Browsing by Subject "Bifurcation"

While alternans in a single cardiac cell appears through a simple

period-doubling bifurcation, in extended tissue the exact nature

of the bifurcation is unclear. In particular, the phase of

alternans can exhibit wave-like spatial dependence, either

stationary or traveling, which is known as discordant

alternans. We study these phenomena in simple cardiac models

through a modulation equation proposed by Echebarria-Karma. In

this dissertation, we perform bifurcation analysis for their

modulation equation.

Suppose we have a cardiac fiber of length l, which is

stimulated periodically at its x=0 end. When the pacing period

(basic cycle length) B is below some critical value B_c,

alternans emerges along the cable. Let a(x,n) be the amplitude

of the alternans along the fiber corresponding to the n-th

stimulus. Echebarria and Karma suppose that a(x,n) varies

slowly in time and it can be regarded as a time-continuous

function a(x,t). They derive a weakly nonlinear modulation

equation for the evolution of a(x,t) under some approximation,

which after nondimensionization is as follows:

&partial_t a = σ a + L a - g a 3,

where the linear operator

L a = &partial_xxa - &partial_x a -Λ-1 ∫ 0 _x a(x',t)dx'.

In the equation, σ is dimensionless and proportional to

B_c - B, i.e. σ indicates how rapid the pacing is,

Λ-1 is related to the conduction velocity (CV) of the

propagation and the nonlinear term -ga3 limits growth after the

onset of linear instability. No flux boundary conditions are

imposed on both ends.

The zero solution of their equation may lose stability, as the

pacing rate is increased. To study the bifurcation, we calculate

the spectrum of operator L. We find that the

bifurcation may be Hopf or steady-state. Which bifurcation occurs

first depends on parameters in the equation, and for one critical

case both modes bifurcate together at a degenerate (codimension 2)

bifurcation.

For parameters close to the degenerate case, we investigate the

competition between modes, both numerically and analytically. We

find that at sufficiently rapid pacing (but assuming a 1:1

response is maintained), steady patterns always emerge as the only

stable solution. However, in the parameter range where Hopf

bifurcation occurs first, the evolution from periodic solution

(just after the bifurcation) to the eventual standing wave

solution occurs through an interesting series of secondary

bifurcations.

We also find that for some extreme range of parameters, the

modulation equation also includes chaotic solutions. Chaotic waves

in recent years has been regarded to be closely related with

dreadful cardiac arrhythmia. Proceeding work illustrated some

chaotic phenomena in two- or three-dimensional space, for instance

spiral and scroll waves. We show the existence of chaotic waves in

one dimension by the Echebarria-Karma modulation equation for

cardiac alternans. This new discovery may provide a different

mechanism accounting for the instabilities in cardiac dynamics.

This dissertation addresses two aspects of dynamical systems arising from biological networks: homeostasis-bifurcation and identifiability.

Homeostasis occurs when a biological quantity does not change very much as a parameter is varied over a wide interval. Local bifurcation occurs when the multiplicity or stability of equilibria changes at a point. Both phenomena can occur simultaneously and as the result of a single mechanism. We show that this is the case in the feedback inhibition network motif. In addition we prove that longer feedback inhibition networks are less stable. Towards understanding interactions between homeostasis and bifurcations, we define a new type of singularity, the homeostasis-bifurcation point. Using singularity theory, the behavior of dynamical systems with homeostasis-bifurcation points is characterized. In particular, we show that multiple homeostatic plateaus separated by hysteretic switches and homeostatic limit cycle periods and amplitudes are common when these singularities occur.

Identifiability asks whether it is possible to infer model parameters from measurements. We characterize the structural identifiability properties for feedforward networks with linear reaction rate kinetics. Interestingly, the set of reaction rates corresponding to the edges of the graph are identifiable, but the assignment of rates to edges is not; Permutations of the reaction rates leads to the same measurements. We show how the identifiability results for linear kinetics can be extended to Michaelis-Menten kinetics using asymptotics.

Computational simulations of a two-dimensional incompressible regularized lid-driven cavity were performed and analyzed to identify the dynamic behavior of the flow through multiple bifurcations which ultimately result in chaotic flow. Pseudo-spectral numerical simulations were performed at Reynolds numbers from 1,000 to 25,000. Traditional as well as novel methods were implemented to characterize the system's behavior. The first critical Reynolds number, near 10,250, is found in agreement with existing literature. An additional bifurcation is observed near a Reynolds number of 15,500. The largest Lyapunov exponent was studied as a potential perspective on chaos characterization but its accurate computation was found to be prohibitive. Phase space and power spectrum analyses yielded comparable conclusions about the flow's progression to chaos. The flow's transition from quasi-periodicity to chaos between Reynolds numbers of 18,000 and 23,000 was observed to be gradual and of the form of a toroidal bifurcation. The concepts of frequency shredding and power capacity are introduced which, paired with an existing understanding of frequency entrainment, can help explain the system's progression through quasi-periodicity to chaos.