Applied Dynamical Systems
Complete Revision & Study Guide
A topic-by-topic guide covering flows and invariant sets, stability and linearization, manifolds,
bifurcation theory, asymptotic methods, Floquet theory, and chaos in continuous dynamical
systems.
Original study notes — independently written summary and explanation
,Contents
• 1. Foundations: Dynamical Systems and Flows
• 2. Invariant Sets, Poincaré Maps & Limit Sets
• 3. Stability of Fixed Points
• 4. Linearization & the Hartman–Grobman Theorem
• 5. Stable and Unstable Manifolds
• 6. Non-Hyperbolic Points & Center Manifolds
• 7. Bifurcations of Fixed Points
• 8. Normal Forms & Near-Identity Transformations
• 9. Asymptotic Methods
• 10. Stability of Periodic Orbits: Floquet Theory
• 11. Bifurcations of Periodic Orbits
• 12. Chaos and Strange Attractors
, 1. Foundations: Dynamical Systems and Flows
1.1 What is a dynamical system?
A dynamical system is a pair (M, φ), where M is the state space (or phase space) — the set
of all possible states the system can be in — and φ is an evolution rule describing how a
state changes over time. A point x ∈ M represents the complete state of the system:
everything you would need to know to determine how it evolves from that moment onward.
There are two common notions of time:
• Discrete time: n ∈ ℤ, with an update rule x_{n+1} = φ(xₙ).
• Continuous time: t ∈ ℝ, with a flow φ_t : ℝ × M → M.
1.2 The flow and its defining properties
For a continuous-time system, the flow φ_t takes an initial point x and traces out a smooth
curve in state space — an orbit. The flow satisfies two defining properties:
φ₀(x) = x
φ_{t+s}(x) = [φ_s ∘ φ_t](x)
The first says that no time evolution leaves you where you started; the second says that
evolving forward by t and then by s is the same as evolving forward directly by t+s. If t is
allowed to run over all of ℝ (not just t ≥ 0), one can show φ_{-t} = φ_t ⁻¹ — meaning the flow
forms a one-parameter family of diffeomorphisms (bijections that are differentiable in both
directions).
1.3 Connecting flows to differential equations
Where do differential equations come in? Consider the flow over an infinitesimally small time
step. The time derivative of the flow at a point defines the generalised velocity field:
d/dt[φ_t(x)] = f(φ_t(x))
Writing this more familiarly as dx/dt = f(x) recovers the standard notation for a system of
ODEs. This is called an autonomous system, since f does not depend explicitly on time. If it
does — dx/dt = f(x, t) — the system is non-autonomous.
Trick: converting a non-autonomous system into an autonomous one
Any non-autonomous system of dimension d can be converted into an autonomous system
of dimension d+1 by introducing an auxiliary variable θ := t, with dθ/dt = 1. Appending this to
the state vector artificially suppresses the explicit time dependence, at the cost of one extra
dimension.
1.4 Worked example: the pendulum
For the (undamped, unforced) pendulum, ẍ = −sin(x). Writing this as a first-order system with
state x = (x, ẋ) gives the generalised velocity field (ẋ, −sin x). The resulting phase portrait
shows the classic alternating pattern of centres and saddles characteristic of pendulum
dynamics.
A useful worked example is x̊ = x(1 − x²): a one-dimensional, nonlinear system. Because the
system is symmetric under x → −x, it suffices to analyse x > 0 and infer the rest — a useful
simplification technique whenever a symmetry is present. Solving directly via separation of
variables gives the flow explicitly, and taking t → ∞ shows solutions converge to x = 1 for
any x > 0, and remain at x = 0 if started there exactly.
Complete Revision & Study Guide
A topic-by-topic guide covering flows and invariant sets, stability and linearization, manifolds,
bifurcation theory, asymptotic methods, Floquet theory, and chaos in continuous dynamical
systems.
Original study notes — independently written summary and explanation
,Contents
• 1. Foundations: Dynamical Systems and Flows
• 2. Invariant Sets, Poincaré Maps & Limit Sets
• 3. Stability of Fixed Points
• 4. Linearization & the Hartman–Grobman Theorem
• 5. Stable and Unstable Manifolds
• 6. Non-Hyperbolic Points & Center Manifolds
• 7. Bifurcations of Fixed Points
• 8. Normal Forms & Near-Identity Transformations
• 9. Asymptotic Methods
• 10. Stability of Periodic Orbits: Floquet Theory
• 11. Bifurcations of Periodic Orbits
• 12. Chaos and Strange Attractors
, 1. Foundations: Dynamical Systems and Flows
1.1 What is a dynamical system?
A dynamical system is a pair (M, φ), where M is the state space (or phase space) — the set
of all possible states the system can be in — and φ is an evolution rule describing how a
state changes over time. A point x ∈ M represents the complete state of the system:
everything you would need to know to determine how it evolves from that moment onward.
There are two common notions of time:
• Discrete time: n ∈ ℤ, with an update rule x_{n+1} = φ(xₙ).
• Continuous time: t ∈ ℝ, with a flow φ_t : ℝ × M → M.
1.2 The flow and its defining properties
For a continuous-time system, the flow φ_t takes an initial point x and traces out a smooth
curve in state space — an orbit. The flow satisfies two defining properties:
φ₀(x) = x
φ_{t+s}(x) = [φ_s ∘ φ_t](x)
The first says that no time evolution leaves you where you started; the second says that
evolving forward by t and then by s is the same as evolving forward directly by t+s. If t is
allowed to run over all of ℝ (not just t ≥ 0), one can show φ_{-t} = φ_t ⁻¹ — meaning the flow
forms a one-parameter family of diffeomorphisms (bijections that are differentiable in both
directions).
1.3 Connecting flows to differential equations
Where do differential equations come in? Consider the flow over an infinitesimally small time
step. The time derivative of the flow at a point defines the generalised velocity field:
d/dt[φ_t(x)] = f(φ_t(x))
Writing this more familiarly as dx/dt = f(x) recovers the standard notation for a system of
ODEs. This is called an autonomous system, since f does not depend explicitly on time. If it
does — dx/dt = f(x, t) — the system is non-autonomous.
Trick: converting a non-autonomous system into an autonomous one
Any non-autonomous system of dimension d can be converted into an autonomous system
of dimension d+1 by introducing an auxiliary variable θ := t, with dθ/dt = 1. Appending this to
the state vector artificially suppresses the explicit time dependence, at the cost of one extra
dimension.
1.4 Worked example: the pendulum
For the (undamped, unforced) pendulum, ẍ = −sin(x). Writing this as a first-order system with
state x = (x, ẋ) gives the generalised velocity field (ẋ, −sin x). The resulting phase portrait
shows the classic alternating pattern of centres and saddles characteristic of pendulum
dynamics.
A useful worked example is x̊ = x(1 − x²): a one-dimensional, nonlinear system. Because the
system is symmetric under x → −x, it suffices to analyse x > 0 and infer the rest — a useful
simplification technique whenever a symmetry is present. Solving directly via separation of
variables gives the flow explicitly, and taking t → ∞ shows solutions converge to x = 1 for
any x > 0, and remain at x = 0 if started there exactly.
