17 research outputs found
Analyzing stability of a delay differential equation involving two delays
Analysis of the systems involving delay is a popular topic among applied
scientists. In the present work, we analyze the generalized equation
involving two delays
viz. and . We use the the stability conditions to
propose the critical values of delays. Using examples, we show that the chaotic
oscillations are observed in the unstable region only. We also propose a
numerical scheme to solve such equations.Comment: 10 pages, 7 figure
Classification of coupled dynamical systems with multiple delays: Finding the minimal number of delays
In this article we study networks of coupled dynamical systems with
time-delayed connections. If two such networks hold different delays on the
connections it is in general possible that they exhibit different dynamical
behavior as well. We prove that for particular sets of delays this is not the
case. To this aim we introduce a componentwise timeshift transformation (CTT)
which allows to classify systems which possess equivalent dynamics, though
possibly different sets of connection delays. In particular, we show for a
large class of semiflows (including the case of delay differential equations)
that the stability of attractors is invariant under this transformation.
Moreover we show that each equivalence class which is mediated by the CTT
possesses a representative system in which the number of different delays is
not larger than the cycle space dimension of the underlying graph. We conclude
that the 'true' dimension of the corresponding parameter space of delays is in
general smaller than it appears at first glance
Comparing Epileptiform Behavior of Mesoscale Detailed Models and Population Models of Neocortex
Two models of the neocortex are developed to study normal and pathologic neuronal activity. One model contains a detailed description of a neocortical microcolumn represented by 656 neurons, including superficial and deep pyramidal cells, four types of inhibitory neurons, and realistic synaptic contacts. Simulations show that neurons of a given type exhibit similar, synchronized behavior in this detailed model. This observation is captured by a population model that describes the activity of large neuronal populations with two differential equations with two delays. Both models appear to have similar sensitivity to variations of total network excitation. Analysis of the population model reveals the presence of multistability, which was also observed in various simulations of the detailed model
Versal unfoldings for linear retarded functional differential equations
We consider parametrized families of linear retarded functional differential
equations (RFDEs) projected onto finite-dimensional invariant manifolds, and
address the question of versality of the resulting parametrized family of
linear ordinary differential equations. A sufficient criterion for versality is
given in terms of readily computable quantities. In the case where the
unfolding is not versal, we show how to construct a perturbation of the
original linear RFDE (in terms of delay differential operators) whose
finite-dimensional projection generates a versal unfolding. We illustrate the
theory with several examples, and comment on the applicability of these results
to bifurcation analyses of nonlinear RFDEs
Existence, uniqueness, and global asymptotic stability of an equilibrium in a multiple unbounded distributed delay network
By employing the notion of M-matrices and Banach's contraction mapping theorem, we provide complete characterisation of the existence and uniqueness of an equilibrium of a Cohen–Grossberg–Hopfield-type neural network endowed with multiple distributed time delays. Invoking similar arguments, and by constructing a suitable Lyapunov functional, we establish sufficient conditions for the global asymptotic stability of the equilibrium, independent of time delays
Existence, uniqueness, and global asymptotic stability of an equilibrium in a multiple unbounded distributed delay network
By employing the notion of M-matrices and Banach’s contraction mapping principle, we provide complete characterisation of the existence and uniqueness of an equilibrium of a Cohen–Grossberg–Hopfield-type neural network endowed with multiple unbounded distributed time delays. Invoking similar arguments, and by constructing a suitable Lyapunov functional, we establish sufficient conditions for the global asymptotic stability of the equilibrium, independent of time delays
Maximum Likelihood Inference for Univariate Delay Differential Equation Models with Multiple Delays
This article presents statistical inference methodology based on maximum likelihoods for delay differential equation models in the univariate setting. Maximum likelihood inference is obtained for single and multiple unknown delay parameters as well as other parameters of interest that govern the trajectories of the delay differential equation models. The maximum likelihood estimator is obtained based on adaptive grid and Newton-Raphson algorithms. Our methodology estimates correctly the delay parameters as well as other unknown parameters (such as the initial starting values) of the dynamical system based on simulation data. We also develop methodology to compute the information matrix and confidence intervals for all unknown parameters based on the likelihood inferential framework. We present three illustrative examples related to biological systems. The computations have been carried out with help of mathematical software: MATLAB® 8.0 R2014b