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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.

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.

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.

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 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.

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 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.

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).

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 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.

The parameter dependence of the solution x of equation f0(x)+u1f1(x)+u2f2(x) = 0 is considered. Our aim is to divide the parameter plane (u1;u2) according to the number of the solutions, that is to construct a bifurcation curve. This curve is given by the singularity set, but in practice it is dicult to depict it, because it is often derived in implicit form. Here we apply the parametric representation method which has the following advantages: (1) the singularity set can be easily constructed as a curve parametrized by x, called D-curve; (2) the solutions belonging to a given parameter pair can be determined by a simple geometric algorithm based on the tangential property; (3) the global bifurcation diagram, that divides the parameter plane according to the number of solutions can be geometrically constructed with the aid of the D-curve. c 1999 Elsevier Science B.V. All rights reserved.

In practical applications of dynamical systems, it is often necessary to determine the number and the stability of the stationary states. The parameric respresentation method is a useful tool in such problems. Consider the two parameter families of functions:f(x) =uo +u1x +g(x), whereuo andu1 are the parameters. We are interested in the number of zeros as well as in the stability. We want to determine the “stable region” on the parameter plane, where the real parts of the roots off are negative. The D-curve (along which the discriminant off is zero) helps us. We applied the method to the cases of cubic and quartic equation, giving pictorial meaning to the root structure. In this respect, the R-curves and the I-curves (along which the sum or difference, respectively, of two zeros is constant) also have a significance. Using these concepts, we established a relation between the (n - 1)th Routh-Hurwitz condition and the Hopf bifurcation.

We show that the solutions of the reversible LVA model are bounded. We give a sufficient condition for the existence of a globally asymptotically stable stationary point. In the case where only the first reaction is reversible in the LVA model, we use Liapunov functions to investigate the global behaviour of the system. A certain parameterL plays an important role in the phase portrait. The stationary point in the axis x is an attractor forL > 1, with a basin containing the open positive quadrant. ForL 1/2 and loses its stability forL < 1/2 via supercritical Hopf bifurcation.

Traveling wave solutions of reaction-diffusion (semilinear parabolic) systems are studied. The number and stability of these solutions can change via different bifurcations: saddle-node, Hopf and transverse instability bifurcations. Conditions for these bifurcations can be determined from the linearization of the reaction-diffusion system. If an eigenvalue (or a pair) of the linearized system has zero real part, then a bifurcation occurs. Discretizing the linear system we obtain a matrix eigenvalue problem. It is known that its eigenvalues tend to the eigenvalues of the original system as the discretization step size goes to zero. Thus to obtain bifurcation curves we have to study the spectra of large matrices. The general bifurcation conditions for the matrices will be derived. These results will be applied to a reaction-diffusion system describing flame propagation.

The steady, spherically symmetric solutions to the reaction-diffusion equations based on a simple autocatalytic reaction followed by the decay of the autocatalyst are considered. Three parameters\char22{}the orders with respect to the autocatalyst in the autocatalysis p and in the decay q and the rate of decay of the autocatalyst relative to its autocatalytic production $K$\char22{}determine the steady concentration profiles. Numerical integrations for a fixed value of the order of the autocatalyst show that the concentration profiles have different forms depending on whether $qlp$ or $qg~p.$ In the former case, there is a critical decay rate ${K}_{\mathrm{crit}}$ for solutions to exist, with multiple solutions for $Kl{K}_{\mathrm{crit}}.$ In the latter case, there is a single solution for each value of K. This difference in the nature of the solution is confirmed by an analysis for p large. The temporal stability of the isothermal flame balls is examined, with temporally stable solutions being possible, provided that the ratio of the diffusion coefficient of the autocatalyst to that of the reactant is sufficiently small. The change in stability appears only when there are multiple solutions and is through a subcritical Hopf bifurcation.

In Section 8.1 we overview the classical theory of heat conduction formulated in thermodynamic terms. The dissipative character of pure heat conduction is manifested in the heat conductional inequality, the maximum principle, and other related properties (Section 8.2). These properties can be stated in general, far beyond the linear theory. The classical heat equation yields an infinite velocity of propagation. The hyperbolic heat equation has been proposed to overcome this paradox. However, the maximum principle is not valid for a hyperbolic equation; a satisfactory theory involving the maximum principle, as well as finite propagation, is not yet known (Section 8.3). The required basic properties may be used as postulates in searching for a new theory. An attempt of this kind is outlined. For a homogeneous one-dimensional-medium it is demonstrated in Section 8.4 that a theory for stationary heat conduction can be derived. The solution of the initial-boundary value problem for the classical linear heat conduction equation of parabolic type has several special characteristic properties, like contractivity in time, nonoscillatory behavior, exponential convergence, and others. In Section 8.5 we list some such properties. As is typical, the continuous problem cannot be solved analytically. In Section 8.6 some numerical processes are applied. This means that we define the approximate solution at the discrete points of a mesh. Obviously, the basic question is the convergence, that is, when refining the mesh, the numerical solution should be convergent to the solution of the original continuous problem. It is no less important to require the preservation of the discrete analogues of the basic qualitative properties of the continuous solution mentioned the above. In Section 8.7 the exact conditions for the preservation of some qualitative properties are established. There are damped traveling wave solutions of the sourceless parabolic heat equation. In Section 8.8 we list all the “shape-preserving signal forms,” that is, all the signal forms that can propagate inside the body without distortion, after a transient period. The solutions of this “shape-preserving” type have an attractive property: the asymptotic solutions belonging to different initial conditions tend to solutions of that kind.

В[5] было дано достаточн ое условие для того, чт обы для коэффициентов Фурье функцийƒ∈L1(G m ) иg∈C(G m ) по системе характеровG m выполнялось равенст во Парсеваля. ЗдесьG m —компактная г руппа типа Виленкина, обладающая ограниче нной структурой. В данной работе устано влена аналогичная те орема для всехG m , т.е. и дляG m без тр ебования ограниченности стру ктуры. Установлены до статочные условия типа Дини-Лип шица для равномерной иL1-сходи мости рядов Фурье по произвольной систем е типа Виленкяна. В «ограниченном» случ ае аналогичные утвер ждения были доказаны в работах [2], [3], [4], [7] и [8]. Исследованы роль с труктурыG m , а также окончательность сфо рмулированных утвер ждений.

2. In this section we introduce some definitions and notations. Let m= =(m0, rnl, ..., rnk, ...) be a sequence of natural numbers, rnk>=2 (kEN:= (0, 1 . . . . }) and denote Z ~ (kEN) the mk th discrete cyclic group where Z~,k is represented by {0, 1, ..., ink--1}. If we define G m a s the complete direct product of Z~ ' s then Gm is a compact Abelian group with Haar measure 1. The elements of Gm are of the form (xQ, x~, ..., Xk, ...) (xkEZ,,k, kEN) and the topology of G,, is completely determined by the sets

The Hardy type inequality $$\left( * \right) \left( {\sum\limits_{k = 1}^\infty {\frac{{\left| {\hat f\left( k \right)} \right|^p }}{{k^{2 - p} }}} } \right)^{I/p} \leqslant C_p \left\| f \right\|_{H_{ * * }^P } \left( {1/2< p \leqslant 2} \right)$$ is proved for functionsf belonging to the Hardy spaceH ** p (Gm) defined by means of a maximal function. We extend (*) for 2 0 provided that the Vilenkin-Fourier coefficients off are monotone.