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We consider a non-autonomous coupled system (x , y) , whose x coordinate satisfies a semilinear parabolic equation, and y coordinate satisfies a differential equation whose solutions do not converge too rapidly. Our aim is to study the persistence of dynamical behavior for such coupled system under a small random perturbation driven by stationary multipilcative noise. We show that for the perturbed system there exists a C 1 invariant manifold S (t , ω) = { (x , y) ∈ X × Y | x = σ (t , y , ω) } under the condition that the linear part of x equation satisfies the exponential dichotomy. Meanwhile, we observe that typically if the linear part of x equation is uniformly attracting or uniformly expanding, then the corresponding invariant manifold shares the same qualitative properties. In the end, as the perturbation tends to 0, we show that the invariant manifold and its derivative in y are approaching to those of the original system, suggesting that such structure is persistent under the small random perturbation. [ABSTRACT FROM AUTHOR]

We study admissible observation operators for perturbed evolution equations using the concept of maximal regularity. We first show the invariance of the maximal L p -regularity under non-autonomous Miyadera–Voigt perturbations. Second, we establish the invariance of admissibility of observation operators under such a class of perturbations. Finally, we illustrate our result with two examples, one on a non-autonomous parabolic system, and the other on an evolution equation subject to a Neumann boundary condition and a non local perturbation. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of non-autonomous competitive systems of difference equations with asymptotically constant coefficients. We are mainly interested in global attractivity results for such systems and the application of such results to the evolutionary population of competition models of two species. [ABSTRACT FROM AUTHOR]

We consider a SIR-type compartmental model divided into two age classes to explain the seasonal exacerbations of bacterial meningitis, especially among children outside of the meningitis belt. We describe the seasonal forcing through time-dependent transmission parameters that may represent the outbreak of the meningitis cases after the annual pilgrimage period (Hajj) or uncontrolled inflows of irregular immigrants. We present and analyse a mathematical model with time-dependent transmission. We consider not only periodic functions in the analysis but also general non-periodic transmission processes. We show that the long-time average values of transmission functions can be used as a stability marker of the equilibrium. Furthermore, we interpret the basic reproduction number in case of time-dependent transmission functions. Numerical simulations support and help visualize the theoretical results. [ABSTRACT FROM AUTHOR]

As wind and solar power penetration increases, more and more conventional power plants are being replaced; as a result, the nature of transient stability of the system evolves where the converter’s behaviour play dominating role during network events. This has necessitated a re-assessment of the nonlinear stability of the system. So far, the energy function-based transient stability method applied to synchronous machines has been applied to the converter-based system. However, there is ambiguity in terms of the damping quantification capturing the non-autonomous behaviour of the wind turbine systems, such as post-fault active current ramp rate control. This work aims to clarify the similarity between the synchronous machine model and a reduced large signal model of a wind turbine, and the difference in terms of the damping characteristics and how this impacts the system’s stability from a nonlinear perspective. A non-autonomous energy function is discussed that analytically proves that a wind turbine system with post-fault active ramp rate control is more stable compared to no ramp rate control. Finally, the stability boundary (region of attraction) is constructed and validated using time-domain simulation studies in PSCAD.

The direct parametrisation method for invariant manifolds is used for model order reduction of forced-damped mechanical structures subjected to geometric nonlinearities. Nonlinear mappings are introduced, allowing one to pass from the degrees of freedom of the finite-element model to the normal coordinates. Arbitrary orders of expansions are considered for the unknown mappings and the reduced dynamics, which are then solved sequentially through the homological equations for both autonomous and time-dependent terms. It is emphasised that the two problems share a similar structure, which can be used for an efficient implementation of the non-autonomous added terms. Special emphasis is also put on the new resonance conditions arising due to the presence of the external forcing frequencies, which allow predicting phenomena such as parametric excitation and isolas formation. The method is then applied to structures of academic and industrial interest. First, the large amplitude vibrations of a forced-damped cantilever beam are studied. This example highlights that high-order non-autonomous terms are compulsory to correctly estimate the maximum vibration amplitude experienced by the structure. The birth of isolated solutions is also illustrated on this example. The cantilever is then used to show how quadratic coupling creates conditions for the excitation of the parametric instability, and that this feature is correctly embedded in the reduction process. A shallow arch excited with multi-modal forcing is then studied to detail different forcing effects. Finally, the approach is validated on a structure of industrial relevance, i.e. a comb-driven micro-electro-mechanical resonator. The accuracy and computational performance reported suggest that the proposed methodology can accurately predict the nonlinear dynamic response of a large class of nonlinear vibratory systems. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global practical uniform h-stability analysis problem for a certain class of nonlinear non-autonomous systems where the associated nominal system is linear that depends on the time t and the perturbation term satisfies some conditions. Moreover, with the help of the new notion of practical h-stable functions, we study the separation principle in the practical sense for two classes of nonlinear non-autonomous systems having a nominal linear part. We propose some classes of memoryless state linear and nonlinear feedback controllers. Furthermore, an illustrative example is given to demonstrate the applicability of main result. [ABSTRACT FROM AUTHOR]

We establish well-posedness results for non-autonomous semilinear input-output systems, the central assumption being the scattering-passivity of the considered semilinear system. Along the way, we also establish global stability estimates. We consider both systems with distributed control and observation and systems with boundary control and observation, and we treat them in a unified manner. Applications are given to nonlinearly controlled collocated systems and to nonlinearly controlled port-Hamiltonian systems. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of non-autonomous competitive systems of difference equations with asymptotically constant coefficients. We are mainly interested in global attractivity results for such systems and the application of such results to the evolutionary population of competition models of two species.

We study the problem of existence of harmonic solutions for some generalisations of the periodically perturbed Liénard equation, where the damping function depends both on the position and the velocity. In the associated phase-space this corresponds to a term of the form f(x, y) instead of the standard dependence on x alone. We introduce suitable autonomous systems to control the orbits behaviour, allowing thus to construct invariant regions in the extended phase-space and to conclude about the existence of the harmonic solution, by invoking the Brouwer fixed point Theorem applied to the Poincaré map. Applications are given to the case of the p -Laplacian and the prescribed curvature equation. [ABSTRACT FROM AUTHOR]

The commonly defined fractional derivatives, like Riemann-Liouville and Caputo ones, are non-local operators which have the long-term memory characteristic, since they are in connection with all historical data. Because of this special property, they may be invalid for modeling some processes and materials with short-term memory phenomena. Motivated by this observation and in order to enlarge the applicability of fractional calculus theories, a fractional derivative with the short-term memory property is defined in this paper. It can be viewed as an extension of the Caputo fractional derivative. Several properties of this short memory fractional derivative are given and proved. Meanwhile, the stability problem for fractional differential equations with such a derivative is studied. By applying fractional Lyapunov direct methods, the stability conditions applicable to the local case and the global case are established respectively. Finally, three numerical examples are provided to demonstrate the correctness and effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

A non-autonomous evolution semi-linear differential system under non-instantaneous impulses, delays, and perturbed by non-local conditions is studied. Its piece-wise continuous solutions belong to a finite dimensional Banach space. The existence and uniqueness of solutions on the interval [-r, τ] are obtained by applying Karakostas' fixed-point theorem. Further results concerning solution prolongation are developed. An example is presented, and several remarks on the infinite-dimensional case are included. [ABSTRACT FROM AUTHOR]

An ecological system comprehended by a pest and its natural enemy, the predator, is considered. Parameters of system are time dependent in order to accompany their variations associated to climate evolutions. Combining the use of pesticides and of extra supply of food to predators, we propose the eradication of pest through optimal control having those two measures as controls. Is established that the resulting problem has a unique solution. Uniqueness is obtained on the whole interval using a recursive argument. The usefulness of model to tackle the pest population is backed by numerical simulation results. [ABSTRACT FROM AUTHOR]

We study long time dynamics of a randomly perturbed non-autonomous coupled system (x , y) , whose x coordinate satisfies a semilinear parabolic equation with an additive noise, and y coordinate satisfies a differential equation whose solutions do not converge too rapidly. The noise is either the white noise induced by a Brownian motion W (t , ω) or a stationary process ζ δ (θ t ω) whose integral is approximating W (t , ω). After addressing certain assumptions for such system, we show that for δ = 0 (resp. δ > 0) with respect to the noise W (t , ω) (resp. integral of ζ δ (θ t ω)) there exists a invariant manifold which is exponentially attracting any other solution outside it. Also, as δ tends to 0, the invariant manifold and its derivative in y for the case δ > 0 are approaching to those for δ = 0. [ABSTRACT FROM AUTHOR]

In this paper, several types of reiterative distributional chaos are concerned in discrete dynamical systems. Some implications between distributional chaos and reiterative distributional chaos are obtained. It is further shown that an equicontinuous non-autonomous system (X , f 1 , ∞) , where f 1 , ∞ = { f i } i ≥ 1 is a sequence of self-maps of a metric space X, exhibits reiterative distributional chaos of type i ( i ∈ { 1 , 1 + , 2 , 2 1 2 , 2 1 2 - } ) if and only if its kth iteration f 1 , ∞ [ k ] exhibits reiterative distributional chaos of type i for any k ≥ 2 . [ABSTRACT FROM AUTHOR]

This paper addresses linear quadratic control optimal problems for non-autonomous linear control systems using strongly continuous quasi semigroups. Riccati equations are implemented to investigate the control optimal problems of cost functional with finite and infinite horizons. The unique optimal pair for the cost functional is determined by the mild solution of the associated closed-loop problem and the feedback control of the solution of the corresponding Riccati equation. In addition for the infinite horizon, the stabilizability of the system is a sufficiency for the solvability to the Riccati equation. An application in a parabolic system is proposed. [ABSTRACT FROM AUTHOR]

This paper develops further and systematically the asymptotic expansion theory that was initiated by Foias and Saut in (Ann Inst H Poincaré Anal Non Linéaire, 4(1):1–47 1987). We study the long-time dynamics of a large class of dissipative systems of nonlinear ordinary differential equations with time-decaying forcing functions. The nonlinear term can be, but not restricted to, any smooth vector field which, together with its first derivative, vanishes at the origin. The forcing function can be approximated, as time tends to infinity, by a series of functions which are coherent combinations of exponential, power and iterated logarithmic functions. We prove that any decaying solution admits an asymptotic expansion, as time tends to infinity, corresponding to the asymptotic structure of the forcing function. Moreover, these expansions can be generated by more than two base functions and go beyond the polynomial formulation imposed in previous work. [ABSTRACT FROM AUTHOR]

In this article we are interested in the boundary stabilization in finite time of one-dimensional linear hyperbolic balance laws with coefficients depending on time and space. We extend the so called "backstepping method" by introducing appropriate time-dependent integral transformations in order to map our initial system to a new one which has desired stability properties. The kernels of the integral transformations involved are solutions to non standard multi-dimensional hyperbolic PDEs, where the time dependence introduces several new difficulties in the treatment of their well-posedness. This work generalizes previous results of the literature, where only time-independent systems were considered. [ABSTRACT FROM AUTHOR]

In this paper, the preimage branch t-entropy and entropy dimension for nonautonomous systems are studied and some systems with preimage branch t-entropy zero are introduced. Moreover, formulas calculating the s-topological entropy of a sequence of equi-continuous monotone maps on the unit circle are given. Finally, examples to show that the entropy dimension of non-autonomous systems can be attained by any positive number s are constructed. [ABSTRACT FROM AUTHOR]

We prove equivalence of different stronger forms of sensitivity under certain conditions for non-autonomous systems and provide examples wherever equivalence is not true in general. We also prove that if a nonminimal, topologically transitive non-autonomous system having the set of almost periodic points dense converges uniformly, then it is thickly syndetically sensitive. Moreover, we introduce the notion of property P for non-autonomous systems, study it in general and on certain induced systems. [ABSTRACT FROM AUTHOR]

The robust control problem for systems with control coefficients without a constant sign and not separated from zero is addressed. It is shown that persistence of excitation conditions for the modulus of the control gain are sufficient to ensure stability of the system and matching conditions for the disturbance are sufficient to ensure finite-time stability, in the discontinuous case. Additionally, it is shown that finite-time convergent homogeneous controllers can guarantee uniform ultimate boundedness of the solutions even in the presence of mismatched disturbances and persistence of excitation conditions for the modulus of the control gain. [ABSTRACT FROM AUTHOR]

Abstract We consider the non-autonomous dynamical system { τ n } , where τ n is a continuous map X → X , and X is a compact metric space. We assume that { τ n } converges uniformly to τ. The inheritance of chaotic properties as well as topological entropy by τ from the sequence { τ n } has been studied in [4,5,10,13,17]. In [16] the generalization of SRB measures to non-autonomous systems has been considered. In this paper we study absolutely continuous invariant measures (acim) for non-autonomous systems. After generalizing the Krylov–Bogoliubov Theorem [7] and Straube's Theorem [14] to the non-autonomous setting, we prove that under certain conditions the limit map τ of a non-autonomous sequence of maps { τ n } with acims has an acim. [ABSTRACT FROM AUTHOR]

We propose using operator learning to approximate the dynamical response of non-autonomous systems, such as nonlinear control systems. Unlike classical function learning, operator learning maps between two function spaces, does not require discretization of the output function, and provides flexibility in data preparation and solution prediction. Particularly, we apply and redesign the Deep Operator Neural Network (DeepONet) to recursively learn the solution trajectories of the dynamical systems. Our approach involves constructing and training a DeepONet that approximates the system's local solution operator. We then develop a numerical scheme that recursively simulates the system's long/medium-term dynamic response for given inputs and initial conditions, using the trained DeepONet. We accompany the proposed scheme with an estimate for the error bound of the associated cumulative error. Moreover, we propose a data-driven Runge–Kutta (RK) explicit scheme that leverages the DeepONet's forward pass and automatic differentiation to better approximate the system's response when the numerical scheme's step size is small. Numerical experiments on the predator–prey, pendulum, and cart pole systems demonstrate that our proposed DeepONet framework effectively learns to approximate the dynamical response of non-autonomous systems with time-dependent inputs. • Recursive DeepONets can approximate the response of non-autonomous systems. • Data-driven RK DeepONets can integrate challenging oscillatory responses. • A theoretical study is performed to investigate the error accumulation for DeepONets. • Recursive DeepONets can effectively tackle control and power engineering tasks. [ABSTRACT FROM AUTHOR]

Most of problems in Topological Dynamics in the theory of general autonomous discrete dynamical systems have been addressed in the non-autonomous setting. In this paper we will review some of them, giving references and stating open questions.

In this paper, we construct the pullback exponential attractors for evolution processes in which the difference of 2 solutions lacks the smoothing property. To do this, by the uniform squeezing property of the corresponding discrete process, we add the points to the pullback attractor such that every new set of it has the finite fractal dimension and pullback exponentially attracts every bounded subset of the phase space. As the applications, we establish the existence of pullback exponential attractors for non‐autonomous reaction‐diffusion equation without any restriction on the growing order of nonlinear term and non‐autonomous strongly damped wave equation in R n with critical nonlinearity. [ABSTRACT FROM AUTHOR]

Given a nonautonomous discrete dynamical system (NDS) ( X , f 1 , ∞ ) we show that transitivity and density of periodic points do not imply sensitivity in general, i.e., in the definition of Devaney chaos there are no redundant conditions for NDS. In addition, we show that if we also assume uniform convergence of the sequence ( f n ) that induces the NDS, then sensitivity follows. Furthermore, in contrast to the autonomous case, we show that there exist minimal NDS which are neither equicontinuous nor sensitive. [ABSTRACT FROM AUTHOR]

Spectral submanifolds (SSMs) have recently been shown to provide exact and unique reduced-order models for nonlinear unforced mechanical vibrations. Here, we extend these results to periodically or quasi-periodically forced mechanical systems, obtaining analytic expressions for forced responses and backbone curves on modal (i.e. two dimensional) time-dependent SSMs. A judicious choice of the parametrization of these SSMs allows us to simplify the reduced dynamics considerably. We demonstrate our analytical formulae on three numerical examples and compare them to results obtained from available normal-form methods. [ABSTRACT FROM AUTHOR]

Reservoir models play an important role in representing fluxes of matter and energy in ecological systems and are the basis of most models in biogeochemistry. These models are commonly used to study the effects of environmental change on the cycling of biogeochemical elements and to predict potential transitions of ecosystems to alternative states. To study critical regime changes of time-dependent, externally forced biogeochemical systems, we analyze the behavior of reservoir models typical for element cycling (e.g., terrestrial carbon cycle) with respect to time-varying signals by applying the mathematical concept of input to state stability (ISS). In particular, we discuss ISS as a generalization of preceding stability notions for non-autonomous, non-linear reservoir models represented by systems of ordinary differential equations explicitly dependent on time and a time-varying input signal. We also show how ISS enhances existing stability concepts, previously only available for linear time variant (LTV) systems, to a tool applicable also in the non-linear case. [ABSTRACT FROM AUTHOR]

In this paper, we study the second order non-autonomous system \begin{eqnarray*} \ddot{u}(t)+A\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0, \ \ \forall t\in\mathbb{R}, \end{eqnarray*} where $A$ is an antisymmetric $N\times N$ constant matrix, $L\in C(\mathbb{R},\mathbb{R}^{N\times N})$ may not be uniformly positive definite for all $t\in\mathbb{R}$, and $W(t,u)$ is allowed to be sign-changing and local superquadratic. Under some simple assumptions on $A$, $L$ and $W$, we establish some existence criteria to guarantee that the above system has at least one homoclinic solution or infinitely many homoclinic solutions by using mountain pass theorem or fountain theorem, respectively. Recent results in the literature are generalized and significantly improved.

Backward compact dynamics is deduced for a non-autonomous Benjamin-Bona-Mahony (BBM) equation on an unbounded 3D-channel. A backward compact attractor is defined by a time-dependent family of backward compact, invariant and pullback attracting sets. The theoretical existence result for such an attractor is derived from the backward flattening property, and this property is proved to be equivalent to the backward asymptotic compactness in a uniformly convex Banach space. Finally, it is shown that the BBM equation has a backward compact attractor in a Sobolev space under some suitable assumptions, such as, backward translation boundedness and backward small-tail. Both spectrum decomposition and cut-off technique are used to give all required backward uniform estimates. [ABSTRACT FROM AUTHOR]

Hybrid systems are dynamical systems with interacting discrete computation and continuous physical processes, which have become more common, more indispensable, and more complicated in our modern life. Particularly, many of them are safety-critical, and therefore are required to meet a critical safety standard. Invariant generation plays a central role in the verification and synthesis of hybrid systems. In the previous work, the fourth author and his coauthors gave a necessary and sufficient condition for a semi-algebraic set being an invariant of a polynomial autonomous dynamical system, which gave a confirmative answer to the open problem. In addition, based on which a complete algorithm for generating all semi-algebraic invariants of a given polynomial autonomous hybrid system with the given shape was proposed. This paper considers how to extend their work to non-autonomous dynamical and hybrid systems. Non-autonomous dynamical and hybrid systems are with inputs, which are very common in practice; in contrast, autonomous ones are without inputs. Furthermore, the authors present a sound and complete algorithm to verify semi-algebraic invariants for non-autonomous polynomial hybrid systems. Based on which, the authors propose a sound and complete algorithm to generate all invariants with a pre-defined template. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of exponential stability for a class of uncertain linear non-autonomous time-delay systems. Here, the parameter uncertainties are time-varying and unknown but norm-bounded and the delays are time-varying. Based on combination of the Riccati equation approach and the use of suitable Lyapunov-Krasovskii functional, new sufficient conditions for the robust stability are obtained in terms of the solution of Riccati-type equations. The approach allows to compute simultaneously the two bounds that characterize the exponential stability rate of the solution. As an application, sufficient conditions for the robust stabilization are derived. Numerical examples illustrated the results are given.

In this paperwe study a non-autonomous lattice dynamical system with delay. Under rather general growth and dissipative conditions on the nonlinear term,we define a non-autonomous dynamical system and prove the existence of a pullback attractor for such system as well. Both multivalued and single-valued cases are considered.

In this paper, we prove the existence of a pullback attractor in higher regularity space for the multi-valued process associated with the 2D-non-autonomous incompressible non-Newtonian fluid with delays and without the uniqueness of solutions. [ABSTRACT FROM AUTHOR]

In this paper, some aspects on the periodic solutions of the extended Duffing–Van der Pol oscillator are discussed. Doing different rescaling of the variables and parameters of the system associated with the extended Duffing–Van der Pol oscillator, we show that it can bifurcate one or three periodic solutions from a two-dimensional manifold filled by periodic solutions of the referred system. For each rescaling we exhibit concrete values for which these bounds are reached. Beyond that we characterize the stability of some periodic solutions. Our approach is analytical and the results are obtained using the averaging theory and some algebraic techniques. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with non-autonomous p-Laplacian problems in bounded domains, which are associated with multivalued processes. Under appropriate conditions, we prove that this class of problems defines a monotone evolution process for which we obtain the existence of a pullback attractor as well as the existence of a global maximal solution and a global minimal solution delimiting this attractor. [ABSTRACT FROM AUTHOR]

In this paper, we study stability of non-autonomous discrete dynamical systems. For a two-sided non-autonomous systems with covering relations determined by a transition matrixA, we show that any smallC0perturbed system has a sequence of compact invariant sets restricted to which the system is topologically semi-conjugate to σA, the two-sided subshift of finite type induced byA. Together with Lyapunov condition of good rate, the semi-conjugacy will become conjugacy. Moreover, if the Lyapunov condition is strict and has perfect rate, then any smallC1perturbed systems is topological conjugate to σA. We also study topological chaos of one-sided systems and systems with limit functions. Lack of hyperbolicity and the time dependence of the rate prevent us from applying classical hyperbolic results or earlier works for autonomous systems: cone condition and Lyapunov function. Two examples are provided to demonstrate the existence of a non-trivial, non-hyperbolic invariant and essential of controlling rate. [ABSTRACT FROM AUTHOR]

The asymptotic behaviour is studied for a class of non-autonomous infinite-dimensional non-linear dissipative systems. This is achieved by using the concept of contraction semi-flow, which is a generalization of contraction non-linear semigroup. Conditions are presented under which the solution of the abstract differential equation converges to the omega limit set (the equilibrium profile, respectively). The general development is applied to semi-linear systems with time-varying non-linearity. Asymptotic behaviour and stability criteria are established on the basis of the conditions given in the early portion of the paper. The theoretical results are applied to a general class of first-order hyperbolic time-varying semi-linear PDEs. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to show that the singularities called impasse points of real implicit ordinary differential equations are generically branch points of the solution, using a complex viewpoint. This research starts from a result asserting that the real solution in most cases behaves like ± x − x 0 at an impasse point ( x 0 , y 0 ) . We extend the notion of regular impasse point defined in previous works, and consider instead transverse impasse point, where the underlying vector field is transverse to the singular locus, even in the case when this hypersurface is not a manifold locally. Here we make use of Puiseux expansions and we show that, under generic hypotheses, the solution is multivalued at such a point. We prove that the Puiseux exponent is related simply to the multiplicity of the impasse point in the singular locus: if M is the total multiplicity of the singularity ( z 0 , y 0 ) , there is a unique solution at this point and it is M + 1 -valued. [ABSTRACT FROM AUTHOR]