nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)

132 Mathematical Appendices
the compact set, but our simulations (with varying initial conditions) seem to verify that
as long as we stick to choosing initial conditions in at least max (M, Ma, Ba, Bn, C)
we are on the safe side (due to the crowding terms and the Micahelis-Menten function
these concentrations will never exceed ∞ and the concentrations cannot be negative).
By these means the solutions to the DuCa model will be unique, and the maximal
interval of existence is 0 ≤ t < ∞.
A.2 The Implicit Function Theorem
The implicit function theorem 1 is the theorem at the heart of bifurcation analysis, since
it provides sufficient conditions for an equilibrium to exist as a parameter is varied.
Formally the theorem can be stated as
Theorem A.2
Let r ∈ R and x ∈ R n , then, if g(r, x) is a C 1 function on R × R n with the properties
and
g : R × R n → R n , g(0, 0) = 0 (A.3)
Det(Dxg(0, 0)) = 0 (A.4)
there exists a unique differentiable function G(r), defined on a neighborhood, U ⊂ R, of
r = 0, such that
G : U → R n , G(0) = 0 (A.5)
and
g(r, G(r)) = 0, for r ∈ U (A.6)
This theorem promises us that: given an equilibrium for a system of differential equations
that depends on a parameter, then if the Jacobian of the system evaluated in that
equilibrium is invertible, i.e. has a nonzero determinant, the equilibrium will persist (in
a neighborhood of the parameter value) as the parameter is changed.
The theorem can be generalized to an m-dimensional parameter, and a general equilibrium
other than (0, 0).
A.3 The Hartman-Grobman Theorem
The Hartman-Grobman theorem provides the link between the behavior of a nonlinear
system and the behavior of its linearized counterpart, when the eigenvalues of the
Jacobian are not purely imaginary or zero. The following formal statement of the
theorem is based on Guckenheimer and Holmes (2002).
Theorem A.3
Let (r, x) be as in Theorem A.2, and let g(r, x) = ˙x be a system of ordinary differential
equations. Assume that (0, 0) is an equilibrium, then if Dxg(0, 0) has no zero or purely
imaginary eigenvalues there is a homeomorphism, h, defined on some neighborhood, V ,
1 The following is loosely based on Crawford (1991).