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```1
Chapter I. Continuous dynamical systems
Let X be a metric space and R the additive group of the reals.
Definition 1. A continuous dynamical system (R-action) on X is
a continuous map ϕ : R × X → X satisfying the two conditions:
1. ϕ(0, x) = x for x ∈ X,
2. ϕ(t2, ϕ(t1, x)) = ϕ(t1 + t2, x) for x ∈ X and t1,t2 ∈ R.
Definition 2. A reverse dynamical system to ϕ : R × X → X is the
map −ϕ : R × X → X defined by ϕ(t, x) = ϕ(−t, x)
For t ∈ R, the map ϕt : X → X is defined by ϕt (x) = ϕ(t, x).
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Definition 3. Let x ∈ X.
1. The set γ(x) = ϕ(R × {x}) is the trajectory of x.
2. The set γ −(x) = {ϕ(t, x) |t ≤ 0} is the negative semi-trajectory
of x.
3. The set γ +(x) = {ϕ(t, x) |t ≥ 0} is the positive semi-trajectory
of x.
A continuous dynamical system, particularly defined on a manifold, is often called a flow. Another name for a continuous dynamical system is directed 1-foliation. It is said that ϕ : R × X → X is
defined on X, even though the domain of the function is R × X. A
trajectory is often called an orbit.
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Proposition 1. For every t ∈ R, ϕt is a homeomorphism.
Proof. The maps ϕt and ϕ−t are continuous and inverse maps of
each other:
(ϕ−t ◦ ϕt )(x) = ϕ(−t, ϕ(t, x)) = ϕ(t − t, x) = ϕ(0, x) = x.
Proposition 2. Let U ⊂ X be an open set. Then the set
S
V = x∈U γ(x) is open.
Proof. V =
x∈U γ(x) = {(t, x) | x ∈ U,t ∈ R} =
S
3
S
t∈R ϕt (U).
Proposition 3. (G, ◦), where
G = {ϕt |t ∈ R}
and ϕs ◦ ϕt is the composition of maps, is an abelian group.
Proof. Since ϕs ◦ ϕt = ϕs+t , the composition is well defined in G.
Associativity holds in the composition of maps. The group identity is the identity map ϕ0. The inverse of ϕt is the map ϕ−t .
We have (ϕ0 ◦ ϕt )(x) = (ϕt ◦ ϕ0)(x) = ϕt (x). G is abelian since
(ϕs ◦ ϕt )(x) = ϕ(s, ϕ(t, x)) = ϕ(t + s, x) = ϕ(s + t, x) =
ϕ(t, ϕ(s, x)) = (ϕt ◦ ϕs)(x).
4
Proposition 4. A trajectory is one of the following sets:
• A one-point set (a singular point). The trajectory as well as
the point are called singular. A singular point is also called a
critical point, rest point, or fixed point.
• A set homeomorphic to S1. Then the trajectory is called periodic or closed.
• A one-to-one image of R.
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Proof. Let x ∈ X. Suppose that γ(x) is not a one-to-one image of
R, i.e., there are s0 and s00 in R, with s0 > s00 such that ϕ(s0, x) =
ϕ(s00, x). Then ϕ(t, x) = x, where t = s0 − s00.
Let t0 = inf{t > 0 | ϕ(t, x) = x}. Then either ϕ(t0, x) = x or there
is a sequence (tn)tn>t0 convergent to t0 such that for each n ∈ N,
ϕ(tn, x) = x. The continuity of ϕ implies that ϕ(t0, x) = x.
Case 1. t0 > 0. For any t ∈ R, t = s + kt0, where s ∈ [0,t0] and
k ∈ Z. The set {ϕ(t, x) |t ∈ [0,t0]} is homeomorphic to S1 and
hence γ(x) is homeomorphic to S1.
Case 2. t0 = 0. Then for any ε > 0, there is a tn such that ϕ(tn, x) =
x and diam({ϕ(t, x) |t ∈ [0,tn]}) < ε. By an argument similar to
the above, γ(x) = ({ϕ(t, x) |t ∈ [0,tn]}) and hence diam(γ(x)) = 0,
so x is a singular point.
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Definition 4. Let γ(x) be a periodic trajectory. If ϕ(t, x) = x and
t > 0, then t is a period of γ(x). The number
t0 = min{t ∈ R+ | ϕ(t, x) = x} is the fundamental period.
Proposition 5. The numbers t and t0 in the above definition do not
depend on the choice of the point of the trajectory.
Example. Let S1 be the unit circle in the complex plane
S1 = {(eθ i ∈ C | 0 ≤ θ < 2π}.
Let ϕ : R × X → X be given by ϕ(t, eθ i) = e(at+θ )i, where a ∈ R+.
Then the fundamental period if the trajectory is 2π
a.
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Proposition 6. The collection of all trajectories is a decomposition of X, i.e., X is the union of trajectories and trajectories are
pairwise disjoint.
Definition 5.
• X and ϕ are called the phase space and the phase map, respectively.
• The phase portrait is a geometric representation of the trajectories of a dynamical system in the phase space.
• The directed phase portrait (in the phase space) is a geometric
representation of the dynamical system restricted to individual
trajectories.
Note that a phase portrait for ϕ is also a phase portrait for −ϕ.
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Figure 1: Brouwer
“lines”
Proofs Based on Geometric Intuition
Krystyna Kuperberg
There is no set L homeomorphic to a line that intersects every
trajectory in one point.
9
Definition 6. Let ϕ : R × X → X and ψ : R × Y → Y on X be
dynamical systems. If there is a homeomorphism h : X → Y such
that the diagram
idR ×h
R×X →
R ×Y
↓ϕ
↓ψ
h
X
→
Y
commutes, then ϕ and ψ are (topologically) conjugate.
Definition 7. Let ϕ : R × X → X and ψ : R × Y → Y on X be
dynamical systems. If there is a homeomorphism h : X → Y such
that for x ∈ X h(γ(x)) = γ(h(x)), then ϕ and ψ are topologically
equivalent.
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Proposition 7. If ϕ and ψ are conjugate, then they are topologically equivalent.
Proof. fI ϕ and ψ are conjugate, then we have
(t, x) 7→ (t, h(x))
↓
↓
ϕ(t, x) 7→ ψ(t, h(x))
Hence h(γ(x)) ⊂ γ(h(x)), the image of a trajectory is contained
in a trajectory. Since Y decomposes into trajectories and h is onto,
h(γ(x)) = γ(h(x)).
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Example. Topological equivalence is a weaker condition than topological conjugacy. Let X = Y = S1 = {(eθ i ∈ C | 0 ≤ θ < 2π},
Define ϕ : R × X → X by ϕ(t, eθ i) = e(t+θ )i, and ψ : R × Y → Y
by ϕ(t, eθ i) = e(2t+θ )i. ϕ and ψ are topologically equivalent but
not topologically conjugate.
12
Definition 8. A set A ⊂ X is invariant if it is the union of trajectories.
Proposition 8. If A and B are invariant then the following sets are
invariant
1. B − A
2. the interior of A, Int(A),
3. the closure of A, Cl(A),
4. the boundary of A, Bd(A).
Proof. Since the trajectories are pairwise disjoint, a trajectory is
either contained in A or in B − A, thus B − A is invariant. Cl(A) is
invariant since if y = limn→∞ xn, for some xn ∈ A, then for t ∈ R
ϕ(t, y) = limn→∞ ϕ(t, xn). By Proposition 1, Int(A) is invariant,
and hence Bd(A)= Cl(A)−Int(A) is invariant as well.
13
Proposition 9. If for λ ∈ Λ, the set Aλ is invariant then
1. the union A =
S
λ ∈Λ Aλ
2. the intersection A =
is invariant,
T
λ ∈Λ Aλ
is invariant.
Definition 9. For x ∈ X define
1. The α-limit set of x, L−(x), as
\
−
Cl γ (ϕ(t, x))
t<0
2. The ω-limit set of x, L+(x), as
\
+
Cl γ (ϕ(t, x))
t>0
14
Proposition 10.
−
T
1. L (x) = n→∞ Cl (γ −(ϕ(tn, x))), where (tn) is any sequence of
real numbers such that limn→∞ = −∞.
+
T
2. L (x) = n→∞ Cl (γ +(ϕ(tn, x))), where (tn) is any sequence of
real numbers such that limn→∞ = ∞.
The notation L−(x) and L+(x) is not standard. For example,
α(x) and ω(x) are often used to denote the α-limit set and the
ω-limit set, respectively.
Proposition 11. If X is compact, then each of the limit sets L−(x)
and L+(x) is compact and connected.
15
Lemma 1. The limit sets L−(x) and L+(x) are invariant.
Proof. Consider the set L+(x). Let y ∈ L+(x) and let (tn), with
limn→∞ tn = ∞, be a sequence such that limn→∞ ϕ(tn, x) = y. Then
for t ∈ R, we have limn→∞ ϕ(t + tn, x) = limn→∞ ϕ(t, ϕ(tn, x)) =
ϕ(t, y), by the continuity of φ , and clearly limn→∞ = ∞. Hence
γ(y) ⊂ L+(x). A similar argument yields that L−(x) is invariant.
The trajectory γ(x) may or may not be contained in L+(x) or
L−(x). If it is contained in a limit set, then it is dense in this
set. Hence if γ(x) is contained in each L+(x) and L−(x), then
L+(x) = L−(x). Another interesting case of the limit sets being
equal is when γ(x) is homoclinic.
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Definition 10. A trajectory then γ(x) is homoclinic, if
L+(x) = L−(x) and it is a singleton.
17
Definition 11. A set A ⊂ X is minimal if it is nonempty, compact,
invariant, and minimal in this respect, i.e., if B ⊂ A is nonempty,
compact, and invariant, then B = A.
Proposition 12. A minimal set is the closure of a trajectory.
The converse is not true. The closure of a trajectory need not be
a minimal set.
Definition 12. A dynamical system is minimal if X is a minimal
set.
Proposition 13. A dynamical system is minimal if every trajectory
is dense.
Definition 13. A trajectory γ(x) is ergodic, if Cl(γ(x)) = X.
Definition 14. A dynamical system is transitive if there is an ergodic trajectory.
18
Proposition 14. If X is compact, then there is a minimal set.
Proof. Let x ∈ X and let C = {Cλ }λ ∈Λ be the collection of all
T
compact, invariant sets containing x. The set λ ∈Λ Cλ is minimal.
The minimal set is not unique. There may be many minimal
sets. Every singular trajectory is a minimal set. Every periodic
trajectory is a minimal set.
19
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