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This paper considers the problem of the instability issues of stochastic differential systems with impulsive effects. Definitions of instability including almost sure exponential instability, pth moment exponential instability and pth moment instability are given. By using the Lyapunov functions, the comparison theorem and the average impulsive interval condition, several instability criteria are established. Three examples are given to illustrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper presents a predator–prey fishery model with impulsive effects, which considers transient/nontransient pulse birth and introduces the harvesting and releasing of the prey fish population at different fixed moments of time. We prove that all solutions of the investigated system are uniformly ultimately bounded. Further, the sufficient conditions for the globally asymptotical stability of the predator–extinction boundary periodic solution and the permanence of the system have been subsequently obtained. Finally, the implications of the investigated model for practical fisheries management are illustrated by numerical simulations. It is shown that releasing and harvesting the prey fish population reasonably allow the fish to persist, and thus ensure sustainable development of the fishery resources. [ABSTRACT FROM AUTHOR]

This research investigates the existence of piecewise asymptotically almost automorphic mild solutions for integrodifferential equations with infinite delay. The existence results are proved by using the Mönch's fixed point theorem, the concept of measures of non-compactness theorem and resolvent operator. Finally, an example is presented to illustrate our obtained results. [ABSTRACT FROM AUTHOR]

The reachable set estimation for two-dimensional (2-D) switched nonlinear positive systems (SNPSs) with bounded disturbances given by the Roesser model is investigated in this paper, in which both the time-varying delays and lagged impulsive effects are taken into account. By applying the average dwell time (ADT) technique, we provide a sufficient condition for the presence of a ball such that any solution of the system converges exponentially within it. An accurate estimate of the convergence rate is provided. We also extend the result to 2-D SNPSs with multi-directional delays, general 2-D switched linear systems, and 2-D SPNSs with heterogeneous delays. Finally, an example is worked out to demonstrate the effectiveness of the main result.

This paper investigates the stability in an exponential sense of complex-valued Bidirectional Associative Memory (BAM) neural networks with time delays under the stochastic and impulsive effects. By utilizing the contracting mapping theorem, the existence and uniqueness of the equilibrium point for the proposed complex-valued neural networks are verified. Moreover, based on the Lyapunov - Krasovskii functional construction, matrix inequality techniques and stability theory, some novel time-delayed sufficient conditions are attained in linear matrix inequalities (LMIs) form, which ensure the exponential stability of the trivial solution for the addressed neural networks. Finally, to illustrate the superiority and effects of our theoretical results, two numerical examples with their simulations are provided via MATLAB LMI control toolbox. [ABSTRACT FROM AUTHOR]

The authors investigate the multiplicity of solutions to a quasilinear periodic boundary value problem with impulsive effects. They use variational methods and some critical points theorems for smooth functionals, due to Ricceri, that are defined on reflexive Banach spaces. They obtain the existence of at least three solutions to the problem. The applicability of the results is illustrated with an example.

This research relies on several kinds of Volterra-type integral differential systems and their associated stability concerns under the impulsive effects of the Volterra integral terms at certain time instants. The dynamics are defined as delay-free dynamics contriobution together with the contributions of a finite set of constant point delay dynamics, plus a Volterra integral term of either a finite length or an infinite one with intrinsic memory. The global asymptotic stability is characterized via Krasovskii–Lyapuvov functionals by incorporating the impulsive effects of the Volterra-type terms together with the effects of the point delay dynamics. [ABSTRACT FROM AUTHOR]

The stabilization problem of stochastic delayed reaction-diffusion Cohen–Grossberg BAM neural networks (SDRDCGBAMNNs) with impulsive effects and boundary control is studied in this paper. By using suitable boundary controllers, Lyapunov–Krasovskii functional, linear matrix inequalities and average impulsive interval method, new sufficient criteria are found to ensure that the SDRDCGBAMNNs achieve boundary stabilization in finite-time. Based on these criteria, the effects of impulsive and boundary controllers on finite-time stability are analyzed. Numerical simulations are performed to demonstrate the feasibility of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of subharmonic solutions for the degenerate periodic systems of Lotka-Volterra type with impulsive effects. In our degenerate model, the variation of the predator and prey populations may vanish on a time interval, which imitates the (real) possibility that the predation is seasonally absent. Our proof is based on the Poincaré-Birkhoff theorem. By using phase plane analysis, we can find the large gap in the rotation numbers between the "small" solutions and the "large" solutions, which guarantees a suitable twist property. By applying the Poincaré-Birkhoff theorem, we then obtain the existence of subharmonic solutions. Our main theorem extends the associated results by J. López-Gómez et al.

In this paper, a brief survey on the stability of differential systems with impulsive effects is provided. A large number of research results on the stability of differential systems with impulsive effects are considered. These systems include impulsive differential systems, stochastic impulsive differential systems and differential systems with several specific impulses (non-instantaneous impulses, delayed impulses, impulses suffered by logic choice and impulse time windows). The stability issues as well as the applications in neural networks are discussed in detail. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and multiplicity of solutions for a class of (p,q)$$ \left(p,q\right) $$‐Laplacian Kirchhoff‐type impulsive fractional differential equations. First, under a weaker condition than the Ambrosetti–Rabinowitz condition and the Miyagaki–Souto condition, the existence of an unbounded sequence of nontrivial solutions follows from the fountain theorem. Then, some new criteria are given to guarantee that the fractional differential equation has at least two nontrivial solutions using the Nehari manifold method combined with the fibering map. [ABSTRACT FROM AUTHOR]

In this paper, the prescribed performance synchronization problem is addressed for a class of neural networks with impulsive effects. According to the prescribed performance control principle and the Lyapunov's second stability theorem, a preset performance control protocol is designed. For neural networks with impulsive effects, the proposed control scheme can not only guarantee the steady-state performance of synchronization errors, but also ensure the transient performance of the synchronization process. This improves the performance of the neural networks effectively. Finally, a numerical simulation is given to illustrate the effectiveness and feasibility of the proposed control scheme. [ABSTRACT FROM AUTHOR]

Summary: In this article, the global and exponential synchronization of Lur'e networks with hybrid time‐varying delays is investigated. For sake of saving control costs, a novel impulsive adaptive controller is presented, where the adaptive feedback control input is utilized to compensate the control effects of the impulsive signal. Meanwhile, suitable control gains are obtained according to the designing on adaptive updating laws. In view of the definition of average impulsive interval, the generalized comparison principle and the formula of parameter variation, sufficient conditions for achieving global and exponential synchronization of Lur'e dynamical networks are eventually derived. Two relevant numerical simulations are provided to illustrate the effectiveness of theoretical analysis and the control scheme. [ABSTRACT FROM AUTHOR]

This paper deals with a class of fractional-order octonion-valued neural networks (FOOVNNs) with impulsive effects. Firstly, although the multiplication of octonion numbers does not satisfy the commutativity and associativity, we don't need to separate an octonion-valued system into eight real-valued systems. Secondly, by applying the appropriate Lyapunov function, and inequality techniques, we obtain the global asymptotical synchronization of FOOVNNs. Finally, we give two illustrative examples to illustrate the feasibility of the proposed method.

Spatiotemporal dynamics of an impulsive eco-epidemiological model with Crowley-Martin type functional responses in a heterogeneous space is studied. The ultimate boundedness of solutions is obtained. The conditions of persistence and extinction under impulsive controls are derived. Furthermore, the existence and globally asymptotic stability of a unique positive periodic solutions are proved. Numerical simulations are also shown to illustrate our theoretical results. Our results show that impulsive harvesting can accelerate the extinction of ecological epidemics.

We herein report a new class of impulsive fractional stochastic differential systems driven by mixed fractional Brownian motions with infinite delay and Hurst parameter H ^ ∈ (1 / 2 , 1) . Using fixed point techniques, a q-resolvent family, and fractional calculus, we discuss the existence of a piecewise continuous mild solution for the proposed system. Moreover, under appropriate conditions, we investigate the approximate controllability of the considered system. Finally, the main results are demonstrated with an illustrative example. [ABSTRACT FROM AUTHOR]

In this paper, we examine the existence of solutions of p-Laplacian fractional differential equations with instantaneous and non-instantaneous impulses. New criteria guaranteeing the existence of infinitely many solutions are established for the considered problem. The problem is reduced to an equivalent form such that the weak solutions of the problem are defined as the critical points of an energy functional. The main result of the present work is established by using a variational approach and a mountain pass lemma. Finally, an example is given to illustrate our main result.

This paper investigates the fixed-time bipartite synchronization issue of nonlinear impulsive time-varying signed networks with time delays. Under the average impulsive interval method, two Lyapunov lemmas are given to ensure the fixed-time stability of nonlinear impulsive time-varying systems. The new settling-time estimations are accurately calculated in terms of the influences of the stabilizing and destabilizing impulses. Furthermore, based on the proposed fixed-time stability theory and the Lyapunov functional method, some sufficient conditions are established to ensure the fixed-time bipartite synchronization of the studied signed networks, and the settling-time estimations are given respectively as well. Moreover, under our theoretical results, an application of coupled time-varying spring-mass-damper systems with impulsive effects is studied. Finally, a numerical example is provided to show the effectiveness of the main results, and the simulation results reveal that synchronizing impulses can shorten the convergence time of the system states while desynchronizing impulses can increase that. • A new theoretical framework for the fixed time stability of nonlinear impulsive time-varying systems is established. • The settling time is fully estimated in terms of stabilizing and destabilizing impulses. • An application of coupled time-varying spring-mass-damper systems with impulsive effects is provided. [ABSTRACT FROM AUTHOR]

In this paper, our aims are to study the stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects. Using Lyapunov function, nonnegative semi-martingale convergence theorem, we obtain the almost sure stability with general decay rate, including the exponential stability and the polynomial stability. Moreover, we give an upper bound of each coefficient at any mode according to the theory of M-matrix. Finally, one example is given to show the effectiveness of the obtained theory.

This article addresses a class of large-scale stochastic impulsive systems with time delay and time-varying input disturbance having bounded magnitude. The main interest is to develop sufficient conditions for the input-to-state stability (ISS) and stabilization in the presence of impulsive effects. The method of Razumikhin–Lyapunov function is used to develop the ISS and stabilization properties. Later, these results are applied to a class of control systems where the controller actuators are susceptible to failures. It should be noted that our results are delay independent, and the designed reliable controller is robust with respect to the actuator failures and to the system uncertainties. It is also observed that if the isolated continuous system is ISS and subjected to bounded impulsive effects, then the resulting impulsive system preserves the ISS property. Moreover, if the isolated continuous subsystems are all ISS and the interconnection amongst them is bounded from above, then the impulsive interconnected system is ISS provided that the degree of stability of each subsystem is larger than the magnitude of interconnection. If the underlying continuous system is unstable, then the input-to-state stabilization of the impulsive system is guaranteed if the stabilizing impulses are applied to the system frequently. As an implication to these results, if the input disturbance is zero, then the input-to-state stability (or stabilization) reduces to the stability (or stabilization) of the equilibrium state of the underlying disturbance-free system. A numerical example and simulations are provided to illustrate the proposed results. [ABSTRACT FROM AUTHOR]

Abstract In this article, we establish a new class of mixed integral fractional delay dynamic systems with impulsive effects on time scales. We investigate the qualitative properties of the considered systems. In fact, the article contains three segments, and the first segment is devoted to investigating the existence and uniqueness results. In the second segment, we study the stability analysis, while the third segment is devoted to investigating the controllability criterion. We use the Leray–Schauder and Banach fixed point theorems to prove our results. Moreover, the obtained results are examined with the help of an example.

The authors give sufficient conditions for the existence of at least three classical solutions to the nonlinear impulsive fractional boundary value problem with a p-Laplacian and Dirichlet conditions D T - α Φ p (c D 0 + α u (t)) + | u (t) | p - 2 u (t) = λ f (t , u (t)) , t ≠ t j , t ∈ (0 , T) , Δ (D T - α - 1 Φ p (c D 0 + α u)) (t j) = I j (u (t j)) , u (0) = u (T) = 0 , where α ∈ (1 p , 1 ] and p > 1 . Their approach is based on variational methods. The main result is illustrated with an example. [ABSTRACT FROM AUTHOR]

In this paper, the problem of global asymptotic stability analysis for stochastic reaction-diffusion bidirectional associative memory neural networks (BAMNNs) with mixed delays and impulsive effects are investigated. The mixed delays consists of time delays in the leakage terms and continuously distributed delays. Based on the Lyapunov–Krasovskii functional (LKF), It o ^ 's differential formula and linear matrix inequality (LMI) method, some sufficient conditions for global asymptotic stability in mean square of the equilibrium point of the systems are derived. The feasibility of the conditions are verified using the MATLAB LMI toolbox. Finally, two examples are provided to illustrate the effectiveness and validity of the derived main results. [ABSTRACT FROM AUTHOR]

Synchronization of a class of nonlinear singularly perturbed complex networks (SPCNs) with semi-Markov jump topologies and impulsive effects is studied in this article. A complex network with a kind of random switching topologies is considered, where the randomness is depicted by a semi-Markov chain. A method is put forward to obtain the upper bound of singularly perturbed parameter (SPP) with different coupling strengths, and the concept of average impulsive interval is introduced to regulate the frequency of impulses. By utilizing the SPP-dependent semi-Markovian Lyapunov function, some sufficient conditions are derived for achieving synchronization of an SPCN. The effectiveness and validity of the proposed synchronization strategy are verified by two numerical examples. [ABSTRACT FROM AUTHOR]

Abstract In this study, we investigate the global exponential stability of Clifford-valued neural network (NN) models with impulsive effects and time-varying delays. By taking impulsive effects into consideration, we firstly establish a Clifford-valued NN model with time-varying delays. The considered model encompasses real-valued, complex-valued, and quaternion-valued NNs as special cases. In order to avoid the issue of non-commutativity of the multiplication of Clifford numbers, we divide the original n-dimensional Clifford-valued model into 2 m n $2^{m}n$ -dimensional real-valued models. Then we adopt the Lyapunov–Krasovskii functional and linear matrix inequality techniques to formulate new sufficient conditions pertaining to the global exponential stability of the considered NN model. Through numerical simulation, we show the applicability of the results, along with the associated analysis and discussion.

In this paper, a new stochastic predator-prey model with impulsive perturbation and Crowley-Martin functional response is proposed. The dynamical properties of the model are systematically investigated. The existence and stochastically ultimate boundedness of a global positive solution are derived using the theory of impulsive stochastic differential equations. Some sufficient criteria are obtained to guarantee the extinction and a series of persistence in the mean of the system. Moreover, we provide conditions for the stochastic permanence and global attractivity of the model. Numerical simulations are performed to support our qualitative results.

Abstract In this paper, the existence of two solutions for superlinear fourth-order impulsive elastic beam equations is obtained. We get two theorems via variational methods and corresponding two-critical-point theorems. Combining with the Newton-iterative method, an example is presented to illustrate the value of the obtained theorems.

In this paper, a class of survival red blood cells model with time-varying delays and impulsive effects is considered. First, some sufficient conditions for the persistence are derived by use of the theory on impulsive differential equations. The persistence describes the persistent survival of the mature red blood cells in the mammal under delay and impulsive perturbations. Then assuming that the coefficients in the model are ω-periodic, some criteria ensuring the existence-uniqueness and global attractivity of positive ω-periodic solution of the addressed model are obtained, which are suitable for survival red blood cells model with any ω∈R+. These global attractivity criteria describe the nonexistence of any dynamic diseases in the mammal. Moreover, our proposed results in this paper extend and improve some recent works in the literature. Finally, two examples and their computer simulations are given to show the effectiveness and advantages of the results.

This paper studies the fixed-time stochastic synchronization problem of multi-weighted complex dynamical networks with impulsive effects. Different from most of previous networks, stochastic disturbances and impulsive effects are considered in this model. The non-chattering controller is utilized, and by devising a novel time-dependent Lyapunov function and dividing the impulsive intervals, a new sufficient criterion for the fixed-time stochastic synchronization (FDTSS) of multi-weighted complex dynamical networks (MWCDNs) is obtained. Secondly, compared with some existing results, this paper proposes a novel means of calculating the settling time by constructing comparison systems and exploiting the method of the classification discussion, and the settling time does not depend on the initial values of the system. In addition, compared with the existing fixed-time and finite-time controllers, the controller devised in this paper is continuous and do not involve sign functions, which non-chattering phenomenon in practical applications. Finally, the Lorenz system is submitted to certify the justifiability of the theoretical results. [ABSTRACT FROM AUTHOR]

The impulsive complex-valued neural networks (CVNNs) with multi-proportional delays (MPDs) are considered. By setting up a suitable Lyapunov–Krasovskii functional (L-KF) and utilizing the matrix inequality skills, several delay-dependent criteria for examining the global polynomial stability (GPS) of the CVNNs are built via linear matrix inequalities (LMIs), which can be verified numerically using the valid YALMIP toolbox in MATLAB. An example with simulations is presented to highlight the potency and the efficiency of the raised criteria. [ABSTRACT FROM AUTHOR]

The primary objective of this work is to investigate the existence of mild solutions for some neutral stochastic integrodifferential systems in Hilbert spaces. These systems have a state-dependent delay and noninstantaneous impulses. In order to achieve this objective, the neutral stochastic integrodifferential system that has been proposed with state-dependent delay and noninstantaneous impulses is converted into an equivalent fixed point problem by means of an appropriate integral operator.Together with the theory of stochastic analysis, we develop several existence results that are founded on the theory of resolvent operators in the sense of Grimmer, Hausdorff's measures of noncompactness, and fixed point theorems.In the last part of this paper, specific examples are used to show how the general conclusions reached in this paper are made. [ABSTRACT FROM AUTHOR]

In this paper, we consider the multiplicity of solutions for some damped nonlinear impulsive differential equations by using Morse theory in combination with the minimax arguments. Under some assumptions, we get some new results on the existence of multiple nontrivial solutions for the problems. Thus, we improve and extend recent results. [ABSTRACT FROM AUTHOR]

This article discusses the fixed-time output synchronization of the coupled neural networks with output coupling based on impulsive effects. Two effective controls involving impulsive effects are proposed, under which sufficient conditions for output synchronization under fixed coupling weights and adaptive coupling weights are derived. Compared with the existing works, the systems investigated herein are with impulsive effects, and the settling time of synchronization is finite and independent of the initial values. Finally, a numerical example is provided to substantiate the validity and feasibility of the results. [ABSTRACT FROM AUTHOR]

In this paper, the inner synchronisation of stochastic impulsive coupled systems with multi-links (SISM) is investigated. The feedback control based on discrete-time state observations, impulsive effects and multiple links are added to stochastic coupled systems simultaneously for the first time. Instead of the mean-square exponential synchronisation, the pth ( p ≥ 2) moment exponential synchronisation is studied. A main theorem is proposed by average impulsive interval approach, graph theory and Lyapunov method to ensure the pth moment exponential synchronisation of SISM. Additionally, we apply the theoretical results to stochastic impulsive Rössler-like circuits with multi-links, which is the first attempt. Finally, a numerical example is provided to illustrate the effectiveness of the developed results. [ABSTRACT FROM AUTHOR]

In this paper, we study a delayed stage‐structured pest management model with periodic releasing infective pest, spraying pesticide and harvesting crop at three different fixed moments. By using impulsive‐type comparison results, small‐amplitude perturbation techniques, and the global attractivity of a one‐order time‐delay system, sufficient condition ensuring the global attractivity of the susceptible pest‐eradication periodic solution is derived. Furthermore, the permanence of the model is also derived. An example and its numerical simulations are given to verify the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we establish a new class of mixed integral fractional delay dynamic systems with impulsive effects on time scales. We investigate the qualitative properties of the considered systems. In fact, the article contains three segments, and the first segment is devoted to investigating the existence and uniqueness results. In the second segment, we study the stability analysis, while the third segment is devoted to investigating the controllability criterion. We use the Leray–Schauder and Banach fixed point theorems to prove our results. Moreover, the obtained results are examined with the help of an example. [ABSTRACT FROM AUTHOR]

In this article, we investigate the global uniform exponential stability problem of linear time‐varying systems with delayed impulsive effects. Two cases are considered: in the first case, the impulse‐free system is uniformly asymptotically stable and in the second case, the impulse‐free system may not be uniformly asymptotically stable. When the impulse‐free system is uniformly asymptotically stable, by the virtue of a uniformly asymptotically stable function, Lyapunov‐based sufficient conditions are derived to ensure the uniform exponential stability of linear time‐varying systems, where destabilizing and stabilizing impulses are considered, respectively. Different from existing results, the Lyapunov function is not assumed to decrease all the time and it is shown that under some conditions, the delayed impulsive effects are stabilizing. For the case that the impulse‐free system is not uniformly asymptotically stable, a time‐varying function is used to characterize the upper bound on the derivative of the Lyapunov function, and it is illustrated that the linear time‐varying systems can be stabilized by the delayed impulsive effects. Moreover, it is also shown that the impulsive input delay‐dependent criteria are less conservative than the delay‐independent ones when the impulsive input delay is small. Some examples are presented to demonstrate the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

Summary: This article is devoted to studying the problem of globally and exponentially adaptive cluster synchronization for a class of Lur'e dynamical networks with multiple time‐varying delays and hybrid couplings. A novel impulsive adaptive pinning feedback control protocol is proposed with fully considering the cluster‐tree topology structures of the networks and only imposed on the Lur'e systems in current cluster which exist directed paths with those Lur'e systems in the other clusters. In view of the concept of average impulsive interval, the comparison principle, the extended parameter variation methods, and the reductio ad absurdum, sufficient conditions are acquired for achieving the cluster synchronization of the derivative coupled Lur'e networks with considering different functions of the impulsive effects, respectively. Furthermore, according to the designed adaptive updating law, suitable feedback control strengths are obtained, which largely save the control costs. Finally, the effectiveness of the control schemes and the theoretical results have been proved by executing two numerical simulation examples. [ABSTRACT FROM AUTHOR]

The averaging method, originally offered by Krylov and Bogolyubov for ordinary differential equations, is one of the most widespread and effective methods for the analysis of nonlinear dynamical systems. Further, the averaging method was developed and applied for investigating of various problems. Impulsive systems of differential equations supply as mathematical models of objects that, during their evolution, they are subjected to the action of short-term forces. Many researches have been devoted to non-fixed impulse problems. For these problems, the existence, stability, and other asymptotic properties of solutions were studied and boundary value problems for impulsive systems were considered. Questions of the existence of periodic and almost periodic solutions to impulsive systems also were examined. In this paper, the averaging method is used to study the existence of two-sided solutions bounding on the axis of impulse systems of differential equations with non-fixed times. It is shown that a one-sided, bounding, asymptotically stable solution to the averaged system generates a two-sided solution to the exact system. The closeness of the corresponding solutions of the exact and averaged systems both on finite and infinite time intervals is substantiated by the first and second theorems of N.N. Bogolyubov.

An Impulsive Multidirectional Associative Memory Neural Network (IMAMNN) with time-varying and leakage delays is proposed. Through the use of a continuation theorem of coincidence degree theory and differential inequality techniques we establish new conditions for the existence and exponential stability of anti-periodic solutions for the model considered in this work. Moreover, two examples and its numerical simulations are presented to show the validity and the effectiveness of the results. [ABSTRACT FROM AUTHOR]

In this work, the problem of hybrid control strategy for delayed neural networks is investigated via an impulsive-based bilateral looped-functional (IBBLF) approach. Firstly, a hybrid controller is introduced, which includes feedback control and impulsive control, and the feedback control plays a vital role in the case of impulsive perturbation. Secondly, an IBBLF is constructed to relax the requirement of positive definiteness and only satisfies it at the impulsive instants, which reduces the conservatism of the stability results. Thirdly, the construction of IBBLF takes full advantage of the state information on both the intervals z (t k +) to z(t) and z(t) to z (t k + 1) , then combining with a proposed lemma, exponential stability criteria with less conservative are obtained. In addition, the obtained results are applied to nonlinear sampled-data systems. Finally, two numerical examples are presented to certify the effectiveness and superiority of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of nonlinear fractional impulsive differential equation involving Sturm-Liouville boundary-value conditions and p-Laplacian operator. By making use of critical point theorem and variational methods, some new criteria are given to guarantee that the considered problem has infinitely many solutions. Our results extend some recent results and the conditions of assumptions are easily verified. Finally, an example is given as an application of our fundamental results. [ABSTRACT FROM AUTHOR]