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We consider time-dependent dynamical systems arising as sequential compositions of self-maps of a probability space. We establish conditions under which the Birkhoff sums for multivariate observations, given a centering and a general normalizing sequence b ( N ) of invertible square matrices, are approximated by a normal distribution with respect to a metric of regular test functions. Depending on the metric and the normalizing sequence b ( N ), the conditions imply that the error in the approximation decays either at the rate O ( N - 1 / 2 ) or the rate O ( N - 1 / 2 log N ) , under the additional assumption that ‖ b ( N ) - 1 ‖ ≲ N - 1 / 2 . The error comes with a multiplicative constant whose exact value can be computed directly from the conditions. The proof is based on an observation due to Sunklodas regarding Stein’s method of normal approximation. We give applications to one-dimensional random piecewise expanding maps and to sequential, random, and quasistatic intermittent systems.

The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its highest curve, the Airy In this paper, we employ the Brownian Gibbs property to make a close comparison between the Airy line ensemble’s curves after affine shift and Brownian bridge, proving the finiteness of a superpolynomially growing moment bound on Radon-Nikodym derivatives. We also determine the value of a natural exponent describing in Brownian last passage percolation the decay in probability for the existence of several near geodesics that are disjoint except for their common endpoints, where the notion of ‘near’ refers to a small deficit in scaled geodesic energy, with the parameter specifying this nearness tending to zero. To prove both results, we introduce a technique that may be useful elsewhere for finding upper bounds on probabilities of events concerning random systems of curves enjoying the Brownian Gibbs property. Several results in this article play a fundamental role in a further study of Brownian last passage percolation in three companion papers (Hammond 2017a,b,c), in which geodesic coalescence and geodesic energy profiles are investigated in scaled coordinates.

Illegal fishing in small-scale fisheries is a contentious issue and resists a straightforward interpretation. Particularly, there is little knowledge regarding cooperative interactions between legal and illegal fishers and the potential effects on fisheries arising from these interactions. Taking the Chilean king crab ( Lithodes santolla ; common name centolla) fishery as a case study, our goal is twofold: (i) to model the effect of illegal-legal fishers’ interactions on the fishery and (ii) analyze how management and social behavior affect fishery’s outcomes. We framed the analysis of this problem within game theory combined with network theory to represent the architecture of competitive interactions. The fishers’ system was set to include registered (legal) fishers and unregistered (illegal) fishers. In the presence of unregistered fishers, legal fishers may decide to cooperate (ignoring the presence of illegal fishers) or defect, which involves becoming a “super fisher” and whitewashing the captures of illegal fishers for a gain. The utility of both players, standard fisher and super fisher depend on the strategy chosen by each of them, as well as on the presence of illegal fishers. The nodes of the network represent the legal fishers (both standard and super fishers) and the links between nodes indicate that these fishers compete for the resource, assumed to be finite and evenly distributed across space. The decision to change (or not) the adopted strategy is modeled considering that fishers are subjected to variable levels of temptation to whitewash the illegal capture and to social pressure to stop doing so. To represent the vital dynamics of the king crab, we propose a model that includes the Allee effect and a term accounting for the crab extraction. We found that the super fisher strategy leads to the decrease of the king crab population under a critical threshold as postulated in the tragedy of the commons hypothesis when there are: (i) high net extraction rates of the network composed of non-competing standard fishers, (ii) high values of the extent of the fishing season, and (iii) high density of illegal fishers. The results suggest that even in the presence of super fishers and illegal fishers, the choice of properly distributed fishing/closure cycles or setting an extraction limit per vessel can prevent the king crab population from falling below a critical threshold. This finding, although controversial, reflects the reality of this fishery that, for decades, has operated under a dynamic in which whitewashing and super fishers have become well established within the system.

We consider the time-dependent dynamical system q¨ a=−Γbca q˙ b q˙ c−ω(t)Qa(q) where ω(t) is a non-zero arbitrary function and the connection coefficients Γbca are computed from the kinetic metric (kinetic energy) of the system. In order to determine the quadratic first integrals (QFIs) I we assume that I=Kab q˙ a q˙ b+Ka q˙ a+K where the unknown coefficients Kab,Ka,K are tensors depending on t,qa and impose the condition dIdt=0 . This condition leads to a system of partial differential equations (PDEs) involving the quantities Kab,Ka,K, ω(t) and Qa(q) . From these PDEs, it follows that Kab is a Killing tensor (KT) of the kinetic metric. We use the KT Kab in two ways: a. We assume a general polynomial form in t both for Kab and Ka ; b. We express Kab in a basis of the KTs of order 2 of the kinetic metric assuming the coefficients to be functions of t. In both cases, this leads to a new system of PDEs whose solution requires that we specify either ω(t) or Qa(q) . We consider first that ω(t) is a general polynomial in t and find that in this case the dynamical system admits two independent QFIs which we collect in a Theorem. Next, we specify the quantities Qa(q) to be the generalized time-dependent Kepler potential V=−ω(t)rν and determine the functions ω(t) for which QFIs are admitted. We extend the discussion to the non-linear differential equation x¨ =−ω(t)xμ+ϕ(t) x˙ (μ≠−1) and compute the relation between the coefficients ω(t),ϕ(t) so that QFIs are admitted. We apply the results to determine the QFIs of the generalized Lane–Emden equation.

Condition spectrum is an essential generalization of the spectrum. This article considers the condition spectrum of bounded linear operators on Banach space and develops certain topological properties. It is observed that the condition spectrum is useful than the spectrum and pseudospectrum for identifying the norm behavior of non-normal matrices. For a bounded linear operator A on a Banach space, we find upper and lower bounds for ‖ e tA ‖ , t ≥ 0 and ‖ A n ‖ , n = 1 , 2 , … using the condition spectrum of A . These bounds are used to identify the transient effect of the quantities appearing in the time dependent linear dynamical system.

This article is concerned with the question of uniqueness of self-similar profiles for Smoluchowski’s coagulation equation which exhibit algebraic decay (fat tails) at infinity. More precisely, we consider a rate kernel Establishing uniqueness of self-similar profiles for Smoluchowski’s coagulation equation is generally considered to be a difficult problem which is still essentially open. Concerning fat-tailed self-similar profiles this article actually gives the first uniqueness statement for a non-solvable kernel.

Contact geometry allows us to describe some thermodynamic and dissipative systems. In this paper we introduce a new geometric structure in order to describe time-dependent contact systems: cocontact manifolds. Within this setting we develop the Hamiltonian and Lagrangian formalisms, both in the regular and singular cases. In the singular case, we present a constraint algorithm aiming to find a submanifold where solutions exist. As a particular case we study contact systems with holonomic time-dependent constraints. Some regular and singular examples are analyzed, along with numerical simulations.

This study investigates the modulation effects on the dynamics of a ring of three double-well Duffing oscillators coupled unidirectionally with time-dependent damping. The aim is to understand how different modulation parameters and damping coefficients influence the behavior of the system. We analyzed three cases of the time-dependent damping coefficient: constant damping ( α = 0.4 ), damping increasing over time ( α ( t ) = t τ ) , and damping decreasing over time ( α ( t ) = 1 t + ϵ ) . Additionally, three values of modulation amplitude ( Γ ) and three values of modulation frequency ( θ ) were studied as function of the coupling strength ( σ ). The dynamics were examined using time series, phase portraits, Poincaré sections, bifurcation diagrams, Lyapunov exponents, and Fourier and Hilbert transformations. For constant damping, the inclusion of modulation altered the dynamics significantly, replacing fixed points with stable oscillations, heteroclinic and chaotic orbits. Smaller modulation parameters resulted in more dynamic changes. For increasing damping, the system transitioned to stable oscillations. Rotating waves were prevented because of high damping. For decreasing damping, the system shifted from dissipative to conservative behavior, exhibiting transient toroidal hyperchaos and stable oscillations at low coupling strengths. The modulation term introduces complex behaviors to the system, such as stable oscillations and confined hyperchaos, which are not present in non-modulated scenarios. The amplitude and frequency of modulation critically impact the system’s dynamics, making it more dynamic compared to previous findings without modulation. The results highlight the significant role of modulation in altering the path to hyperchaos and the overall dynamic behavior of the coupled Duffing oscillators.

In this paper, we investigate two members of the Kadomtsev–Petviashvili (KP) hierarchy, each with time-dependent coefficients. We use the Painlevé analysis and the WTC–Kruskal method to study the compatibility conditions to ensure the integrability of each equation. We use the simplified Hirota’s method to derive multiple soliton solutions for each equation.

By means of Euler–Bernoulli beam theory and the Lagrange equation of the first kind, the dynamic equation of a beam with an axially harmonic reciprocating moving mid-support (beam-AHRMS) is established. Using two modal coordinates, the equation of the beam-AHRMS is reduced to a special form of Hill’s equation, the particularity of which is that its excitation, i.e., time-dependent variable coefficients in front of the state variable of the equation, is a nonlinear function of the excitation in Mathieu’s equation. Using the perturbation method, the approximate solution for the transition curves between the stable and unstable regions of the system is obtained. The most different behaviour of the special form of Hill’s equation is that there exists a knot in the principal unstable region, where the bandwidth of the unstable region becomes zero for any frequency, and the principal parameter resonance is suppressed. A theorem for the occurrence and location of knots in the principal unstable region is proposed and proven. All the results of the analysis and the theorem are verified via a numerical method.