## Publications

57 results found

Oljaca L, Ashwin P, Rasmussen M, 2022, Measure and statistical attractors for nonautonomous dynamical systems (Sep, 10.1007/s10884-022-10196-5, 2022), *Journal of Dynamics and Differential Equations*, Pages: 1-1, ISSN: 1040-7294

Oljača L, Ashwin P, Rasmussen M, 2022, Measure and statistical attractors for nonautonomous dynamical systems, *Journal of Dynamics and Differential Equations*, ISSN: 1040-7294

Various inequivalent notions of attraction for autonomous dynamical systems have been proposed, each of them useful to understand specific aspects of attraction. Milnor’s notion of a measure attractor considers invariant sets with positive measure basin of attraction, while Ilyashenko’s weaker notion of a statistical attractor considers positive measure points that approach the invariant set in terms of averages. In this paper we propose generalisations of these notions to nonautonomous evolution processes in continuous time. We demonstrate that pullback/forward measure/statistical attractors can be defined in an analogous manner and relate these to the respective autonomous notions when an autonomous system is considered as nonautonomous. There are some subtleties even in this special case–we illustrate an example of a two-dimensional flow with a one-dimensional measure attractor containing a single point statistical attractor. We show that the single point can be a pullback measure attractor for this system. Finally, for the particular case of an asymptotically autonomous system (where there are autonomous future and past limit systems) we relate pullback (respectively, forward) attractors to the past (respectively, future) limit systems.

Chappelle G, Hastings A, Rasmussen M, 2022, Occupancy times for time-dependent stage-structured models, *Journal of Mathematical Biology*, Vol: 84, ISSN: 0303-6812

During their lifetimes, individuals in populations pass through different states, and the notion of an occupancy time describes the amount of time an individual spends in a given set of states. Questions related to this idea were studied in a recent paper by Roth and Caswell for cases where the environmental conditions are constant. However, it is truly important to consider the case where environments are changing randomly or in directional way through time, so the transition probabilities between different states change over time, motivating the use of time-dependent stage-structured models. Using absorbing inhomogenous Markov chains and the discrete-time McKendrick–von Foerster equation, we derive explicit formulas for the occupancy time, its expectation, and its higher-order moments for stage-structured models with time-dependent transition rates. The results provide insights into the dynamics of long lived plant or animal populations where individuals transition in both directions between reproductive and non reproductive stages. We apply our approach to study a specific time-dependent model of the Southern Fulmar, and obtain insights into how the number of breeding attempts depends on external conditions that vary through time.

Castro MM, Chemnitz D, Chu H, et al., 2022, The Lyapunov spectrum for conditioned random dynamical systems, Pages: 1-36

We establish the existence of a full spectrum of Lyapunov exponents for memoryless random dynamical systems with absorption. To this end, we crucially embed the process conditioned to never being absorbed, the Q-process, into the framework of random dynamical systems, allowing us to study multiplicative ergodic properties. We show that the finite-time Lyapunov exponents converge in conditioned probability and apply our results to iterated function systems and stochastic differential equations.

Longo IP, Nunez C, Obaya R,
et al., 2021, Rate-induced tipping and saddle-node bifurcation for quadratic differential equations with nonautonomous asymptotic dynamics, *SIAM Journal on Applied Dynamical Systems*, Vol: 20, Pages: 500-540, ISSN: 1536-0040

An in-depth analysis of nonautonomous bifurcations of saddle-node type for scalar differential equations x′=−x2+q(t)x+p(t), where q:R→Randp:R→Rare bounded and uniformly continuous, is fundamental to explain the absence or occurrence of rate-induced tipping for the differential equation y′= (y−(2/π) arctan(ct))2+p(t) as the rate c varies on [0,∞). A classical attractor-repeller pair, whose existence for c= 0 is assumed, may persist for any c >0, or disappear for a certain critical rate c=c0, giving rise to rate-induced tipping. A suitable example demonstrates that one can have more than one critical rate, and the existence of the classical attractor-repeller pair may return when c increases.

Ndour M, Padberg-Gehle K, Rasmussen M, 2021, Spectral early-warning signals for sudden changes in time-dependent flow patterns, *Fluids*, Vol: 6, Pages: 49-49, ISSN: 2311-5521

Lagrangian coherent sets are known to crucially determine transport and mixing processes in non-autonomous flows. Prominent examples include vortices and jets in geophysical fluid flows. Coherent sets can be identified computationally by a probabilistic transfer-operator-based approach within a set-oriented numerical framework. Here, we study sudden changes in flow patterns that correspond to bifurcations of coherent sets. Significant changes in the spectral properties of a numerical transfer operator are heuristically related to critical events in the phase space of a time-dependent system. The transfer operator approach is applied to different example systems of increasing complexity. In particular, we study the 2002 splitting event of the Antarctic polar vortex.

Colonius F, Rasmussen M, 2021, Quasi-ergodic limits for finite absorbing Markov chains, *Linear Algebra and its Applications*, Vol: 609, Pages: 253-288, ISSN: 0024-3795

We present formulas for quasi-ergodic limits of finite absorbing Markov chains. Since the irreducible case has been solved in 1965 by Darroch and Seneta [6], we focus on the reducible case, and our results are based on a very precise asymptotic analysis of the (exponential and polynomial) growth behaviour along admissible paths.

Giesl P, Hamzi B, Rasmussen M,
et al., 2020, Approximation of Lyapunov functions from noisy data, *Journal of Computational Dynamics*, Vol: 7, Pages: 57-81, ISSN: 2158-2491

Methods have previously been developed for the approximation of Lyapunovfunctions using radial basis functions. However these methods assume that theevolution equations are known. We consider the problem of approximating a givenLyapunov function using radial basis functions where the evolution equationsare not known, but we instead have sampled data which is contaminated withnoise. We propose an algorithm in which we first approximate the underlyingvector field, and use this approximation to then approximate the Lyapunovfunction. Our approach combines elements of machine learning/statisticallearning theory with the existing theory of Lyapunov function approximation.Error estimates are provided for our algorithm.

Engel M, Lamb J, Rasmussen M, 2019, Conditioned Lyapunov exponents for random dynamical systems, *Transactions of the American Mathematical Society*, Vol: 372, Pages: 6343-6370, ISSN: 0002-9947

We introduce the notion of Lyapunov exponents for random dynamical systems, conditioned to tra-jectories that stay within a bounded domain for asymptotically long times. This is motivated by thedesire to characterize local dynamical properties in the presence of unbounded noise (when almost alltrajectories are unbounded). We illustrate its use in the analysis of local bifurcations in this context.The theory of conditioned Lyapunov exponents of stochastic differential equations builds on thestochastic analysis of quasi-stationary distributions for killed processes and associated quasi-ergodic dis-tributions. We show that conditioned Lyapunov exponents describe the asymptotic stability behaviourof trajectories that remain within a bounded domain and – in particular – that negative conditionedLyapunov exponents imply local synchronisation. Furthermore, a conditioned dichotomy spectrum isintroduced and its main characteristics are established.

Engel M, Lamb J, Rasmussen M, 2019, Bifurcation analysis of a stochastically driven limit cycle, *Communications in Mathematical Physics*, Vol: 365, Pages: 935-942, ISSN: 0010-3616

We establish the existence of a bifurcation from an attractive random equilibrium to shear-inducedchaos for a stochastically driven limit cycle, indicated by a change of sign of the first Lyapunov exponent.This relates to an open problem posed by Kevin Lin and Lai-Sang Young in [11, 16], extending resultsby Qiudong Wang and Lai-Sang Young [14] on periodically kicked limit cycles to the stochastic context.

Doan TS, Engel M, Lamb J,
et al., 2018, Hopf bifurcation with additive noise, *Nonlinearity*, Vol: 31, ISSN: 0951-7715

We consider the dynamics of a two-dimensional ordinary differential equation exhibiting a Hopf bifurcation subject to additive white noise and identify three dynamical phases: (I) a random attractor with uniform synchronisation of trajectories, (II) a random attractor with non-uniform synchronisation of trajectories and (III) a random attractor without synchronisation of trajectories. The random attractors in phases (I) and (II) are random equilibrium points with negative Lyapunov exponents while in phase (III) there is a so-called random strange attractor with positive Lyapunov exponent.We analyse the occurrence of the different dynamical phases as a function of the linear stability of the origin (deterministic Hopf bifurcation parameter) and shear (amplitude-phase coupling parameter). We show that small shear implies synchronisation and obtain that synchronisation cannot be uniform in the absence of linear stability at the origin or in the presence of sufficiently strong shear. We provide numerical results in support of a conjecture that irrespective of the linear stability of the origin, there is a critical strength of the shear at which the system dynamics loses synchronisation and enters phase (III).

Kuehn C, Malavolta G, Rasmussen M, 2018, Early-warning signals for bifurcations in random dynamical systems with bounded noise, *Journal of Mathematical Analysis and Applications*, Vol: 464, Pages: 58-77, ISSN: 0022-247X

We consider discrete-time one-dimensional random dynamical systems withbounded noise, which generate an associated set-valued dynamical system. Weprovide necessary and sufficient conditions for a discontinuous bifurcation ofa minimal invariant set of the set-valued dynamical system in terms of thederivatives of the so-called extremal maps. We propose an algorithm forreconstructing the derivatives of the extremal maps from a time series that isgenerated by iterations of the original random dynamical system. We demonstratethat the derivative reconstructed for different parameters can be used as anearly-warning signal to detect an upcoming bifurcation, and apply the algorithmto the bifurcation analysis of the stochastic return map of the Koper model,which is a three-dimensional multiple time scale ordinary differential equationused as prototypical model for the formation of mixed-mode oscillationpatterns. We apply our algorithm to data generated by this map to detect anupcoming transition.

Rasmussen M, Rieger J, Webster KN, 2018, A reinterpretation of set differential equations as differential equations in a Banach space, *Proceedings of the Royal Society of Edinburgh Section A-Mathematics*, Vol: 148, Pages: 429-446, ISSN: 1473-7124

Set differential equations are usually formulated in terms of theHukuhara differential. As a consequence, the theory of set differentialequations is perceived as an independent subject, in which all resultsare proved within the framework of the Hukuhara calculus.We propose to reformulate set differential equations as ordinarydifferential equations in a Banach space by identifying the convex andcompact subsets ofRdwith their support functions. Using this rep-resentation, standard existence and uniqueness theorems for ordinarydifferential equations can be applied to set differential equations. Weprovide a geometric interpretation of the main result, and we demon-strate that our approach overcomes the heavy restrictions the use ofthe Hukuhara differential implies for the nature of a solution.

Callaway M, Doan TS, Lamb JSW,
et al., 2017, The dichotomy spectrum for random dynamical systems and pitchfork bifurcations with additive noise, *Annales de l’Institut Henri Poincaré Probabilités et Statistiques*, Vol: 53, Pages: 1548-1574, ISSN: 0246-0203

We develop the dichotomy spectrum for random dynamical systems and demonstrate its use in the characterization of pitchfork bifurcations for random dynamical systems with additive noise.Crauel and Flandoli (J. Dynam. Differential Equations10 (1998) 259–274) had shown earlier that adding noise to a system with a deterministic pitchfork bifurcation yields a unique attracting random equilibrium with negative Lyapunov exponent throughout, thus “destroying” this bifurcation. Indeed, we show that in this example the dynamics before and after the underlying deterministic bifurcation point are topologically equivalent.However, in apparent paradox to (J. Dynam. Differential Equations10 (1998) 259–274), we show that there is after all a qualitative change in the random dynamics at the underlying deterministic bifurcation point, characterized by the transition from a hyperbolic to a non-hyperbolic dichotomy spectrum. This breakdown manifests itself also in the loss of uniform attractivity, a loss of experimental observability of the Lyapunov exponent, and a loss of equivalence under uniformly continuous topological conjugacies.Nous développons le spectre de dichotomie pour les systèmes dynamiques aléatoires et nous démontrons son utilité pour la caractérisation des bifurcations de fourches dans des systèmes dynamiques aléatoires avec du bruit additif.Crauel et Flandoli (J. Dynam. Differential Equations10 (1998) 259–274) ont précédemment montré que l’ajout de bruit additif à un système comprenant une bifurcation de fourche déterministe produit un unique équilibre aléatoire attractif avec un exposant de Lyapunov négatif partout, « détruisant » ainsi cette bifurcation. En effet, nous montrons dans cet exemple que la dynamique avant et après le point de bifurcation déterministe sous-jacent sont t

Cherubini AM, Lamb JSW, Rasmussen M,
et al., 2017, A random dynamical systems perspective on stochastic resonance, *Nonlinearity*, Vol: 30, Pages: 2835-2853, ISSN: 1361-6544

We study stochastic resonance in an over-damped approximation of the stochastic Duffing oscillator from a random dynamical systems point of view. We analyse this problem in the general framework of random dynamical systems with a nonautonomous forcing. We prove the existence of a unique global attracting random periodic orbit and a stationary periodic measure. We use the stationary periodic measure to define an indicator for the stochastic resonance.

Luo Y, Shi S, Lu X,
et al., 2017, Transient dynamics of terrestrial carbon storage: mathematical foundation and its applications, *Biogeosciences*, Vol: 14, Pages: 145-161, ISSN: 1726-4189

Terrestrial ecosystems have absorbed roughly 30 % of anthropogenic CO2 emissions over the past decades, but it is unclear whether this carbon (C) sink will endure into the future. Despite extensive modeling and experimental and observational studies, what fundamentally determines transient dynamics of terrestrial C storage under global change is still not very clear. Here we develop a new framework for understanding transient dynamics of terrestrial C storage through mathematical analysis and numerical experiments. Our analysis indicates that the ultimate force driving ecosystem C storage change is the C storage capacity, which is jointly determined by ecosystem C input (e.g., net primary production, NPP) and residence time. Since both C input and residence time vary with time, the C storage capacity is time-dependent and acts as a moving attractor that actual C storage chases. The rate of change in C storage is proportional to the C storage potential, which is the difference between the current storage and the storage capacity. The C storage capacity represents instantaneous responses of the land C cycle to external forcing, whereas the C storage potential represents the internal capability of the land C cycle to influence the C change trajectory in the next time step. The influence happens through redistribution of net C pool changes in a network of pools with different residence times.Moreover, this and our other studies have demonstrated that one matrix equation can replicate simulations of most land C cycle models (i.e., physical emulators). As a result, simulation outputs of those models can be placed into a three-dimensional (3-D) parameter space to measure their differences. The latter can be decomposed into traceable components to track the origins of model uncertainty. In addition, the physical emulators make data assimilation computationally feasible so that both C flux- and pool-related datasets can be used to better constrain model predictions of land C

Rasmussen M, Hastings A, Smith MJ,
et al., 2016, Transit times and mean ages for nonautonomous and autonomous compartmental systems, *Journal of Mathematical Biology*, Vol: 73, Pages: 1379-1398, ISSN: 1432-1416

We develop a theory for transit times and mean ages for nonautonomous compartmental systems. Using the McKendrick–von Förster equation, we show that the mean ages of mass in a compartmental system satisfy a linear nonautonomous ordinary differential equation that is exponentially stable. We then define a nonautonomous version of transit time as the mean age of mass leaving the compartmental system at a particular time and show that our nonautonomous theory generalises the autonomous case. We apply these results to study a nine-dimensional nonautonomous compartmental system modeling the terrestrial carbon cycle, which is a modification of the Carnegie–Ames–Stanford approach model, and we demonstrate that the nonautonomous versions of transit time and mean age differ significantly from the autonomous quantities when calculated for that model.

Rasmussen M, Rieger J, Webster KN, 2016, Approximation of reachable sets using optimal control and support vector machines, *Journal of Computational and Applied Mathematics*, Vol: 311, Pages: 68-83, ISSN: 0377-0427

We propose and discuss a new computational method for the numerical approximation of reachable sets for nonlinear control systems. It is based on the support vector machine algorithm and represents the set approximation as a sublevel set of a function chosen in a reproducing kernel Hilbert space. In some sense, the method can be considered as an extension to the optimal control algorithm approach recently developed by Baier, Gerdts and Xausa. The convergence of the method is illustrated numerically for selected examples.

Doan TS, Palmer KJ, Rasmussen M, 2016, The Bohl spectrum for nonautonomous differential equations, *Journal of Dynamics and Differential Equations*, Vol: 29, Pages: 1459-1485, ISSN: 1572-9222

We develop the Bohl spectrum for nonautonomous lineardifferential equation on a half line, which is a spectral concept that liesbetween the Lyapunov and the Sacker–Sell spectrum. We prove thatthe Bohl spectrum is given by the union of finitely many intervals, andwe show by means of an explicit example that the Bohl spectrum doesnot coincide with the Sacker–Sell spectrum in general even for boundedsystems. We demonstrate for this example that any higher-order nonlinearperturbation is exponentially stable (which is not evident from theSacker–Sell spectrum), but we show that in general this is not true. Wealso analyze in detail situations in which the Bohl spectrum is identicalto the Sacker–Sell spectrum.

Wang YP, Jiang J, Chen-Charpentier B,
et al., 2016, Responses of two nonlinear microbial models to warming and increased carbon input, *Biogeosciences*, Vol: 13, Pages: 887-902, ISSN: 1726-4189

A number of nonlinear microbial models of soil carbon decomposition have been developed. Some of them have been applied globally but have yet to be shown to realistically represent soil carbon dynamics in the field. A thorough analysis of their key differences is needed to inform future model developments. Here we compare two nonlinear microbial models of soil carbon decomposition: one based on reverse Michaelis–Menten kinetics (model A) and the other on regular Michaelis–Menten kinetics (model B). Using analytic approximations and numerical solutions, we find that the oscillatory responses of carbon pools to a small perturbation in their initial pool sizes dampen faster in model A than in model B. Soil warming always decreases carbon storage in model A, but in model B it predominantly decreases carbon storage in cool regions and increases carbon storage in warm regions. For both models, the CO2 efflux from soil carbon decomposition reaches a maximum value some time after increased carbon input (as in priming experiments). This maximum CO2 efflux (Fmax) decreases with an increase in soil temperature in both models. However, the sensitivity of Fmax to the increased amount of carbon input increases with soil temperature in model A but decreases monotonically with an increase in soil temperature in model B. These differences in the responses to soil warming and carbon input between the two nonlinear models can be used to discern which model is more realistic when compared to results from field or laboratory experiments. These insights will contribute to an improved understanding of the significance of soil microbial processes in soil carbon responses to future climate change.

Lamb JSW, Rasmussen M, Rodrigues CS, 2015, Topological bifurcations of minimal invariant sets for set-valued dynamical systems, *Proceedings of the American Mathematical Society*, Vol: 143, Pages: 3927-3937, ISSN: 1088-6826

We discuss the dependence of set-valued dynamical systems on parameters. Under mild assumptions which are naturally satisfied for random dynamical systems with bounded noise and control systems, we establish the fact that topological bifurcations of minimal invariant sets are discontinuous with respect to the Hausdorff metric, taking the form of lower semi-continuous explosions and instantaneous appearances. We also characterise these transitions by properties of Morse-like decompositions.

Doan TS, Rasmussen M, Kloeden PE, 2015, The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor, *Discrete and Continuous Dynamical Systems - Series B*, Vol: 20, Pages: 875-887, ISSN: 1531-3492

The dichotomy spectrum is introduced for linear mean-square random dynamical systems, and it is shown that for finite-dimensional mean-field stochastic differential equations, the dichotomy spectrum consists of finitely many compact intervals. It is then demonstrated that a change in the sign of the dichotomy spectrum is associated with a bifurcation from a trivial to a non-trivial mean-square random attractor.

Wang YP, Chen BC, Wieder WR,
et al., 2014, Oscillatory behavior of two nonlinear microbial models of soil carbon decomposition, *Biogeosciences*, Vol: 11, Pages: 1817-1831, ISSN: 1726-4189

A number of nonlinear models have recently beenproposed for simulating soil carbon decomposition. Theirpredictions of soil carbon responses to fresh litter input andwarming differ significantly from conventional linear models.Using both stability analysis and numerical simulations,we showed that two of those nonlinear models (a two-poolmodel and a three-pool model) exhibit damped oscillatory responsesto small perturbations. Stability analysis showed thefrequency of oscillation is proportional to qε−1 − 1 Ks/Vsin the two-pool model, and to qε−1 − 1 Kl/Vlin the threepoolmodel, where ε is microbial growth efficiency, Ks andKl are the half saturation constants of soil and litter carbon,respectively, and Vs and Vl are the maximal rates of carbondecomposition per unit of microbial biomass for soil and littercarbon, respectively. For both models, the oscillation hasa period of between 5 and 15 years depending on other parametervalues, and has smaller amplitude at soil temperaturesbetween 0 and 15 ◦C. In addition, the equilibrium poolsizes of litter or soil carbon are insensitive to carbon inputs inthe nonlinear model, but are proportional to carbon input inthe conventional linear model. Under warming, the microbialbiomass and litter carbon pools simulated by the nonlinearmodels can increase or decrease, depending whether ε varieswith temperature. In contrast, the conventional linear modelsalways simulate a decrease in both microbial and litter carbonpools with warming. Based on the evidence available,we concluded that the oscillatory behavior and insensitivityof soil carbon to carbon input are notable features in thesenonlinear models that are somewhat unrealistic. We recommendthat a better model for capturing the soil carbon dynamicsover decadal to centennial timescales would combinethe sensitivity of the conventional models to carbon influxwith the flexible response to warming of the nonlinear model.

Pereira T, Eldering J, Rasmussen M,
et al., 2014, Towards a theory for diffusive coupling functions allowing persistent synchronization, *Nonlinearity*, Vol: 27, Pages: 501-525, ISSN: 0951-7715

Wang YP, Chen BC, Wieder WR,
et al., 2013, Oscillatory behavior of two nonlinear microbial models of soil carbon decomposition, *Biogeosciences Discussions*, Vol: 10, Pages: 19661-19700, ISSN: 1810-6285

A number of nonlinear models have recently been proposed for simulating soil carbon decomposition. Their predictions of soil carbon responses to fresh litter input and warming differ significantly from conventional linear models. Using both stability analysis and numerical simulations, we showed that two of those nonlinear models (a two-pool model and a three-pool model) exhibit damped oscillatory responses to small perturbations. Stability analysis showed the frequency of oscillation is proportional to √ (ϵ −1−1)Ks/Vs in the two-pool model, and to √ (ϵ −1−1)Kl/Vl in the three-pool model, where ϵ is microbial growth efficiency, Ks and Kl are the half saturation constants of soil and litter carbon, respectively, and Vs and Vl are the maximal rates of carbon decomposition per unit of microbial biomass for soil and litter carbon, respectively. For both models, the oscillation has a period between 5 and 15 yr depending on other parameter values, and has smaller amplitude at soil temperatures between 0 °C to 15 °C. In addition, the equilibrium pool sizes of litter or soil carbon are insensitive to carbon inputs in the nonlinear model, but are proportional to carbon input in the conventional linear model. Under warming, the microbial biomass and litter carbon pools simulated by the nonlinear models can increase or decrease, depending whether ϵ varies with temperature. In contrast, the conventional linear models always simulate a decrease in both microbial and litter carbon pools with warming. Based on the evidence available, we concluded that the oscillatory behavior and insensitivity of soil carbon to carbon input in the nonlinear models are unrealistic. We recommend that a better model for capturing the soil carbon dynamics over decadal to centennial timescales would combine the sensitivity of the conventional models to carbon influx with the flexible response to warming of the nonlinear model.

Kloeden PE, Poetzsche C, Rasmussen M, 2013, Discrete-Time Nonautonomous Dynamical Systems, STABILITY AND BIFURCATION THEORY FOR NON-AUTONOMOUS DIFFERENTIAL EQUATIONS, CETRARO, ITALY 2011, Publisher: SPRINGER-VERLAG BERLIN, Pages: 35-102, ISBN: 978-3-642-32905-0

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Giesl P, Rasmussen M, 2012, Areas of attraction for nonautonomous differential equations on finite time intervals, *JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS*, Vol: 390, Pages: 27-46, ISSN: 0022-247X

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Kloeden PE, Poetzsche C, Rasmussen M, 2012, Limitations of pullback attractors for processes, *JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS*, Vol: 18, Pages: 693-701, ISSN: 1023-6198

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- Citations: 22

Kloeden PE, Rasmussen M, 2011, Nonautonomous Dynamical Systems, Publisher: Amer Mathematical Society, ISBN: 9780821868713

Giesl P, Rasmussen M, 2011, A NOTE ON ALMOST PERIODIC VARIATIONAL EQUATIONS, *COMMUNICATIONS ON PURE AND APPLIED ANALYSIS*, Vol: 10, Pages: 983-994, ISSN: 1534-0392

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