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bulk sample The sample resulting from the the bundle morphism is usually required to
planned aggregation or combination or sample preserve that structure (e.g., to be linear on
units. fibers).
A bundle morphism (), φ) is called a strong
morphism if φ is a diffeomorphism. It is called
bundle A triple (E,B,π), consisting of two
vertical if M = M and φ = id . If )

topological spaces E and B and a continous M
is surjective, (), φ) is called a bundle epimor-
surjective map π : E → B. E is called the phism.If ) is injective, (), φ) is called a bundle
total space, B the base space, and π −1 (x) the
monomorphism.If ) is a diffeomorphism, then φ
fiber at x ∈ B.A trivial bundle is of the form −1 −1
is also a diffeomorphism and () ,φ ) is a bun-
π : B × E → B with π = pr the projec- dle morphism, called the inverse morphism.In
1
tion onto the first factor. If all fibers π −1 (x) are
that case (), φ) is called a bundle isomorphism.
homeomorphic to a space F, and the bundle is µ i µ i
If (x ,y ), (x ,y ) are fibered coordinates
locally trivial; i.e., there are homeomorphisms on B and B , the local expression of a fibered

ψ : U × F → π −1 (U ) ( U an open cover of morphism is:
i
i
i
i
B) with transition maps which are homeomorph-
isms, then the bundle is called a fiber bundle and x µ = f (x)
µ
F is called the typical fiber. If the typical fiber y = Y (x, y)
i
i
F is a vector space, the bundle is called a vector
bundle. If the spaces E, B are smooth mani-
2
Burger’s equation u = u + u .
folds and all the maps above are smooth, then the t xx x
bundles are called smooth fiber bundles. Exam-
bursting Some biological cells exhibit brief
ples are the tangent bundle and cotangent bundle
bursts of oscillations in their membrane electric
of a smooth manifold B.A principal fiber bun-
potential interspersed with quiescent periods
dle consists of a smooth fiber bundle (E,B,π)
and a Lie group G acting freely on E on the during which the membrane potential changes
only slowly. The first mathematical model for
right (u, g) ∈ E × G → ug ∈ E satisfying
this phenomenon was proposed by T.R. Chay
ψ (x, gh) = ψ (x, g)h , x ∈ U ,g,h ∈ G.
i i i
and J. Keizer in terms of five coupled nonlin-
ear ordinary differential equations in which a
bundle morphisms If B = (B,M,π,F) slow oscillator modulates a high frequency oscil-

and B = (B ,M ,π ,F ) are two fiber bun- lation. When the slow oscillating variable (S)

dles,a bundle morphism is a pair of maps (), φ) passes some numerical value µ, the high fre-

such that ) : B → B , φ : M → M and quency oscillation occurs; while when the slow
π ◦ ) = φ ◦ π; i.e., ) sends fibers into fibers. oscillating variable is below the critical value, the

One usually summarizes this property by saying fast oscillation disappears and the corresponding
that the following diagram: variable F changes slowly. Hence the overall
dynamics of F shows bursts of high frequency
)
B −→ B oscillations when S> µ, interspered with qui-
↓ ↓ escent periods when S< µ.(cf. J. Rinzel, Burst
φ
M −→ M oscillations in an excitable membrane model.
In: Ordinary and Partial Differential Equations,
is commutative. If the bundles are endowed with Eds. B.D. Sleeman and R.J. Jarvis, Springer-
some additional structure (e.g., vector bundles) Verlag, New York, 1985).

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