We consider linear reaction systems with slow and fast reactions, which can be interpreted as master equations or Kolmogorov forward equations for Markov processes on a finite state space. We investigate their limit behavior if the fast reaction rates tend to infinity, which leads to a coarse-grained model where the fast reactions create microscopically equilibrated clusters, while the exchange mass between the clusters occurs on the slow time scale. Assuming detailed balance the reaction system can be written as a gradient flow with respect to the relative entropy. Focusing on the physically relevant cosh-type gradient structure we show how an effective limit gradient structure can be rigorously derived and that the coarse-grained equation again has a cosh-type gradient structure. We obtain the strongest version of convergence in the sense of the Energy-Dissipation Principle (EDP), namely EDP-convergence with tilting.

In the present paper we advocate the Howland-Evans approach to solution of the abstract non-autonomous Cauchy problem (non-ACP) in a separable Banach space X. The main idea is to reformulate this problem as an autonomous Cauchy problem (ACP) in a new Banach space L p (I, X), p ∈ [1, ∞), consisting of X-valued functions on the time-interval I. The fundamental observation is a one-to-one correspondence between solution operators (propagators) for a non-ACP and the corresponding evolution semigroups for ACP in L p (I, X). We show that the latter also allows to apply a full power of the operator-theoretical methods to scrutinise the non-ACP including the proof of the Trotter product approximation formulae with operatornorm estimate of the rate of convergence. The paper extends and improves some recent results in this direction in particular for Hilbert spaces.where {C(t)} t∈I is a one-parameter (time-dependent) family of closed linear operators in the separable Banach space X. Here the time-interval I := [0, T ] ⊂ R and we also introduce I 0 := (0, T ]. To solve the non-autonomous Cauchy problem (non-ACP) (1.1) means to find a so-called solution operator (or propagator ): {U (t, s)} (t,s)∈∆ , ∆ = {(t, s) ∈ I 0 × I 0 : 0 < s ≤ t ≤ T }, with the property that u(t) = U (t, s)u s , (t, s) ∈ ∆, is in a certain sense a solution of the problem (1.1) for an appropriate set of initial data u s .By definition, propagator {U (t, s)} (t,s)∈∆ is supposed to be a strongly continuous operator-valued func-

Phase response curve (PRC) is an extremely useful tool for studying the response of oscillatory systems, e.g. neurons, to sparse or weak stimulation. Here we develop a framework for studying the response to a series of pulses which are frequent or/and strong so that the standard PRC fails. We show that in this case, the phase shift caused by each pulse depends on the history of several previous pulses. We call the corresponding function which measures this shift the phase response function (PRF). As a result of the introduction of the PRF, a variety of oscillatory systems with pulse interaction can be reduced to phase systems. The main assumption of the classical PRC model, i.e. that the effect of the stimulus vanishes before the next one arrives, is no longer a restriction in our approach. However, as a result of the phase reduction, the system acquires memory, which is not just a technical nuisance but an intrinsic property relevant to strong stimulation. We illustrate the PRF approach by its application to various systems, such as Morris-Lecar, Hodgkin-Huxley neuron models, and others. We show that the PRF allows predicting the dynamics of forced and coupled oscillators even when the PRC fails. Thus, the PRF provides an effective tool that may be used for simulation of neural, chemical, optic oscillators, etc.A variety of physical, chemical, biological, and other systems exhibit periodic behaviors.The state of such a system can be naturally determined by its phase [1], that is, the single variable indicating the position of the system within its cycle. The concept of the phase proved to be exceptionally useful for the study of driven and coupled oscillators [1][2][3].In order to describe the response of oscillators to an external force or coupling the so- called phase response curve (PRC) is widely used. The PRC defines the oscillator's response to a single short stimulus (pulse). The PRC can be calculated numerically or measured experimentally for oscillatory systems of different origin [4]. These properties made it a useful tool for the study of forced or coupled oscillators [5][6][7][8][9][10][11], and it is especially effective in neuroscience where the interactions are mediated by pulses. If the pulse arrivals are separated by sufficiently long time intervals, the transient caused by a pulse vanishes before the next one comes. From the theoretical point of view, it means that the system returns to the vicinity of its stable limit cycle before the next pulse arrives, see Fig. 1(a). In this case the effect of each pulse can be described by the classical PRC Z(ϕ), which determines the resulting phase shift given that the pulse arrived at the phase ϕ. Another case when the PRCs are useful is when the forcing is continuous in time but weak ( Fig. 1(b)). In this case the system remains close to the limit cycle, and the phase dynamics can be described by the so-called infinitesimal phase response curve [12].Therefore, the PRC-based approach is applicable for either weak or sparse stimulation.However, in many re...

We revise the operator-norm convergence of the Trotter product formula for a pair {A, B} of generators of semigroups on a Banach space. Operator-norm convergence holds true if the dominating operator A generates a holomorphic contraction semigroup and B is a A-infinitesimally small generator of a contraction semigroup, in particular, if B is a bounded operator. Inspired by studies of evolution semigroups it is shown in the present paper that the operator-norm convergence generally fails even for bounded operators B if A is not a holomorphic generator. Moreover, it is shown that operator norm convergence of the Trotter product formula can be arbitrary slow.

We improve some recent estimates of the rate of convergence for product approximations of solution operators for linear non-autonomous Cauchy problem. The Trotter product formula approximation is proved to converge to the solution operator in the operator-norm. We estimate the rate of convergence of this approximation. The result is applied to diffusion equation perturbed by a time-dependent potential.