ISSN:
1531-3492
eISSN:
1553-524X
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Discrete & Continuous Dynamical Systems - B
May 2004 , Volume 4 , Issue 2
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2004, 4(2): 349-390
doi: 10.3934/dcdsb.2004.4.349
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Abstract:
When the steady states at infinity become unstable through a pattern forming bifurcation, a travelling wave may bifurcate into a modulated front which is time-periodic in a moving frame. This scenario has been studied by B. Sandstede and A. Scheel for a class of reaction-diffusion systems on the real line. Under general assumptions, they showed that the modulated fronts exist and are spectrally stable near the bifurcation point. Here we consider a model problem for which we can prove the nonlinear stability of these solutions with respect to small localized perturbations. This result does not follow from the spectral stability, because the linearized operator around the modulated front has essential spectrum up to the imaginary axis. The analysis is illustrated by numerical simulations.
When the steady states at infinity become unstable through a pattern forming bifurcation, a travelling wave may bifurcate into a modulated front which is time-periodic in a moving frame. This scenario has been studied by B. Sandstede and A. Scheel for a class of reaction-diffusion systems on the real line. Under general assumptions, they showed that the modulated fronts exist and are spectrally stable near the bifurcation point. Here we consider a model problem for which we can prove the nonlinear stability of these solutions with respect to small localized perturbations. This result does not follow from the spectral stability, because the linearized operator around the modulated front has essential spectrum up to the imaginary axis. The analysis is illustrated by numerical simulations.
2004, 4(2): 391-406
doi: 10.3934/dcdsb.2004.4.391
+[Abstract](1076)
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We discuss a two-parameter family of maps that generalize piecewise linear, expanding maps of the circle. One parameter measures the effect of a non-linearity which bends the branches of the linear map. The second parameter rotates points by a fixed angle. For small values of the nonlinearity parameter, we compute the invariant measure and show that it has a singular density to first order in the nonlinearity parameter. Its Fourier modes have forms similar to the Weierstrass function. We discuss the consequences of this singularity on the Lyapunov exponents and on the transport properties of the corresponding multibaker map. For larger non-linearities, the map becomes non-hyperbolic and exhibits a series of period-adding bifurcations.
We discuss a two-parameter family of maps that generalize piecewise linear, expanding maps of the circle. One parameter measures the effect of a non-linearity which bends the branches of the linear map. The second parameter rotates points by a fixed angle. For small values of the nonlinearity parameter, we compute the invariant measure and show that it has a singular density to first order in the nonlinearity parameter. Its Fourier modes have forms similar to the Weierstrass function. We discuss the consequences of this singularity on the Lyapunov exponents and on the transport properties of the corresponding multibaker map. For larger non-linearities, the map becomes non-hyperbolic and exhibits a series of period-adding bifurcations.
2004, 4(2): 407-417
doi: 10.3934/dcdsb.2004.4.407
+[Abstract](1440)
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Abstract:
The Intra Venous Glucose Tolerance Test (IVGTT) is a simple and established experimental procedure in which a challenge bolus of glucose is administered intra-venously and plasma glucose and insulin concentrations are then frequently sampled. The modeling of the measured concentrations has the goal of providing information on the state of the subject's glucose/insulin control system: an open problem is to construct a model representing simultaneously the entire control system with a physiologically believable qualitative behavior. A previously published single-distributed-delay differential model was shown to have desirable properties (positivity, boundedness, global stability of solutions) under the hypothesis of a specific, square-wave delay integral kernel. The present work extends the previous results to a family of models incorporating a generic non- negative, square integrable normalized kernel. Every model in this family describes the rate of glucose concentration variation as due to both insulin-dependent and insulin-independent net glucose tissue uptake, as well as to constant liver glucose production. The rate of variation of plasma insulin concentration depends on insulin catabolism and on pancreatic insulin secretion. Pancreatic insulin secretion at time $t$ is assumed to depend on the earlier effects of glucose concentrations, up to time $t$ (distributed delay). We consider a non-negative, square integrable normalized weight function $\omega$ on $R^+ =[0, \infty)$ as the fraction of maximal pancreatic insulin secretion at a given glucose concentration. No change in local asymptotic stability is introduced by the time delay. Considering an appropriate Lyapunov functional, it is found that the system is globally asymptotically stable if the average time delay has a parameter- dependent upper bound. An example of good model fit to experimental data is shown using a specific delay kernel.
The Intra Venous Glucose Tolerance Test (IVGTT) is a simple and established experimental procedure in which a challenge bolus of glucose is administered intra-venously and plasma glucose and insulin concentrations are then frequently sampled. The modeling of the measured concentrations has the goal of providing information on the state of the subject's glucose/insulin control system: an open problem is to construct a model representing simultaneously the entire control system with a physiologically believable qualitative behavior. A previously published single-distributed-delay differential model was shown to have desirable properties (positivity, boundedness, global stability of solutions) under the hypothesis of a specific, square-wave delay integral kernel. The present work extends the previous results to a family of models incorporating a generic non- negative, square integrable normalized kernel. Every model in this family describes the rate of glucose concentration variation as due to both insulin-dependent and insulin-independent net glucose tissue uptake, as well as to constant liver glucose production. The rate of variation of plasma insulin concentration depends on insulin catabolism and on pancreatic insulin secretion. Pancreatic insulin secretion at time $t$ is assumed to depend on the earlier effects of glucose concentrations, up to time $t$ (distributed delay). We consider a non-negative, square integrable normalized weight function $\omega$ on $R^+ =[0, \infty)$ as the fraction of maximal pancreatic insulin secretion at a given glucose concentration. No change in local asymptotic stability is introduced by the time delay. Considering an appropriate Lyapunov functional, it is found that the system is globally asymptotically stable if the average time delay has a parameter- dependent upper bound. An example of good model fit to experimental data is shown using a specific delay kernel.
2004, 4(2): 419-434
doi: 10.3934/dcdsb.2004.4.419
+[Abstract](1267)
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Abstract:
In order to describe excitable reaction-diffusion systems, we derive a two-dimensional model with a Hopf and a semilocal saddle-node homoclinic bifurcation. This model gives the theoretical framework for the analysis of the saddle-node homoclinic bifurcation as observed in chemical experiments, and for the concepts of excitability and excitability threshold. We show that if diffusion drives an extended system across the excitability threshold then, depending on the initial conditions, wave trains, propagating solitary pulses and propagating pulse packets can exist in the same extended system. The extended model shows chemical turbulence for equal diffusion coefficients and presents all the known types of topologically distinct activity waves observed in chemical experiments. In particular, the approach presented here enables to design experiments in order to decide between excitable systems with sharp and finite width thresholds.
In order to describe excitable reaction-diffusion systems, we derive a two-dimensional model with a Hopf and a semilocal saddle-node homoclinic bifurcation. This model gives the theoretical framework for the analysis of the saddle-node homoclinic bifurcation as observed in chemical experiments, and for the concepts of excitability and excitability threshold. We show that if diffusion drives an extended system across the excitability threshold then, depending on the initial conditions, wave trains, propagating solitary pulses and propagating pulse packets can exist in the same extended system. The extended model shows chemical turbulence for equal diffusion coefficients and presents all the known types of topologically distinct activity waves observed in chemical experiments. In particular, the approach presented here enables to design experiments in order to decide between excitable systems with sharp and finite width thresholds.
2004, 4(2): 435-456
doi: 10.3934/dcdsb.2004.4.435
+[Abstract](963)
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Abstract:
Two-dimensional mappings obtained by coupling two piecewise increasing expanding maps are considered. Their dynamics is described when the coupling parameter increases in the expanding domain. By introducing a coding and by analysing an admissibility condition, upper and lower bounds of the corresponding symbolic systems are obtained. As a consequence, the topological entropy is located between two decreasing step functions of the coupling parameter. The analysis firstly applies to mappings with piecewise affine local maps which allow explicit expressions and, in a second step, is extended by continuity to mappings with piecewise smooth local maps.
Two-dimensional mappings obtained by coupling two piecewise increasing expanding maps are considered. Their dynamics is described when the coupling parameter increases in the expanding domain. By introducing a coding and by analysing an admissibility condition, upper and lower bounds of the corresponding symbolic systems are obtained. As a consequence, the topological entropy is located between two decreasing step functions of the coupling parameter. The analysis firstly applies to mappings with piecewise affine local maps which allow explicit expressions and, in a second step, is extended by continuity to mappings with piecewise smooth local maps.
2004, 4(2): 457-464
doi: 10.3934/dcdsb.2004.4.457
+[Abstract](1084)
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Abstract:
The bifurcations of strange nonchaotic attractors in quasi-periodically forced systems are poorly understood. A simple two-parameter example is introduced which unifies previous observations of non-smooth pitchfork bifurcations and blowout bifurcations of strange nonchaotic attractors. The corresponding bifurcation curves can be calculated analytically. The example shows how these bifurcations are organized around a codimension two point in parameter space.
The bifurcations of strange nonchaotic attractors in quasi-periodically forced systems are poorly understood. A simple two-parameter example is introduced which unifies previous observations of non-smooth pitchfork bifurcations and blowout bifurcations of strange nonchaotic attractors. The corresponding bifurcation curves can be calculated analytically. The example shows how these bifurcations are organized around a codimension two point in parameter space.
2004, 4(2): 465-478
doi: 10.3934/dcdsb.2004.4.465
+[Abstract](1258)
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Abstract:
The equations of mean field magnetohydrodynamics with constant mean velocity are proved to posses solutions bounded in the $H^{1}$-norm for all time, and a compact attractor whose dimension is estimated. It is shown that depending on the functional form of the so-called alpha term the attractor may reduce to zero or be a larger set. If, as usual in physical situations, there exists a set of solutions with a minimum size $N$, the dimension of this set decreases rapidly with increasing $N$. Finally, the dependence of the dimension on the magnetic diffusivity is analyzed, suggesting that the evolution of a magnetic field under the mean field equation is much more restricted than the one deduced from the full magnetohydrodynamic system.
The equations of mean field magnetohydrodynamics with constant mean velocity are proved to posses solutions bounded in the $H^{1}$-norm for all time, and a compact attractor whose dimension is estimated. It is shown that depending on the functional form of the so-called alpha term the attractor may reduce to zero or be a larger set. If, as usual in physical situations, there exists a set of solutions with a minimum size $N$, the dimension of this set decreases rapidly with increasing $N$. Finally, the dependence of the dimension on the magnetic diffusivity is analyzed, suggesting that the evolution of a magnetic field under the mean field equation is much more restricted than the one deduced from the full magnetohydrodynamic system.
2004, 4(2): 479-495
doi: 10.3934/dcdsb.2004.4.479
+[Abstract](1205)
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Abstract:
Vaccination that gives partial protection for both newborns and susceptibles is included in a transmission model for a disease that confers no immunity. A general form of the vaccine waning function is assumed, and the interplay of this together with the vaccine efficacy and vaccination rates is discussed. The integro-differential system describing the model is studied for a constant vaccine waning rate, in which case it reduces to an ODE system, and for a constant waning period, in which case it reduces to a system of delay differential equations. For some parameter values, the model is shown to exhibit a backward bifurcation, leading to the existence of subthreshold endemic equilibria. Numerical examples are presented that demonstrate the consequence of this bifurcation in terms of epidemic control. The model can alternatively be interpreted as one consisting of two social groups, with education playing the role of vaccination.
Vaccination that gives partial protection for both newborns and susceptibles is included in a transmission model for a disease that confers no immunity. A general form of the vaccine waning function is assumed, and the interplay of this together with the vaccine efficacy and vaccination rates is discussed. The integro-differential system describing the model is studied for a constant vaccine waning rate, in which case it reduces to an ODE system, and for a constant waning period, in which case it reduces to a system of delay differential equations. For some parameter values, the model is shown to exhibit a backward bifurcation, leading to the existence of subthreshold endemic equilibria. Numerical examples are presented that demonstrate the consequence of this bifurcation in terms of epidemic control. The model can alternatively be interpreted as one consisting of two social groups, with education playing the role of vaccination.
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