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Let $ (X, T) $ (X , T) be a linear dynamical system defined on a Banach space X and $ {L_T}:{K_{{X^ * }}} \to {K_{{X^ * }}} $ L T : K X ∗ → K X ∗ be defined by $ {L_T}S = TS $ L T S = TS , where $ {K_{{X^ * }}} = \overline {span\{ \langle \cdot,{y^ * } \rangle x:{y^ * } \in {X^ * },x \in X\} } $ K X ∗ = span { 〈 ⋅ , y ∗ 〉 x : y ∗ ∈ X ∗ , x ∈ X } ¯ . This paper first obtains the equivalence of the (syndetically) weak multi-sensitive compactness between $ (X, T) $ (X , T) and $ ({K_{{X^ * }}},{L_T}) $ (K X ∗ , L T). Then, it is shown that (1) there exists a non-mean-sensitive linear dynamical system $ (X ,T) $ (X , T) satisfying that $ ({K_{{X^ * }}},{L_T}) $ (K X ∗ , L T) is mean sensitive; (2) There exists a non-topologically-transitive and weakly multi-sensitive compact system, which is not syndetically weakly multi-sensitive compact. Besides, it is proved that every weakly mixing linear dynamical system defined on a separable Banach space is weakly multi-sensitive compact. [ABSTRACT FROM AUTHOR]

Let (X , T) be a linear dynamical system, where X is a separable Banach space and T : X → X is a bounded linear operator. We show that if (X , T) is invertible, then (X , T) is weakly sensitive compact if and only if (X , T) is thickly weakly sensitive compact; and that there exists a system (X × Y , T × S) such that: (1) (X × Y , T × S) is cofinitely weakly sensitive compact; (2) (X , T) and (Y , S) are weakly sensitive compact; and (3) (X , T) and (Y , S) are not syndetically weakly sensitive compact. We also show that if (X , T) is weakly sensitive compact, where X is a complex Banach space, then the spectrum of T meets the unit circle. [ABSTRACT FROM AUTHOR]

In this work, the global dissipativity and exponential dissipativity of quaternion‐valued memristor‐based neutral‐type neural networks (QVNTMNNs) with time‐varying delays are studied. It is worth noting that the QVNTMNN is explored as a single entity without decomposition. Considering the QVNTMNNs as a switching system and utilizing Lyapunov theorem as well as inequality techniques, several sufficient criteria depending on the state‐dependent switching parameters to guarantee global dissipativity and exponential dissipativity of QVNTMNNs with time‐varying delays are obtained in the form of quaternion‐valued linear matrix inequalities (QVLMIs) and corresponding complex‐valued liner matrix inequalities (CVLMIs). Moreover, the positive invariant set, global attractive set, and exponential attractive set of QVNTMNNs with time‐varying delays can also be derived. Finally, two numerical examples are given to substantiate the reliability and validity of our theoretical criteria presented in this work. [ABSTRACT FROM AUTHOR]

We study syndetic sensitivity and mean sensitivity for linear dynamical systems. For the syndetic sensitivity aspect, we obtain some properties of syndetic sensitivity for adjoint operators and left multiplication operators. We also show that there exists a linear dynamical system (X × Y , T × S) such that (X × Y , T × S) is cofinitely sensitive but (X , T) and (Y , S) are not syndetically sensitive. For the mean sensitivity aspect, we show that if (Y , S) is sensitive and not mean sensitive, where Y is a complex Banach space, the spectrum of T meets the unit circle. We also obtain some results regarding mean sensitive perturbations. [ABSTRACT FROM AUTHOR]

This paper investigates the state feedback control design problem to avoid players going bankrupt for a class of networked evolutionary games. First, an algebraic expression is formulated for the given networked evolutionary games by using the semi‐tensor product of matrices. Then, based on the algebraic form, a necessary and sufficient condition for the existence of state feedback control, which can guarantee that no players will go bankrupt, is presented. Finally, an illustrative example demonstrates the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

This paper addresses the problems of robust-output-controllability and robust optimal output control for incomplete Boolean control networks with disturbance inputs. First, by resorting to the semi-tensor product technique, the system is expressed as an algebraic form, based on which several necessary and sufficient conditions for the robust output controllability are presented. Second, the Mayer-type robust optimal output control issue is studied and an algorithm is established to find a control scheme which can minimize the cost functional regardless of the effect of disturbance inputs. Finally, a numerical example is given to demonstrate the effectiveness of the obtained new results. [ABSTRACT FROM AUTHOR]

By using weak Lyapunov exponent, some sufficient conditions ensuring a system ([a, b], f) to be sensitively dependent on the initial value x0 ∈ [a, b] are obtained. [ABSTRACT FROM AUTHOR]

In this paper it is proved that the backward shift operator on the Köthe sequence space admits a pair which is not asymptotic, if and only if it has an uncountable invariant ϵ-scrambled set for some ϵ > 0, if and only if it has an ϵ-scrambled subspace for some ϵ > 0, if and only if it has an invariant scrambled linear manifold. An analogous result for distributional chaos of type 2 is also obtained. [ABSTRACT FROM AUTHOR]

In this paper, various notions of chaos in the Biebutov system are studied. First, the Biebutov system is generalized to the weighted Biebutov system. Meanwhile, we introduce the concept of continuous distributional chaos for the weighted Biebutov system and prove that it exhibits continuous distributional chaos and Li-Yorke sensitivity. Finally, we prove that the weighted Biebutov system is mixing and Devaney chaotic. [ABSTRACT FROM AUTHOR]

In this paper, chaos generated by a class of nonconstant weighted shift operators is studied. First, we prove that for the weighted shift operator Bμ : Σ(X) → Σ(X) defined by Bμ(x0, x1, ...) = (μ(0)x1, μ(1)x2, ...), where X is a normed linear space (not necessarily complete), weak mix, transitivity (hypercyclity) and Devaney chaos are all equivalent to separability of X and this property is preserved under iterations. Then we get that is distributionally chaotic and Li-Yorke sensitive for each positive integer N. Meanwhile, a sufficient condition ensuring that a point is k-scrambled for all integers k > 0 is obtained. By using these results, a simple example is given to show that Corollary 3.3 in [Fu & You, 2009] does not hold. Besides, it is proved that the constructive proof of Theorem 4.3 in [Fu & You, 2009] is not correct. [ABSTRACT FROM AUTHOR]