# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

August 2012 , Volume 32 , Issue 8

Special issue in honor of John Guckenheimer
on the occasion of his 65th birthday

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2012, 32(8): i-iii doi: 10.3934/dcds.2012.32.8i +[Abstract](968) +[PDF](99.3KB)
Abstract:
The biosciences provide rich grounds for mathematical problems, and many questions require the development of new mathematical theory and algorithms. This special issue gives particular attention to new ideas and developments of the theory of and algorithms for dynamical systems that are directly or implicitly motivated by real-world applications in the biosciences. It documents the outcomes of the Current topic workshop (CTW) "New Developments in Dynamical Systems Arising from the Biosciences" held at the Mathematical Biosciences Institute (MBI) in Columbus, Ohio, in March 2011. MBI seeks to reinforce existing research efforts and inspire young talent to tackle such mathematical challenges with the aim of fostering innovations and the development of new areas in the mathematical sciences motivated by important questions in the biosciences. The workshop covered four themes to showcase how the biosciences have been inspiring recent progress: systems with delays, systems with multiple scales, dynamics of networks, and stochastic bifurcation theory. The goal was to discuss new directions of fundamental research in each of the four themes, how they are connected, and how they contribute to the understanding of specific questions in bioscience applications. Indeed, several of the talks touched on multiple themes.

2012, 32(8): 2607-2651 doi: 10.3934/dcds.2012.32.2607 +[Abstract](1653) +[PDF](686.1KB)
Abstract:
In this paper we prove that periodic boundary-value problems (BVPs) for delay differential equations are locally equivalent to finite-dimensional algebraic systems of equations. We rely only on regularity assumptions that follow those of the review by Hartung et al. (2006). Thus, the equivalence result can be applied to differential equations with state-dependent delays, transferring many results of bifurcation theory for periodic orbits to this class of systems. We demonstrate this by using the equivalence to give an elementary proof of the Hopf bifurcation theorem for differential equations with state-dependent delays. This is an extension of the Hopf bifurcation theorem by Eichmann (2006), along with an alternative proof.
2012, 32(8): 2653-2673 doi: 10.3934/dcds.2012.32.2653 +[Abstract](1870) +[PDF](603.2KB)
Abstract:
We consider two identical oscillators with time delayed coupling, modelled by a system of delay differential equations. We reduce the system of delay differential equations to a phase model where the time delay enters as a phase shift. By analyzing the phase model, we show how the time delay affects the stability of phase-locked periodic solutions and causes stability switching of in-phase and anti-phase solutions as the delay is increased. In particular, we show how the phase model can predict when the phase-flip bifurcation will occur in the original delay differential equation model. The results of the phase model analysis are applied to pairs of Morris-Lecar oscillators with diffusive or synaptic coupling and compared with numerical studies of the full system of delay differential equations.
2012, 32(8): 2675-2699 doi: 10.3934/dcds.2012.32.2675 +[Abstract](1357) +[PDF](354.2KB)
Abstract:
The subject of this paper is the analysis of the equibria of a SIR type epidemic model, which is taken as a case study among the wide family of dynamical systems of infinite dimension. For this class of systems both the determination of the stationary solutions and the analysis of their local asymptotic stability are often unattainable theoretically, thus requiring the application of existing numerical tools and/or the development of new ones. Therefore, rather than devoting our attention to the SIR model's features, its biological and physical interpretation or its theoretical mathematical analysis, the main purpose here is to discuss how to study its equilibria numerically, especially as far as their stability is concerned. To this end, we briefly analyze the construction and solution of the system of nonlinear algebraic equations leading to the stationary solutions, and then concentrate on two numerical recipes for approximating the stability determining values known as the characteristic roots. An algorithm for the purpose is given in full detail. Two applications are presented and discussed in order to show the kind of results that can be obtained with these tools.
2012, 32(8): 2701-2727 doi: 10.3934/dcds.2012.32.2701 +[Abstract](2393) +[PDF](2188.2KB)
Abstract:
We study the dynamics of a linear scalar delay differential equation $$\epsilon \dot{u}(t)=-\gamma u(t)-\sum_{i=1}^N\kappa_i u(t-a_i-c_iu(t)),$$ which has trivial dynamics with fixed delays ($c_i=0$). We show that if the delays are allowed to be linearly state-dependent ($c_i\ne0$) then very complex dynamics can arise, when there are two or more delays. We present a numerical study of the bifurcation structures that arise in the dynamics, in the non-singularly perturbed case, $\epsilon=1$. We concentrate on the case $N=2$ and $c_1=c_2=c$ and show the existence of bistability of periodic orbits, stable invariant tori, isola of periodic orbits arising as locked orbits on the torus, and period doubling bifurcations.
2012, 32(8): 2729-2757 doi: 10.3934/dcds.2012.32.2729 +[Abstract](1922) +[PDF](1411.3KB)
Abstract:
Some neurons in the nervous system do not show repetitive firing for steady currents. For time-varying inputs, they fire once if the input rise is fast enough. This property of phasic firing is known as Type III excitability. Type III excitability has been observed in neurons in the auditory brainstem (MSO), which show strong phase-locking and accurate coincidence detection. In this paper, we consider a Hodgkin-Huxley type model (RM03) that is widely-used for phasic MSO neurons and we compare it with a modification of it, showing tonic behavior. We provide insight into the temporal processing of these neuron models by means of developing and analyzing two reduced models that reproduce qualitatively the properties of the exemplar ones. The geometric and mathematical analysis of the reduced models allows us to detect and quantify relevant features for the temporal computation such as nearness to threshold and a temporal integration window. Our results underscore the importance of Type III excitability for precise coincidence detection.
2012, 32(8): 2759-2803 doi: 10.3934/dcds.2012.32.2759 +[Abstract](1337) +[PDF](1045.6KB)
Abstract:
In [C. W. Gear, T. J. Kaper, I. G. Kevrekidis and A. Zagaris, Projecting to a slow manifold: Singularly perturbed systems and legacy codes, SIAM J. Appl. Dyn. Syst. 4 (2005), 711--732], we developed the family of constrained runs algorithms to find points on low-dimensional, attracting, slow manifolds in systems of nonlinear differential equations with multiple time scales. For user-specified values of a subset of the system variables parametrizing the slow manifold (which we term observables and denote collectively by $u$), these iterative algorithms return values of the remaining system variables $v$ so that the point $(u,v)$ approximates a point on a slow manifold. In particular, the $m-$th constrained runs algorithm ($m = 0, 1, \ldots$) approximates a point $(u,v_m)$ that is the appropriate zero of the $(m+1)-$st time derivative of $v$. % The accuracy with which $(u,v_m)$ approximates the corresponding point on the slow manifold with the same value of the observables has been established in [A. Zagaris, C. W. Gear, T. J. Kaper and I. G. Kevrekidis, Analysis of the accuracy and convergence of equation-free projection to a slow manifold, ESAIM: M2AN 43(4) (2009) 757--784] for systems for which the observables $u$ evolve exclusively on the slow time scale. There, we also determined explicit conditions under which the $m-$th constrained runs scheme converges to the fixed point $(u,v_m)$ and identified conditions under which it fails to converge. Here, we consider the questions of stability and stabilization of these iterative algorithms for the case in which the observables $u$ are also allowed to evolve on a fast time scale. The stability question in this case is more complicated, since it involves a generalized eigenvalue problem for a pair of matrices encoding geometric and dynamical characteristics of the system of differential equations. We determine the conditions under which these schemes converge or diverge in a series of cases in which this problem is explicitly solvable. We illustrate our main stability and stabilization results for the constrained runs schemes on certain planar systems with multiple time scales, and also on a more-realistic sixth order system with multiple time scales that models a network of coupled enzymatic reactions. Finally, we consider the issue of stabilization of the $m-$th constrained runs algorithm when the functional iteration scheme is divergent or converges slowly. In that case, we demonstrate on concrete examples how Newton's method and Broyden's method may replace functional iteration to yield stable iterative schemes.
2012, 32(8): 2805-2823 doi: 10.3934/dcds.2012.32.2805 +[Abstract](1750) +[PDF](2069.3KB)
Abstract:
Hopf bifurcation in systems with multiple time scales takes several forms, depending upon whether the bifurcation occurs in fast directions, slow directions or a mixture of these two. Hopf bifurcation in fast directions is influenced by the singular limit of the fast time scale, that is, when the ratio $\epsilon$ of the slowest and fastest time scales goes to zero. The bifurcations of the full slow-fast system persist in the layer equations obtained from this singular limit. However, the Hopf bifurcation of the layer equations does not necessarily have the same criticality as the corresponding Hopf bifurcation of the full slow-fast system, even in the limit $\epsilon \to 0$ when the two bifurcations occur at the same point. We investigate this situation by presenting a simple slow-fast system that is amenable to a complete analysis of its bifurcation diagram. In this model, the family of periodic orbits that emanates from the Hopf bifurcation accumulates onto the corresponding family of the layer equations in the limit as $\epsilon \to 0$; furthermore, the stability of the orbits is dictated by that of the layer equation. We prove that a torus bifurcation occurs $O(\epsilon)$ near the Hopf bifurcation of the full system when the criticality of the two Hopf bifurcations is different.
2012, 32(8): 2825-2851 doi: 10.3934/dcds.2012.32.2825 +[Abstract](1848) +[PDF](1276.0KB)
Abstract:
Global bifurcations involving saddle periodic orbits have recently been recognized as being involved in various new types of organizing centers for complicated dynamics. The main emphasis has been on heteroclinic connections between saddle equilibria and saddle periodic orbits, called EtoP orbits for short, which can be found in vector fields in $\mathbb{R}^3$. Thanks to the development of dedicated numerical techniques, EtoP orbits have been found in a number of three-dimensional model vector fields arising in applications.
We are concerned here with the case of heteroclinic connections between two saddle periodic orbits, called PtoP orbits for short. A homoclinic orbit from a periodic orbit to itself is an example of a PtoP connection, but is generically structurally stable in a phase space of any dimension. The issue that we address here is that, until now, no example of a concrete vector field with a non-structurally stable PtoP connection was known. We present an example of a PtoP heteroclinic cycle of codimension one between two different saddle periodic orbits in a four-dimensional vector field model of intracellular calcium dynamics. We first show that this model is a good candidate system for the existence of such a PtoP cycle and then demonstrate how a PtoP cycle can be detected and continued in system parameters using a numerical setup that is based on Lin's method.
2012, 32(8): 2853-2877 doi: 10.3934/dcds.2012.32.2853 +[Abstract](1367) +[PDF](877.7KB)
Abstract:
A great deal of work has gone into classifying bursting oscillations, periodic alternations of spiking and quiescence modeled by fast-slow systems. In such systems, one or more slow variables carry the fast variables through a sequence of bifurcations that mediate transitions between oscillations and steady states. A rigorous classification approach is to characterize the bifurcations found in the neighborhood of a singularity; a measure of the complexity of the bursting oscillation is then given by the smallest codimension of the singularities near which it occurs. Fold/homoclinic bursting, along with most other burst types of interest, has been shown to occur near a singularity of codimension three by examining bifurcations of a cubic Liénard system; hence, these types of bursting have at most codimension three. Modeling and biological considerations suggest that fold/homoclinic bursting should be found near fold/subHopf bursting, a more recently identified burst type whose codimension has not been determined yet. One would expect that fold/subHopf bursting has the same codimension as fold/homoclinic bursting, because models of these two burst types have very similar underlying bifurcation diagrams. However, no codimension-three singularity is known that supports fold/subHopf bursting, which indicates that it may have codimension four. We identify a three-dimensional slice in a partial unfolding of a doubly-degenerate Bodganov-Takens point, and show that this codimension-four singularity gives rise to almost all known types of bursting.
2012, 32(8): 2879-2912 doi: 10.3934/dcds.2012.32.2879 +[Abstract](1453) +[PDF](2400.1KB)
Abstract:
Mixed mode oscillations (MMOs) are complex oscillatory waveforms that naturally occur in physiologically relevant dynamical processes. MMOs were studied in a model of electrical bursting in a pituitary lactotroph [34] where geometric singular perturbation theory and bifurcation analysis were combined to demonstrate that the MMOs arise from canard dynamics. In this work, we extend the analysis done in [34] and consider bifurcations of canard solutions under variations of key parameters. To do this, a global return map induced by the flow of the equations is constructed and a qualitative analysis given. The canard solutions act as separatrices in the return maps, organising the dynamics along the Poincaré section. We examine the bifurcations of the return maps and demonstrate that the map formulation allows for an explanation of the different MMO patterns observed in the lactotroph model.
2012, 32(8): 2913-2935 doi: 10.3934/dcds.2012.32.2913 +[Abstract](1546) +[PDF](541.7KB)
Abstract:
This paper discusses feed-forward chains near points of synchrony-breaking Hopf bifurcation. We show that at synchrony-breaking bifurcations the center manifold inherits a feed-forward structure and use this structure to provide a simplified proof of the theorem of Elmhirst and Golubitsky that there is a branch of periodic solutions in such bifurcations whose amplitudes grow at the rate of $\lambda^{\frac{1}{6}}$. We also use this center manifold structure to provide a method for classifying the bifurcation diagrams of the forced feed-forward chain where the amplitudes of the periodic responses are plotted as a function of the forcing frequency. The bifurcation diagrams depend on the amplitude of the forcing, the deviation of the system from Hopf bifurcation, and the ratio $\gamma$ of the imaginary part of the cubic term in the normal form of Hopf bifurcation to the real part. These calculations generalize the results of Zhang on the forcing of systems near Hopf bifurcations to three-cell feed-forward chains.
2012, 32(8): 2937-2950 doi: 10.3934/dcds.2012.32.2937 +[Abstract](1770) +[PDF](284.5KB)
Abstract:
The metabolic network of a living cell involves several hundreds or thousands of interconnected biochemical reactions. Previous research has shown that under realistic conditions only a fraction of these reactions is concurrently active in any given cell. This is partially determined by nutrient availability, but is also strongly dependent on the metabolic function and network structure. Here, we establish rigorous bounds showing that the fraction of active reactions is smaller (rather than larger) in metabolic networks evolved or engineered to optimize a specific metabolic task, and we show that this is largely determined by the presence of thermodynamically irreversible reactions in the network. We also show that the inactivation of a certain number of reactions determined by irreversibility can generate a cascade of secondary reaction inactivations that propagates through the network. The mathematical results are complemented with numerical simulations of the metabolic networks of the bacterium Escherichia coli and of human cells, which show, counterintuitively, that even the maximization of the total reaction flux in the network leads to a reduced number of active reactions.
2012, 32(8): 2951-2970 doi: 10.3934/dcds.2012.32.2951 +[Abstract](2083) +[PDF](829.5KB)
Abstract:
Translationally invariant integro-differential equations are a common choice of model in neuroscience for describing the coarse-grained dynamics of cortical tissue. Here we analyse the propagation of travelling wavefronts in models of neural media that incorporate some form of modulation or randomness such that translational invariance is broken. We begin with a study of neural architectures in which there is a periodic modulation of the neuronal connections. Recent techniques from two-scale convergence analysis are used to construct a homogenized model in the limit that the spatial modulation is rapid compared with the scale of the long range connections. For the special case that the neuronal firing rate is a Heaviside we calculate the speed of a travelling homogenized front exactly and find how the wave speed changes as a function of the amplitude of the modulation. For this special case we further show how to obtain more accurate results about wave speed and the conditions for propagation failure by using an interface dynamics approach that circumvents the requirement of fast modulation. Next we turn our attention to forms of disorder that arise via the variation of firing rate properties across the tissue. To model this we draw parameters of the firing rate function from a distribution with prescribed spatial correlations and analyse the corresponding fluctuations in the wave speed. Finally we consider generalisations of the model to incorporate adaptation and stochastic forcing and show how recent numerical techniques developed for stochastic partial differential equations can be used to determine the wave speed by minimising the $L^2$ norm of a travelling disordered activity profile against a fixed test function.
2012, 32(8): 2971-2995 doi: 10.3934/dcds.2012.32.2971 +[Abstract](1324) +[PDF](828.7KB)
Abstract:
We consider a pair of uncoupled conditional oscillators near a subcritical Hopf bifurcation that are driven by two weak white noise sources, one intrinsic and one common. In this context the noise drives oscillations in a setting where the underlying deterministic dynamics are quiescent. Synchronization of these noise-driven oscillations is considered, where the noise is also driving synchronization. We first derive the envelope equations of the noise-driven oscillations using a stochastic multiple scales method, providing access to phase and amplitude information. Using both a linearized approximation and an asymptotic analysis of the nonlinear system, we obtain approximations for the probability density of the phase difference of the oscillators. It is found that common noise increases the degree of synchrony in the pair of oscillators, which can be characterized by the ratio of intrinsic to common noise. Asymptotic expressions for the phase difference density provide explicit parametric expressions for the probability of observing different phase dynamics: in-phase synchronization, phase shifted oscillations, and non-synchronized states. Computational results are provided to support analytical conclusions.
2012, 32(8): 2997-3007 doi: 10.3934/dcds.2012.32.2997 +[Abstract](2105) +[PDF](445.7KB)
Abstract:
We study Hopf-Andronov bifurcations in a class of random differential equations (RDEs) with bounded noise. We observe that when an ordinary differential equation that undergoes a Hopf bifurcation is subjected to bounded noise then the bifurcation that occurs involves a discontinuous change in the Minimal Forward Invariant set.

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