and we establish that, in fact, they approximate zero with an explicit rate. Our approach is based on investigating quantitative density of orbits of higher-rank abelian groups.

We describe a spectrally-filtered discrete-in-time downscaling data assimilation algorithm and prove, in the context of the two-dimensional Navier--Stokes equations, that this algorithm works for a general class of interpolants, such as those based on local spatial averages as well as point measurements of the velocity. Our algorithm is based on the classical technique of inserting new observational data directly into the dynamical model as it is being evolved over time, rather than nudging, and extends previous results in which the observations were defined directly in terms of an orthogonal projection onto the large-scale (lower) Fourier modes. In particular, our analysis does not require the interpolant to be represented by an orthogonal projection, but requires only the interpolant to satisfy a natural approximation of the identity., Comment: 27 pages, 1 appendix

Author(s): Han, H; Sanfelice, RG | Abstract: In this paper, we introduce operators, semantics, and conditions that, when possible, are solution-independent to guarantee basic temporal logic specifications for hybrid dynamical systems. Employing sufficient conditions for forward invariance and finite time attractivity of sets for such systems, we derive such sufficient conditions for the satisfaction of formulas involving temporal operators and atomic propositions. Furthermore, we present how to certify formulas that have more than one operator. Academic examples illustrate the results throughout the paper.