two parameter bifurcations in a network of two neurons with multiple delays abstract we consider a network of two coupled neurons with delayed feedback we show that the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena regarding eigenvalues of the connection matrix as bifurcation parameters we obtain codimension 1 bifurcations including a fold bifurcation and a hopf bifurcation and codimension 2 bifurcations including fold hopf bifurcations and hopf hopf bifurcations we also give concrete formulae for the normal form coefficients derived via the center manifold reduction that give detailed information about the bifurcation and stability of various bifurcated solutions in particular we obtain stable or unstable equilibria periodic solutions quasi periodic solutions and sphere like surfaces of solutions we also show how to evaluate critical normal form coefficients from the original system of delay differential equations without computing the corresponding center manifolds

regular dynamics in a delayed network of two neurons with all or none activation functions we consider a delayed network of two neurons with both self feedback and interaction described by an all or none threshold function the model describes a combination of analog and digital signal processing in the network and takes the form of a system of delay differential equations with discontinuous nonlinearity we show that the dynamics of the network can be understood in terms of the iteration of a one dimensional map and we obtain simple criteria for the convergence of solutions the existence multiplicity and attractivity of periodic solutions

bifurcations stability and monotonicity properties of a delayed neural network model abstract a delay differential equation modelling an artificial neural network with two neurons is investigated a linear stability analysis provides parameter values yielding asymptotic stability of the stationary solutions these can lose stability through either a pitchfork or a hopf bifurcation which is shown to be supercritical at appropriate parameter values an interaction takes place between the pitchfork and hopf bifurcations conditions are also given for the set of initial conditions that converge to a stable stationary solution to be open and dense in the functional phase space analytic results are illustrated with numerical simulations

restrictions and unfolding of double hopf bifurcation in functional differential equations abstract the normal form of a vector field generated by scalar delay differential equations at nonresonant double hopf bifurcation points is investigated using the methods developed by faria and magalhaes j differential equations 122 1995 181 we show that 1 there exists linearly independent unfolding parameters of classes of delay differential equations for a double hopf point which generically map to linearly independent unfolding parameters of the normal form equations ordinary differential equations 2 there are generically no restrictions on the possible flows near a double hopf point for both general and z 2 symmetric first order scalar equations with two delays in the nonlinearity and 3 there always are restrictions on the possible flows near a double hopf point for first order scalar delay differential equations with one delay in the nonlinearity and in nth order scalar delay differential equations n 2 with one delay feedback

periodic solutions in an inhibitory two neuron network in this paper we consider a delayed network of two neurons with self feedback and interaction described by an all or none threshold function the discontinuity of signal function makes it difficult to apply directly dynamical system we show that the dynamics of the network can be understood in terms of the iterations of a one dimensional map and we obtain the existence and attractivity of periodic solutions moreover because the network is a limiting case of the corresponding smooth system as the parameter tends to infinity the above results can act as the guide to the rich dynamics of the smooth system therefore our results have important significance in both theory and applications

quantum homogeneous spaces as quantum quotient spaces it is shown that certain embeddable homogeneous spaces of a quantum group that do not correspond to a quantum subgroup still have the structure of quantum quotient spaces a construction of quantum fibre bundles on such spaces is proposed the quantum plane and the general quantum two spheres are discussed in detail

stability and bifurcation in a neural network model with two delays abstract a simple neural network model with two delays is considered linear stability of the model is investigated by analyzing the associated characteristic transcendental equation for the case without self connection it is found that the hopf bifurcation occurs when the sum of the two delays varies and passes a sequence of critical values the stability and direction of the hopf bifurcation are determined by applying the normal form theory and the center manifold theorem an example is given and numerical simulations are performed to illustrate the obtained results

existence results for some fourth order nonlinear elliptic problems of local superlinearity and sublinearity in this paper we study the existence of positive solutions for the problem 0 1 2u c u f x u in u 0 u 0in u u 0on where c 1 and f x u satisfies the local superlinearity and sublinearity condition

codimension one and two bifurcations in a symmetrical ring network with delay abstract in this paper we discussed all the possible codimension one and partial codimension two bifurcations existing in a symmetrical ring network using the symmetric bifurcation theory of delay differential equations coupled with the representation theory of lie groups and dynamical analysis methods we have not only figured out the pattern of each bifurcation but also predicted the directions and stability analysis of the bifurcated solutions according to structure of the system numerical simulations were then given to verify our theoretical analysis and investigate the complexity of codimension two mode intersections comments were also presented to highlight some future work

stability and bifurcations of equilibria in a multiple delayed differential equation the influence of multiple negative delayed feedback loops on the stability of a single action mechanism are considered a characteristic equation for the linearized stability of the equilibrium is completely analyzed as a function of two parameters describing a delay in one loop and a ratio of the gains in the two feedback loops the bifurcations occurring as the linear stability is lost are analyzed by the construction of a centre manifold in particular the nature of hopf and more degenerate higher codimension bifurcations are explicitly determined

attractivity multistability and bifurcation in delayed hopfieldʼs model with non monotonic feedback abstract for a system of delayed neural networks of hopfield type we deal with the study of global attractivity multistability and bifurcations in general we do not assume monotonicity conditions in the activation functions for some architectures of the network and for some families of activation functions we get optimal results on global attractivity our approach relies on a link between a system of functional differential equations and a finite dimensional discrete dynamical system for it we introduce the notion of strong attractor for a discrete dynamical system which is more restrictive than the usual concept of attractor when the dimension of the system is higher than one our principal result shows that a strong attractor of a discrete map gives a globally attractive equilibrium of a corresponding system of delay differential equations our abstract setting is not limited to applications in systems of neural networks we illustrate its use in an equation with distributed delay motivated by biological models we also obtain some results for neural systems with variable coefficients

stability bifurcation and multistability in a system of two coupled neurons with multiple time delays a system of delay differential equations representing a model for a pair of neurons with time delayed connections between the neurons and time delayed feedback from each neuron to itself is studied conditions for the linear stability of the trivial solution of this system are represented in a parameter space consisting of the sum of the time delays between the elements and the product of the strengths of the connections between the elements it is shown that the trivial fixed point may lose stability via a pitchfork bifurcation a hopf bifurcation or one of three types of codimension two bifurcations multistability near these latter bifurcations is predicted using center manifold analysis and confirmed using numerical simulations

double hopf bifurcation and quasi periodic attractors in delay coupled limit cycle oscillators abstract we investigate the double hopf bifurcation at zero equilibrium point firstly we give the critical values of hopf and double hopf bifurcations secondly we implement the normal form method and the center manifold theory for delay coupled limit cycle oscillators and derive the universal unfolding and a complete bifurcation diagram of the system thirdly many interesting phenomena such as attractive periodic motion and three dimensional invariant torus are observed using numerical simulation finally the normal forms of several strong resonant cases are listed

dynamics of a delayed two coupled oscillator with excitatory to excitatory connection in this paper we consider a neural network model consisting of two coupled oscillators with delayed feedback and excitatory to excitatory connection we study how the strength of the connections between the oscillators affects the dynamics of the neural network we give a full classification of all equilibria in the parameter space and obtain its linear stability by analyzing the characteristic equation of the linearized system we also investigate the spatio temporal patterns of bifurcated periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of lie groups moreover the stability and bifurcation direction of the bifurcated periodic solutions are obtained by employing center manifold reduction and normal form theory some numerical simulations are provided to illustrate the theoretical results

on a planar system modelling a neuron network with memory abstract we consider a delay differential equation modelling a network of two neurons with memory the situations without self connections and two delays and with self connections and two equal delays are addressed in terms of local stability and bifurcation analysis in the first case the dynamical behavior is studied by taking one of the delays as the bifurcating parameter namely the hopf bifurcation whose existence was proven in former works occurs as the delay crosses some critical values and is completely described conditions ensuring the stability of the periodic cycles are given when self connections are present hopf and pitchfork bifurcations co exist and the equation describing the flow on the center manifold is derived

on the analysis of the double hopf bifurcation in machining processes via centre manifold reduction the single degree of freedom model of orthogonal cutting is investigated to study machine tool vibrations in the vicinity of a double hopf bifurcation point centre manifold reduction and normal form calculations are performed to investigate the long term dynamics of the cutting process the normal form of the four dimensional centre subsystem is derived analytically and the possible topologies in the infinite dimensional phase space of the system are revealed it is shown that bistable parameter regions exist where unstable periodic and in certain cases unstable quasi periodic motions coexist with the equilibrium taking into account the non smoothness caused by loss of contact between the tool and the workpiece the boundary of the bistable region is also derived analytically the results are verified by numerical continuation the possibility of transient chaotic motions in the global non smooth dynamics is shown

eidetic wolf search algorithm with a global memory structure abstract a recently proposed metaheuristics called wolf search algorithm wsa has demonstrated its efficacy for various hard to solve optimization problems in this paper an improved version of wsa namely eidetic wsa with a global memory structure gms or just ewsa is presented ewsa makes use of gms for improving its search for the optimal fitness value by preventing mediocre visited places in the search space to be visited again in future iterations inherited from swarm intelligence search agents in ewsa and the traditional wsa merge into an optimal solution although the agents behave and make decisions autonomously heuristic information gathered from collective memory of the swarm search agents is stored in gms the heuristics eventually leads to faster convergence and improved optimal fitness the concept is similar to a hybrid metaheuristics based on wsa and tabu search ewsa is tested with seven standard optimization functions rigorously in particular ewsa is compared with two state of the art metaheuristics ant colony optimization aco and particle swarm optimization pso ewsa shares some similarity with both approaches with respect to directed random search the similarity with aco is however stronger as aco uses pheromones as global information references that allow a balance between using previous knowledge and exploring new solutions under comparable experimental settings identical population size and number of generations ewsa is shown to outperform both aco and pso with statistical significance when dedicating the same computation time only aco can be outperformed due to a comparably long run time per iteration of ewsa

patterns in hierarchical networks of neuronal oscillators with d3 z3 symmetry abstract in this paper we consider a hierarchically organized network composed of three interacting systems each of which consists of three coupled oscillators this hierarchical network is equivariant under the symmetry group d 3 z 3 using the lattice of isotropy subgroups we study the reduced equations restricted to invariant fixed point subspaces and prove that it is possible for the oscillator network to have 4 distinct equilibria or 45 distinct periodic solutions with maximal isotropy subgroups these are classified and their bifurcation directions are determined in terms of the quadratic coefficients and relevant quantities a center manifold reduction from the hierarchical network to the normal form equations is then performed in order to investigate the codimension two bifurcations using this reduction we find a great variety of equilibria periodic and quasi periodic oscillation patterns of maximal and submaximal symmetry which can be classified in a two level pattern hierarchy

global attractivity of a synchronized periodic orbit in a delayed network abstract we consider an artificial neural network where the signal transmission is of a digital mcculloch pitts nature and is delayed due to the finite switching speed of neurons amplifiers for a particular connection topology we show that all solutions starting from nonoscillatory initial states will be eventually synchronized and stabilized at a unique limit cycle and hence such a network can be used as a synchronized oscillator

frustration stability and delay induced oscillations in a neural network model the effect of time delays on the linear stability of equilibria in an artificial neural network of hopfield type is analyzed the possibility of delay induced oscillations occurring is characterized in terms of properties of the not necessarily symmetric connection matrix of the network such oscillations are possible exactly when the network is frustrated equivalently when the signed digraph of the matrix does not require the perron property nonlinear analysis centre manifold computation of a three unit frustrated network is presented giving the nature of the bifurcations taking place a supercritical hopf bifurcation is shown to occur and a codimension two bifurcation is unfolded

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented