The perspective taken here and in the works discussed above is to see the system via dynamically defined cylinder sets, which makes it essentially a ‘symbolic approach’. We note that outside this context (for example for an interval map, considering hits to balls rather than cylinders), much less is known in the **random** setting. However, in [AFV], a Poisson distribution was shown for first hitting times to balls in the setting of certain **random** **dynamical** **systems**. We note that this was for **systems** which were all close to a certain well-behaved system, so the randomness could be interpreted as (additive) noise. Moreover, this was an annealed law rather than a quenched one.

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The current paper is devoted to **random** dynamics of stochastic partly dissipative **systems** perturbed by Lévy noise. By the technique of dissipative in probability and multivalued **random** **dynamical** **systems** (MRDS), the existences of **random** attractor for MRDS generated by the stochastically perturbed partly dissipative **systems** are provided, both the weaker restrictions and stronger restrictions on the coefficients of Lévy noise respectively.

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Over the last decade there has been a great deal of work on **random** **dynamical** **systems**. A **random** system is given by a finite family of maps defined on the same state space and a probabilistic regime for choosing one of these maps at each time step. The study of condi- tions that guarantee the existence of an invariant measure for such **systems** was initiated by Morita ([Mor85]) and Pelikan ([Pel84]). They consider the case where each map in the **random** system is piecewise smooth with respect to some finite partition and expanding on average. In [GB03] and [BG05] these results are extended to when the probabilistic regime is position dependent. These results were further generalised by Inoue ([Ino12]) to more general underlying partitions, including ones with countably many elements. In [ANV] limit theorems are studied for **random** **systems** consisting of countably many maps. Their examples include families of maps that are piecewise smooth with respect to some finite partition and are expanding on average. Rousseau and Todd studied hitting time statistics for **random** maps [RT15].

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The paper is organized as follows. In the next section, we review the pullback **random** attractor theory for **random** **dynamical** **systems** and some lemmas. In Section , we deﬁne a **random** **dynamical** system for the stochastic viscous coupled Camassa-Holm equation. Then we derive the uniform estimates of solutions in Section . These estimates are nec- essary for proving the existence of bounded **random** absorbing sets and the asymptotic compactness of the **random** **dynamical** system and prove the existence of a pullback ran- dom attractor in H ([, T ]) × H ([, T ]). We conclude that the global attractor persists

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It is well known that Crauel and Flandoli originally introduced the **random** attractor for the infinite-dimensional RDS [8] [9]. A **random** attractor of RDS is a measurable and compact invariant **random** set attracting all orbits. It is the appropriate generalization of the now classical attractor exists, it is the smallest attracting compact set and the largest invariant set [10]. Zhou et al. [11] studied **random** attractor for damped nonlinear wave equation with white noise. Fan [12] proved **random** attractor for a damped stochastic wave equation with multiplicative noise. These abstract results have been successfully applied to many stochastic dissipative partial differential equations. The existence of a **random** attractors for the wave equations has been investigated by several authors [8] [9] [10].

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In this work, we present some elements for a general framework of inference in **dynamical** **systems**, written in the language of Bayesian probability. The first is the master equation, which is shown as a direct consequence of the laws of probability. Next we develop the treatment of inference over paths from which we obtain the continuity equation and Cauchy’s equation for fluid dynamics, and discuss their range of applicability. Finally we close with some concluding remarks.

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Uncertainty is ubiquitous in physics, chem- istry, bioscience and especially in the social sci- ence. It is often classified into two types. One is epistemic (or reducible) uncertainty, which can possibly be reduced by improved measure- ments. The other is aleatory (or irreducible) uncertainty, which is the result of intrinsic variability/stochasticity of the system. Uncer- tainty propagation through a dynamic system has enjoyed considerable research interests dur- ing the past decade due to the wide applica- tions of mathematical models in studying the **dynamical** behavior of the system.

Producing some theory on the effects of control on strongly dependent **systems** is a challenge. It is not even clear how to understand their dynamics in the autonomous case. Here is a starting point on the latter. Restrict attention to the extremal phases, i.e. those which are not a convex combination of phases. They are mutually singular, i.e. for each phase µ there is a subset A µ of state space with full µ-measure and measure 0 for all

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Sensitivity analysis of **dynamical** **systems** is of interest to numerous researchers be- cause the resulting sensitivity functions have many applications in optimization and design [12, 24, 32, 37], computation of standard errors [5, 6, 17, 18, 35], information theory [15], and parameter estimation and inverse problems [2, 5, 6, 10, 38]. In recent years, due to a continuous need to better describe physical reality as well as a tremendous improvement in computing technology, these models have become more complex, with increased num- bers of parameters. This has naturally introduced questions involving the determination of important parameters, the effects on model output due to changing parameters, etc., with answers that are not obvious for large, complex models [34]. Sensitivity analysis offers answers to these questions for problems of interest, yielding an improved understanding of the underlying mathematical model as well as better results.

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In this chapter we will start looking at various properties that can be used to characterize a system. We will initially illustrate these concepts as much as possible with examples from circuits and mechanical **systems**. However, since these are general concepts we will begin to explore abstract **systems** described only by algebraic, integral, or differential equations. Our goal is to be able to be able to determine whether or not a mathematical model of a system possesses these properties, and to develop the necessary vocabulary.

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Drake et al.(1991) and Stahl (1992) seemed to have found the evidence for directed mutation convincing, since claims for such emerged from some powerful experimental **systems** as the original demonstrations of **random** mutation. However, two important aspects of the classic experiments have been overlooked by directed mutation proponents. First, the authors of the classic experiments were careful about the assumptions of their tests. For example, in its simplest form, the sib selection experiment assumes that putative mutants and their progenitors grow at equal rates (are equally fit) in the absence of selective agents. If , instead, mutants grow more slowly (less fit), then the results of the experiment will deviate from randomness in a manner consistent with directed mutation. Rather than immediately invoke directed mutation on such evidence, Cavalli-Sforza and Lederberg (1956) considered and quantitatively tested the alternative hypothesis of differential growth rates. Second, the observation that some mutations occur after cells are exposed to a selective agent does not indicate that mutations are caused by selection. To imply that postselection per se challenges the Darwinian view of adaptation is to confuse the method of the classic experiments( variation arises before the imposition of selection) with the logical interpretation of the results (variation is not caused by selection).

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The existing scenario based design methodologies for dynamic embedded **systems** can be divided into multiple classes, depending on the way the common working situations are identiﬁed and pre- dicted and by the type of the utilized switching mechanism. In one class of design methodologies the working situations are extracted by analyzing the diﬀerent usage episodes of the system in terms of user actions and the system operations. For instance, this is how the ’use-case scenarios’ are identi- ﬁed in [13] and ’workload scenarios’ in [3]. These working situations do not take into consideration the resources required by the system to meet its constraints and may therefore produce suboptimal designs with higher total exploitation costs.

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Theorem 2.1. Let φ be a continuous **random** **dynamical** system with state space X over ( Ω , , , ( ) θ t t ∈ ) . If there is a closed **random** absorbing set B ( ) ω of φ and φ is asymptotically compact in X , then A ( ) → is a **random** attrac- tor of φ , where

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The main purpose of the paper is to extend the results of Ellerman (Int. J. Semant. Comput. 7:121–145, 2013) to the case of **dynamical** **systems**. We deﬁne the logical entropy and conditional logical entropy of ﬁnite measurable partitions and derive the basic properties of these measures. Subsequently, the suggested concept of logical entropy of ﬁnite measurable partitions is used to deﬁne the logical entropy of a **dynamical** system. It is proved that two metrically isomorphic **dynamical** **systems** have the same logical entropy. Finally, we provide a logical version of the Kolmogorov–Sinai theorem on generators. So it is shown that by replacing the Shannon entropy function by the logical entropy function we obtain the results analogous to the case of classical Kolmogorov–Sinai entropy theory of **dynamical** **systems**.

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for visualization purposes to help better understand the underlying flow geometry of unsteady sys- tems. In such cases, it is often sufficient to visualize these structures, which in and of itself can be very revealing since the location of these structures is usually not obvious from viewing the velocity field or even the trajectories. However there is strong motivation to extract these LCS numerically. One reason is to speed computational efficiency. For example, the main cost associate with com- puting FTLE fields is the integration of the FTLE grid since each point must be advected by the flow. Therefore, it is desirable to develop an automated algorithm such that a coarse FTLE grid can be used to obtain approximate locations of LCS and the FTLE grid could then be adaptively refined near LCS locations to produce a interactively better estimate of the LCS location. This is also desirable in the sense that it will produce a final FTLE grid that is better resolved around the LCS, which would facilitate algorithms to extract the unique curve (surface in high dimensions) representing the LCS. This approach becomes even more compelling for higher-dimensional **systems** since the number of grid points in the FTLE grid increases exponentially with the dimension of the system.

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1. Introduction. The governing equations we use to model complex phenomena are often approximate. For example, we may not know exactly the initial and/or boundary conditions necessary to integrate these equations. Other parameters entering these equations can also be uncertain, either because we are not sure of the model itself or because these parameters may vary from situations to situations in a way that is difficult to predict in detail. The question then becomes whether we can quantify how our imperfect knowledge of the system’s param- eters impact its behavior. This question lends itself naturally to a probabilistic formulation. Consider for example the case of a **dynamical** system whose state at time t can be specified by some u(t) which can be a vector or a field and satisfies

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The QMC **dynamical** entropy given in Equation (7.3) has been considered by many in different contexts. For the AF entropy ([6]) and AOW entropy ([3]), the authors consider only those Θ’s which are ∗-automorphism, and hence the joint densities are given by Equation (4.12). For AOW entropy the authors made the further restriction that γ be simply a partition of unity and considered the transition expectation given in Remark 4.3.1; when considering the transition expectations in Remark 4.3.1, the resulting joint densities in Equation (7.2) are diagonal, as we will see in Section 7.2. In 1999 the KOW entropy was introduced in [38] and the restriction to ∗-automorphisms was not imposed. Moreover, the authors of [38] allow for more generality by introduc- ing an additional Hilbert space to represent noise. Our main result (Theorem 7.3.9) of this chapter makes use of some of the ideas introduced in that paper, but we will not present the full generality here.

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independent of t) with multiplicative and additive white noise, the existence and upper semi-continuity of the **random** attractor have been studied by some authors; see [5–12]. But as is well known, there is no result concerning the dimension of **random** attractor and existence of **random** exponential attractors for second order autonomous and non- autonomous stochastic lattice system (1). Based on the ideas of [3, 5, 9], in this paper, we aim to prove that under certain conditions, the system (1) possesses a **random** expo- nential attractor in weighted spaces of sequences, which implies that the corresponding autonomous and non-autonomous stochastic **systems** (1) have the **random** attractors with ﬁnite fractal dimension.

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Abstract —Biological organisms exist within environments in which complex, non-linear dynamics are ubiquitous. They are coupled to these environments via their own complex, **dynamical** networks of enzyme-mediated reactions, known as biochemical networks. These networks, in turn, control the growth and behaviour of an organism within its environment. In this paper, we consider computational models whose structure and function are motivated by the organisation of biochemical networks. We refer to these as artificial biochemical networks, and show how they can be evolved to control trajectories within three behaviourally diverse complex **dynamical** **systems**: the Lorenz system, Chirikov’s standard map, and legged robot locomotion. More generally, we consider the notion of evolving **dynamical** **systems** to control **dynamical** **systems**, and discuss the advantages and disadvantages of using higher order coupling and config- urable **dynamical** modules (in the form of discrete maps) within artificial biochemical networks. We find both approaches to be advantageous in certain situations, though note that the relative trade-offs between different models of artificial biochemical network strongly depend on the type of **dynamical** **systems** being controlled.

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The geometric study of **dynamical** **systems** is an important chapter of contemporary mathematics due to its applications in Mechanics, Theoretical Physics. If M is a differentiable manifold that corresponds to the configuration space, a **dynamical** system , which are called equations of whose integral curves, ( ) are given The theory of **dynamical** **systems** deals with the integration of such **systems**. One Equations with Two Almost Complex Structures on Symplectic metry It has been used in this paper using two complex structures, examined mechanical **systems** on symplectic geometry. In this paper, we study **dynamical** **systems** with Three Almost Complex Structures . After Introduction in Section 1, we consider . Section 2 deals with the study Almost Complex Structures. Section 3 is devoted to study

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