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The dynamical behaviour of continuous time recurrent neural network models is studied with emphasis on global stability of a unique equilibrium. First we show in a unified context two Liapunov functions that were introduced in the nineties by Hopfield, Grossberg, Matsouka and Forti. Then we introduce a class of networks for which the model becomes a special cooperative system with a unique globally stable steady state. Finally, we show that periodic orbits may occur when the sufficient conditions for the existence of Liapunov functions are violated.

The pairwise ODE model for SIS epidemic propagation on an adaptive network with link number preserving rewiring is studied. The model, introduced by Gross et al. (Phys Rev Lett 96:208701, 2006), consists of four ODEs and contains three parameters, the infection rate $$\tau $$ , the recovery rate $$\gamma $$ and the rewiring rate w. It is proved that transcritical, saddle-node and Andronov–Hopf bifurcations may occur. These bifurcation curves are determined analytically in the $$(\tau , w)$$ parameter plane by using the parametric representation method, together with the two co-dimensional Takens–Bogdanov bifurcation point. It is shown that this parameter plane is divided into four regions by the above bifurcation curves. The possible behaviours are as follows: (a) globally stable disease-free steady state, (b) stable disease-free steady state with two unstable endemic equilibria and a stable periodic orbit, (c) stable disease-free steady state with a stable and an unstable endemic equilibrium and (d) a globally stable endemic equilibrium. Numerical evidence is shown that homoclinic bifurcation, giving rise to an unstable periodic orbit, and cycle-fold bifurcation also occur.

Final epidemic size relations play a central role in mathematical epidemiology. These can be written in the form of an implicit equation which is not analytically solvable in most of the cases. While final size relations were derived for several complex models, including multiple infective stages and models in which the durations of stages are arbitrarily distributed, the solvability of those implicit equations have been less studied. In this paper the SIR homogeneous mean-field and pairwise models and the heterogeneous mean-field model are studied. It is proved that the implicit equation for the final epidemic size has a unique solution, and that through writing the implicit equation as a fixed point equation in a suitable form, the iteration of the fixed point equation converges to the unique solution. The Markovian SIR epidemic model on finite networks is also studied by using the generation-based approach. Explicit analytic formulas are derived for the final size distribution for line and star graphs of arbitrary size. Iterative formulas for the final size distribution enable us to study the accuracy of mean-field approximations for the complete graph.

Deterministic limit of a class of continuous time Markov chains is considered based purely on differential equation techniques. Starting from the linear system of master equations, ordinary differential equations for the moments and a partial differential equation, called Fokker–Planck equation, for the distribution is derived. Introducing closures at the level of the second and third moments, mean-field approximations are introduced. The accuracy of the mean-field approximations and the Fokker–Planck equation is investigated by using two differential equation-based and an operator semigroup-based approach.

Several network processes exhibiting transcritical bifurcation have globally stable steady states. Their dynamical behaviour is captured by a simple property of the right hand side of the corresponding system of ODEs. Based on this property, a class of dynamical systems is introduced, for which the local stability of the trivial steady state determines the global behaviour of the system. It is shown that this condition is satisfied by three network models, namely the individual-based and degree-based ODE approximations of SIS epidemic propagation on networks and the Hopfield model with non-negative weights. The general result enables us to describe the global behaviour of these systems that was not available for the first and third models and was proved in a significantly more complicated way for the second.

In this paper we provide the derivation of a super compact pairwise model with only 4 equations in the context of describing susceptible-infected-susceptible (SIS) epidemic dynamics on heterogenous networks. The super compact model is based on a new closure relation that involves not only the average degree but also the second and third moments of the degree distribution. Its derivation uses an a priori approximation of the degree distribution of susceptible nodes in terms of the degree distribution of the network. The new closure gives excellent agreement with heterogeneous pairwise models that contain significantly more differential equations.

In this paper we make the first steps to bridge the gap between classic control theory and modern, network-based epidemic models. In particular, we apply nonlinear model predictive control (NMPC) to a pairwise ODE model which we use to model a susceptible-infectious-susceptible (SIS) epidemic on non-trivial contact structures. While classic control of epidemics concentrates on aspects such as vaccination, quarantine and fast diagnosis, our novel setup allows us to deliver control by altering the contact network within the population. Moreover, the ideal outcome of control is to eradicate the disease while keeping the network well connected. The paper gives a thorough and detailed numerical investigation of the impact and interaction of system and control parameters on the controllability of the system. The analysis reveals, that for certain set parameters it is possible to identify critical control bounds above which the system is controllable. We foresee, that our approach can be extended to even more realistic or simulation-based models with the aim to apply these to real-world situations., Comment: 20 pages, 8 figures

In this paper we use comparison theorems from classical ODE theory in order to rigorously show that the N-Intertwined Mean-Field Approximation (NIMFA) model provides an upper estimate on the exact stochastic process. The proof of the results relies on the observation that the epidemic process is negatively correlated (in the sense that the probability of an edge being in the susceptible-infected state is smaller than the product of the probabilities of the nodes being in the susceptible and infected states, respectively), which we also prove rigorously. The results in the paper hold for arbitrary weighted and directed networks. We cast the results in a more general framework where alternative closures, other than that assuming the independence of nodes connected by an edge, are possible and provide a succinct summary of the stability analysis of the resulting more general mean-field models., Comment: 18 pages

Early studies of non-Markovian epidemics focused on SIR dynamics on fully connected networks, or homogeneously mixing populations, with the infection process being Markovian but with the infectious period taken from a general distribution [8, 278, 292, 293]. These approaches use probability theory arguments and typically focus on characterising the distribution of final epidemic sizes for finite populations, or on the average size in the infinite population limit. Similarly, the quasi-stationary distribution in a stochastic SIS model, again in a fully connected network, has been the subject of many studies [66, 230, 231]. More recently, it has been shown that one can readily apply results from queueing [19] or branching process [233] theory, or use martingales [65] to cast the same questions within a different framework and obtain results more readily.

In previous chapters, it was shown that dynamics on networks can be described by continuous-time Markov chains, where probabilities of states are determined by master equations. While limiting mean-field ODE models can provide a good approximation of the expected value of certain random variables, a PDE-based approach is needed if information on the probability distribution of these is desired. The aim of this chapter is to develop and study mathematical methods to estimate the accuracy of mean-field ODE and PDE approximations. This will be carried out for continuous-time Markov chains with state space {0, 1, …, N}, for which transition from state k is possible only to states k − 1 and k + 1. These processes are called birth–death or one-step processes.

We consider the susceptible – infected – susceptible (SIS) epidemic on a dynamic network model with addition and deletion of links depending on node status. We analyse the resulting pairwise model using classical bifurcation theory to map out the spectrum of all possible epidemic behaviours. However, the major focus of the chapter is on the evolution and possible equilibria of the resulting networks. Whereas most studies are driven by determining system-level outcomes, e.g., whether the epidemic dies out or becomes endemic, with little regard for the emerging network structure, here, we want to buck this trend by augmenting the system-level results with mapping out of the structure and properties of the resulting networks. We find that depending on parameter values the network can become disconnected and show bistable-like behaviour whereas the endemic steady state sees the emergence of networks with qualitatively different degree distributions. In particular, we observe de-phased oscillations of both prevalence and network degree during which there is role reversal between the level and nature of the connectivity of susceptible and infected nodes. We conclude with an attempt at describing what a potential bifurcation theory for networks would look like.

In this paper, an SIS (susceptible-infected-susceptible)-type epidemic propagation is studied on a special class of 3-regular graphs, called modified cycle graphs. The modified cycle graph is constructed from a cycle graph with N nodes by connecting node i to the node i + d in a way that every node has exactly three links. Monte-Carlo simulations show that the propagation process depends on the value of d in a non-monotone way. A new theoretical model is developed to explain this phenomenon. This reveals a new relation between the spreading process and the average path length in the graph.

We extrapolate from the exact master equations of epidemic dynamics on fully connected graphs to non-fully connected by keeping the size of the state space N + 1, where N is the number of nodes in the graph. This gives rise to a system of approximate ODEs (ordinary differential equations) where the challenge is to compute/approximate analytically the transmission rates. We show that this is possible for graphs with arbitrary degree distributions built according to the configuration model. Numerical tests confirm that: (a) the agreement of the approximate ODEs system with simulation is excellent and (b) that the approach remains valid for clustered graphs with the analytical calculations of the transmission rates still pending. The marked reduction in state space gives good results, and where the transmission rates can be analytically approximated, the model provides a strong alternative approximate model that agrees well with simulation. Given that the transmission rates encompass information both about the dynamics and graph properties, the specific shape of the curve, defined by the transmission rate versus the number of infected nodes, can provide a new and different measure of network structure, and the model could serve as a link between inferring network structure from prevalence or incidence data.

The bifurcations in a three-variable ODE model describing the oxygen reduction reaction on a platinum surface is studied. The investigation is motivated by the fact that this reaction plays an important role in fuel cells. The goal of this paper is to determine the dynamical behaviour of the ODE system, with emphasis on the number and type of the stationary points, and to find the possible bifurcations. It is shown that a non-trivial steady state can appear through a transcritical bifurcation, or a stable and an unstable steady state can arise as a result of saddle-node bifurcation. The saddle-node bifurcation curve is determined by using the parametric representation method, and this enables us to determine numerically the parameter domain where bistability occurs, which is important from the chemical point of view.

For a model of gasless combustion with heat loss, we use geometric singular perturbation theory to show existence of traveling combustion fronts. We show that the fronts are nonlinearly stable in an appropriate sense if an Evans function criterion, which can be verified numerically, is satisfied. For a solid reactant and exothermicity parameter that is not too large, we verify numerically that the criterion is satisfied.

Recently, research that focuses on the rigorous understanding of the relation between simulation and/or exact models on graphs and approximate counterparts has gained lots of momentum. This includes revisiting the performance of classic pairwise models with closures at the level of pairs and/or triples as well as effective-degree-type models and those based on the probability generating function formalism. In this paper, for a fully connected graph and the simple SIS (susceptible-infected-susceptible) epidemic model, a novel closure is introduced. This is done via using the equations for the moments of the distribution describing the number of infecteds at all times combined with the empirical observations that this is well described/approximated by a binomial distribution with time dependent parameters. This assumption allows us to express higher order moments in terms of lower order ones and this leads to a new closure. The significant feature of the new closure is that the difference of the exact system, given by the Kolmogorov equations, from the solution of the newly defined approximate system is of order 1/N 2. This is in contrast with the $\mathcal{O}(1/N)$ difference corresponding to the approximate system obtained via the classic triple closure. The fully connected nature of the graph also allows us to interpret pairwise equations in terms of the moments and thus treat closures and the two approximate models within the same framework. Finally, the applicability and limitations of the new methodology is discussed in detail.

In this paper, the rigorous linking of exact stochastic models to mean-field approximations is studied. Using a continuous-time Markov chain, we start from the exact formulation of a simple epidemic model on a certain class of networks, including completely connected and regular random graphs, and rigorously derive the well-known mean-field approximation that is usually justified based on biological hypotheses. We propose a unifying framework that incorporates and discusses the details of two existing proofs and we put forward a new ordinary differential equation (ODE)-based proof. The more well-known proof is based on a first-order partial differential equation approximation, while the other, more technical one, uses Martingale and Semigroup theory. We present the main steps of both proofs to investigate their applicability in different modelling contexts and to make these ideas more accessible to a broader group of applied researchers. The main result of the paper is a new ODE-based proof that may serve as a building block to prove similar convergence results for more complex networks. The new proof is based on deriving a countable system of ODEs for the moments of a distribution of interest and proving a perturbation theorem for this infinite system.

The bifurcations in a four-variable ODE model of an SIS type epidemic on an adaptive network are studied. The model describes the propagation of the epidemic on a network where links (or edges) of different type (i.e. SI;II and SS ) can be activated or deleted according to a simple rule consisting of random link activation and deletion. In the case when II links cannot be neither deleted nor created it is proved that the system can have at most three steady states with the trivial, disease-free steady state being one of them. It is shown that a stable endemic steady state can appear through a transcritical bifurcation, or a stable and an unstable endemic steady state arise as a result of saddle-node bifurcation. Moreover, at the endemic steady state a Hopf bifurcation may occur giving rise to stable oscillation. The bifurcation curves in the parameter space are determined analytically using the parametric representation method. For certain parameter regimes or bifurcation types, analytical results based on the ODE model show good agreement when compared to results based on individual- based network simulations. When agreement between the two modelling approaches holds, the ODE-based model provides a faster and more reliable tool that can be used to explore full spectrum of model behaviour.

Motivated by industrial applications, we investigate the so-called Dynamic Matrix Control (DMC) strategy for single-input single-output linear continuous-time time-invariant systems. DMC is a type of Model Predictive Control based on the step response model of the process. We show that if the process is governed by a one-dimensional stable dynamical system, then the method drives the output of the sampled system into the desired setpoint as time goes to infinity, that is, the system is asymptotically output controllable with DMC. For two-dimensional systems, sufficient condition on the asymptotic output controllability is given. Mathematics subject classification (2010): Dynamic Matrix Control, step response, asymptotic output controllability, discrete-time Hurwitz matrix.

Many if not all models of disease transmission on networks can be linked to the exact state-based Markovian formulation. However the large number of equations for any system of realistic size limits their applicability to small populations. As a result, most modelling work relies on simulation and pairwise models. In this paper, for a simple SIS dynamics on an arbitrary network, we formalise the link between a well known pairwise model and the exact Markovian formulation. This involves the rigorous derivation of the exact ODE model at the level of pairs in terms of the expected number of pairs and triples. The exact system is then closed using two different closures, one well established and one that has been recently proposed. A new interpretation of both closures is presented, which explains several of their previously observed properties. The closed dynamical systems are solved numerically and the results are compared to output from individual-based stochastic simulations. This is done for a range of networks with the same average degree and clustering coefficient but generated using different algorithms. It is shown that the ability of the pairwise system to accurately model an epidemic is fundamentally dependent on the underlying large-scale network structure. We show that the existing pairwise models are a good fit for certain types of network but have to be used with caution as higher-order network structures may compromise their effectiveness.

The exact multiplicity of positive solutions of the singular semilinear equation Δu + λ f (u) = 0 with Dirichlet boundary condition is studied. The nonlinearity f tends to infinity at zero, it is the linear combination of the functions u −α and u p with α, p > 0. The number λ is a positive parameter. The goal is to determine all possible bifurcation diagrams that can occur for different values of α and p. These kinds of equations have been widely studied on general domains, here we focus on the exact number of radial solutions on balls that can be investigated by the shooting method. It is shown how the so-called time-map can be introduced by the shooting method, and its properties determining the exact number of the positive solutions of the boundary value problem are studied. The exact multiplicity of positive solutions is given in the one-dimensional case, for which all the possible bifurcation diagrams are listed. In the higher dimensional case the bifurcation diagrams are not known for all values of α and p, in this case some open problems are collected.

The aim of this paper is to carry out the full bifurcation analysis of the two-parameter two-dimensional oscillatory cellular neural network (CNN) model (3)–(4) in Chap. 8 of the recent monograph of Chua and Roska (Cellular Neural Networks and Visual Computing, Cambridge University Press, [2002]). The main tool is an averaged divergence inequality implying that—regardless the dimension of the phase space—compact invariant sets are of zero Lebesgue measure.

The maximal operators for Cesàro or (C,α) and Riesz summability with respect to Walsh–Fourier series are investigated as mappings between dyadic Hardy and Lebesgue spaces. It is well known that they are bounded from Hp to Lp for all 1/(α+1)

The number of solutions of the differential equation x(t)+g(x(t)) = 0 subject to the boundary conditions x(t 1 ) = x 1 , x(t 2 ) = x 2 is studied. The set of the points (t 1 , t 2 , x 1 , x 2 ) for which the boundary value problem has i solutions is denoted by Σ i (g). These sets for i = 0, 1, 2,... form a partition of R 4 which is called a focal decomposition. Two differential equations are called focally equivalent if there is a homeomorphism of R 4 mapping their focal decompositions into each other. Here the equations with convex positive g are classified according to focal equivalence. It is shown that there are at least four classes, and the focal decomposition is determined by the asymptotic properties of g at infinity.

A two variable model describing the circadian fluctuation of two proteins (PER and TIM) in cells is considered. The original model was set up by Leloup and Goldbeter [1998], the present form was developed by Tyson et al. [1999]. Periodic solutions with 24-h period were investigated in those papers. Here the possible phase portraits and bifurcations are studied in detail. The saddle-node and Hopf bifurcation curves are determined in the plane of two parameters by using the parametric representation method [Simon et al., 1999]. It is shown that there are four cases according to their mutual position (as the remaining parameters of the system are varied). Using these curves the number and type of the stationary points are determined in all four cases. The global bifurcation diagram, which is a system of bifurcation curves that divide the given parameter plane into regions according to topological equivalence of global phase portraits, is also determined. Finally, the so-called constant period curves are computed numerically. These curves consist of those parameter pairs on the parameter plane for which the system has a limit cycle with a given period. It turns out that in a wide range of parameters the system has limit cycles with period close to 24-h.

A large system of ordinary differential equations is approximated by a parabolic partial differential equation with dynamic boundary condition and a different one with Robin boundary condition. Using the theory of differential operators with Wentzell boundary conditions and similar theories, we give estimates on the order of approximation. The theory is demonstrated on a voter model where the Fourier method applied to the PDE is of great advantage., Comment: reviewer requests incorporated, to appear in "Operators and Matrices"

The propagation of a premixed laminar flame supported by an exothermic chemical reaction under adiabatic conditions but subject to inhibition through a parallel endothermic chemical process is considered. The temporal stability to longitudinal perturbations of any resulting flames is investigated. The heat loss through the endothermic reaction, represented by the dimensionless parameter a, has a strong quenching effect on wave propagation. The wave speed-cooling parameter (α, c) curves are determined for a range of values of the other parameters. These curves can be monotone decreasing or S-shaped, depending on the values of the parameters β, representing the rate at which inhibitor is consumed relative to the consumption of fuel, μ, the ratio of the activation energies of the reactants and the Lewis numbers. This gives the possibility of having either one, two or three different flame velocities for the same value of the cooling parameter α. For Lewis numbers close to unity, when there are three solutions, two of them are stable and one is unstable, with two saddle-node bifurcation points on the (a, c) curve. For larger values of the Lewis numbers there is a Hopf bifurcation point on the curve, dividing it into a stable and an unstable branch. The saddle-node and Hopf bifurcation curves are also determined. The two curves have a common, Takens-Bogdanov bifurcation point.

The stability of a premixed laminar flame supported by a general combustion reaction system is considered using the Evans function method. The spectrum of the linearised second-order differential operator is investigated in detail. The special structure of the differential equations due to an Arrhenius temperature dependence is exploited. It is shown that, for certain combustion systems, the limit of the Jacobian of the reaction terms as the travelling wave coordinate approaches the front and rear of the flame is a lower triangular matrix. For this type of system a simple geometrical method is shown for the study of the essential spectrum of the linearised operator, and for determining the domain of the Evans function. The results are applied to some representative combustion reactions.

The Walsh system will be investigated in the Kaczmarz rearrangement. In an earlier paper we have shown that the maximal operator of the (C, 1)-means of the Walsh-Kaczmarz-Fourier series is bounded from the dyadic Hardy space Hp into Lp for every ½ < p ≤ 1. In the present work, we extend this result to the (C, α) means when 0 < α ≤ 1 and prove their maximal operator σα : Hp → Lp is bounded for all 1/(α + 1) < p ≤ 1. By known results on interpolation we get from this theorem that σα is of weak type (1,1) and bounded from Lq into Lq if 1 < q ≤ ∞. Moreover, the (C, α) means of an integrable function f converge to f a.e.

The linear stability of the travelling wave solutions of a general reaction-diffusion system is investigated. The spectrum of the corresponding second order differential operator $L$ is studied. The problem is reduced to an asymptotically autonomous first order linear system. The relation between the spectrum of $L$ and the corresponding first order system is dealt with in detail. The first order system is investigated using exponential dichotomies. A self-contained short presentation is shown for the study of the spectrum, with elementary proofs. An algorithm is given for the determination of the exact position of the essential spectrum. The Evans function method for determining the isolated eigenvalues of $L$ is also presented. The theory is illustrated by three examples: a single travelling wave equation, a three variable combustion model and the generalized KdV equation.

The steady propagation of a planar laminar premixed flame, with a one-step exothermic reaction and linear heat loss, is studied. The corresponding travelling wave equations are solved numerically, and the temporal stability to longitudinal perturbations of any resulting flames is investigated using the Evans function. The dependence of the flame velocity on the heat loss parameter is determined for different values of the Lewis number. These curves have a turning point, as obtained previously by asymptotic expansions for large activation energy. For Lewis numbers close to unity the upper branch of the curve gives stable flames, the lower branch unstable flames, and the turning point is a saddle-node bifurcation point. For larger values of the Lewis number there is a Hopf-bifurcation point on the upper branch of the curve, dividing it into stable and unstable sections. The saddle-node and Hopf-bifurcation curves are also determined. The two curves have a common, Takens–Bogdanov, bifurcation point.

The exact number of positive solutions of a degenerate quasilinear two-point boundary value problem is investigated. For the generalization of earlier results concerning the nondegenerate case with convex nonlinearity, suitably defined p-convex nonlinearities are considered. Strictly p-convex C2 functions having a non-negative root are classified according to the shape of the bifurcation diagram of positive solutions versus the length of the interval. We have uniqueness when f(0) ≤ 0 and 0, 1 or 2 solutions when f(0) > 0 (similarly to the non-degenerate case), provided that the number of solutions is finite. However, now there may also occur a continuum of solutions, connected to a dead core type phenomenon. The proof of our results relies on the shooting method for the characterization of the shape of the time-map. In contrast to the non-degenerate case, the shooting method does not determine directly the number of solutions, owing to the lack of uniqueness of the corresponding IVP. Exact conditions on the uniqueness of the IVP and, in the case of non-uniqueness, the number and types of its local solutions are given. Based on this, all the positive solutions of the BVP can be compiled.

Two-parameter Vilenkin systems will be investigated. First we give a general sufficient condition for multipliers to be bounded between two-dimensional Hardy spaces Hq(0 ‖f‖{Hq} ~ ‖Uf‖q (f ∈Hq)

The steady propagation of a planar laminar premixed flame, with a one-step exothermic reaction and linear heat loss, is studied. The corresponding travelling wave equations are solved numerically. The dependence of the flame velocity on the heat loss parameter is determined and compared with known results obtained by asymptotic expansion and other approximations. Due to the introduction of an ignition temperature the problem can be reduced to a bounded interval (of length L) and the graph of flame speed versus heat loss parameter can be parametrised by L. The numerical method is tested in the case of a step function nonlinearity when the exact solution of the differential equations can also be calculated.

An adaptive network model using SIS epidemic propagation with link-type dependent link activation and deletion is considered. Bifurcation analysis of the pairwise ODE approximation and the network-based stochastic simulation is carried out, showing that three typical behaviours may occur; namely, oscillations can be observed besides disease-free or endemic steady states. The oscillatory behaviour in the stochastic simulations is studied using Fourier analysis, as well as through analysing the exact master equations of the stochastic model. A compact pairwise approximation for the dynamic network case is also developed and, for the case of link-type independent rewiring, the outcome of epidemics and changes in network structure are concurrently presented in a single bifurcation diagram. By going beyond simply comparing simulation results to mean-field models, our approach yields deeper insights into the observed phenomena and help better understand and map out the limitations of mean-field models.

We investigate the stability of nontrivial nonnegative stationary solutions of semilinear initial-boundary value problems with convex or concave nonlinearity. In the convex case f(0)⩽0 implies instability, in the concave case f(0)⩾0 implies stability. We also discuss the necessity of the sign condition on f(0).

An analogue of the so—called Sunouchi operator with respect to the Walsh—Kaczmarz system will be investigated. We show the boundedness of this operator if we take it as a map from the dyadic Hardy space H p to L p for all 0

The aim of this paper is to prove Paley type inequalities for two-parameter Vilenkin system. Our main result is the following estimate: $$(\mathop \Sigma \limits_{n,k}^\infty \left( {p_n q_k } \right)^{^{1 - 2/p} } \left( {P_n Q_k } \right)^{^{2 - 2/p} } \mathop \Sigma \limits_{j = 1}^{p_{_n - 1} } \mathop \Sigma \limits_{l = 1}^{q_{_k - 1} } \left| {\hat f\left( {jP_n ,lQ_k } \right)} \right|^2 )^{1/2} {\text{ }} \leqslant C_p \left\| f \right\|_{H^p } $$ for martingales f ∈ H p (G p × G q ) (0 < p ≤ 1). Here G p and G q are Vilenkin groups generated by the sequences p = (p n ) and q = (q n ), respectively, and f^(u, v) (u, v ∈ N) is the (u,v)th (two-parameter) Vilenkin-Fourier coefficient of f. The Hardy space H p (G p × G q ) is defined by means of a usual martingal maximal function.

We prove existence of solutions for parabolic initial value problems $\partial_t u = \Delta u + f(u)$ on $R^N$ , where $f : R \rightarrow R$ is a bounded, but possibly discontinuous function.

The parameter dependence of the number and type of the stationary points of an ODE is considered. The number of the stationary points is determined by the saddle-node (SN) bifurcation set and their type (e.g., stability) is given by another bifurcation diagram (e.g., Hopf bifurcation set). The relation between these bifurcation curves on the parmeter plane is investigated. It is shown that the ‘cross-shaped diagram’, when the Hopf bifurcation curve makes a loop around a cusp point of the SN curve, is typical in some sense. It is proved that the two bifurcation curves meet tangentially at their common points (Takens–Bogdanov point), and these common points persist as a third parameter is varied. An example is shown that exhibits two different types of 3-codimensional degenerate Takens–Bogdanov bifurcation.

The one- and two-parameter Walsh system will be considered in the Paley as well as in the Kaczmarz rearrangement. We show that in the two-dimensional case the restricted maximal operator of the Walsh–Kaczmarz (C, 1)-means is bounded from the diagonal Hardy space H p to L p for every . To this end we consider the maximal operator T of a sequence of summations and show that the p-quasi-locality of T implies the same statement for its two-dimensional version T α. Moreover, we prove that the assumption is essential. Applying known results on interpolation we get the boundedness of T α as mapping from some Hardy–Lorentz spaces to Lorentz spaces. Furthermore, by standard arguments it will be shown that the usual two-parameter maximal operators of the (C, 1)-means are bounded from L p spaces to L p if . As a consequence, the a.e. convergence of the (C, 1)-means will be obtained for functions such that their hybrid maximal function is integrable. Of course, our theorems from the two-dimensional case can be extended to higher dimension in a simple way.

The Walsh system will be considered in the Kaczmarz rearrangement. We show that the maximal operator σ* of the (C,1)-means of the Walsh–Kaczmarz–Fourier series is bounded from the dyadic Hardy space Hp into Lp for every 1/2

The so-called Vilenkin systems and the Hardy spaces H p (0 p ≤ 1) with respect to Vilenkin groups will be considered. We investigate certain means of the partial sums of Vilenkin–Fourier series. It will be shown that these means, as operators from H p to L p , are bounded.

As main result we prove that certain means of the partial sums of two-parameter Vilenkin-Fourier series are uniformly bounded operators from HP to Lp (0 < p ≦ 1). The Hardy space Hp (0 < p ≦ 1) will be defined by means of a diagonal maximal function. As a consequence we obtain a so-called strong convergence theorem for the Vilenkin-Fourier partial sums. Some dual inequalities are also verified for BMO spaces.

After a brief introduction in the first theoretic part of this work the geometrical wave theory and its application for rotating waves are discussed. Here the waves are rotating around a circular obstacle which is surrounded by two homogeneous wave conducting regions with different wave velocities. The interface of the inner slow and the outer fast region is also a circle but the two circles (the obstacle and interface) are not concentric. The various asymmetric cases are classified and described theoretically. In the second experimental part chemical waves rotating in a so-called moderately asymmetric reactor are studied. A piecewise homogeneous wave conducting medium is created applying a novel reactor design. All the three theoretical cases of the moderately asymmetric arrangement are realized experimentally and qualitative and quantitative comparison of these results with the theoretical predictions show a good agreement.