Biomedical Engineering Reference
In-Depth Information
9.3 Nonlinear Dynamics
9.3.1 A Toy Nonlinear Oscillator
The observations described in the previous section motivate us to ask two
questions: Why should these temporal patterns be present in forced nonlinear
oscillators? If so, what is the oscillator in this system? We begin this section
by addressing the first question. We shall do this by solving an example, and
illustrating how these temporal patterns emerge. Because biological oscilla-
tors are usually stable nonlinear oscillators [Glass 2001], many characteristics
of the interactions between a periodic input and an ongoing rhythm can be
understood in terms of simple toy models, without paying much attention to
the details of the system [Glass and Mackey 1988, Keener and Sneyd 1998].
For this reason, we can choose to explore the origins of our strange temporal
patterns in terms of a simple system, learning about the tools and processes
that take place in order to gain intuition about the particular system we
might be interested in.
The toy model that we shall use is known as the Poincare oscillator,
named after the French mathematician Henri Poincare, who used it precisely
to illustrate general mechanisms of nonlinear systems. This abstract oscillator
is described in terms of a radial variable ρ and an angular variable φ ,whose
dynamics are governed by the following system of equations:
dt
= λρ (1
ρ ) ,
dt
= ω (mod 2 π ) ,
(9.5)
where “mod 2 π ” denotes a normalization of the phase dynamics: when
φ =2 π , the oscillator has completed a turn in the angular coordinate, and
starts again from φ = 0. The dynamics of this system of equations are easy to
understand: the phase of the oscillator increases monotonically at a constant
rate (of value ω ). The behavior of the radial part of the equation is also sim-
ple: for initial conditions close to the origin, the radial component will grow
with time, as long as λ> 0. On the other hand, for initial conditions far away
from the origin (i.e., with a radial part much larger than 1), the system will
evolve so that the radial coordinate decreases (at a rate given approximately
by λρ 2 ). The “equilibrium” is reached for the radial part when ρ =1.Inthis
case, the dynamics are reduced to a monotonic increase of the phase, and
after a given amount of time ( t =2 π/ω ), the solution repeats itself. This
solution is known as a limit cycle . The term “limit” in this case refers to
the asymptotic nature of the periodic solution: an initial condition not in the
cycle (i.e., either outside or inside) will evolve towards it. Notice that the
larger the value of λ , the faster the time evolution of the system will
be towards the limit cycle, for initial conditions not in the limit cycle. For
our description of forced nonlinear oscillators we shall assume that λ is large,
so convergence is fast.
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