Introduction to Dynamical Systems

Introduction

Dynamical systems theory studies how states evolve over time under fixed rules. The rules may be discrete (iterated maps) or continuous (differential equations). The main goals are qualitative: long-term behavior, stability, sensitivity to initial conditions, and structural changes as parameters vary.

Core questions:

• Where do trajectories go?

• Are equilibria stable?

• How do behaviors change with parameters?

• Can deterministic rules produce unpredictable motion?

Discrete and Continuous Systems

Discrete system:

$x_{n+1}=f\left(x_n\right)$

Continuous system:

$\frac{dx}{dt}=F\left(x\right)$

The state space $X\subseteq R^n$ contains all possible configurations. Dynamics define a flow (continuous time) or iteration (discrete time).

Fixed Points

A fixed point $x^{\ast }$ satisfies:

• Discrete: $f\left(x^{\ast }\right)=x^{\ast }$

• Continuous: $F\left(x^{\ast }\right)=0$

These represent equilibria.

Logistic Map Example

$f\left(x\right)=rx(1-x)$

Fixed points solve:

$r{x}^{\ast }(1-x^{\ast })=x^{\ast }$

Solutions:

$x^{\ast }=0,x^{\ast }=1-\frac{1}{r}(r>1)$

Stability

A fixed point is:

• Stable if nearby states remain nearby.

• Asymptotically stable if nearby states converge to it.

• Unstable if perturbations grow.

Linear Stability

Linearize near $x^{\ast }$.

For continuous systems:

$J=DF\left(x^{\ast }\right)$

Asymptotically stable if all eigenvalues of $J$ have negative real part.

For discrete systems:

Stable if all eigenvalues of $Df\left(x^{\ast }\right)$ satisfy: $\left|\lambda \right|<1$

One-Dimensional Case

Continuous:

$f^{\prime }\left(x^{\ast }\right)<0$ $\Rightarrow$ stable, $f^{\prime }\left(x^{\ast }\right)>0$ $\Rightarrow$ unstable

Discrete:

$|f^{\prime }\left(x^{\ast }\right)|<1\Rightarrow$ stable

Orbits and Attractors

Orbit:

• Discrete: $\{x_0,x_1,x_2,\dots \}$

• Continuous: trajectory curve

Attractor: a set that attracts nearby trajectories.

Types:

• Fixed point

• Periodic orbit (limit cycle)

• Strange attractor

Basin of attraction: set of initial conditions converging to an attractor.

Periodic Orbits

Discrete:

${f\left(x^{\ast }\right)}^n=x^{\ast }$ with minimal period $n$.

Continuous:

Closed attracting orbit $=$ limit cycle.

Bifurcations

Qualitative change as parameter varies.

Saddle-Node

$\dot{x}=r+x^2$

Two equilibria merge and disappear.

Transcritical

$\dot{x}=rx-x^2$

Fixed points exchange stability.

Pitchfork

$\dot{x}=rx-x^3$

One equilibrium splits into two.

Hopf

Complex eigenvalues cross imaginary axis $\to$ limit cycle emerges.

Period-Doubling

Orbit of period $n$ replaced by one of period $2n$. Common route to chaos.

Chaos

Deterministic but unpredictable due to sensitive dependence on initial conditions. Small perturbations grow exponentially.

Lyapunov Exponent

$\delta \left(t\right)\approx {\delta }_0{e}^{\lambda t}$

If largest $\lambda >0$, system is chaotic.

Logistic Map

For $r\gtrsim 3.57$, chaotic behavior appears via period-doubling cascade.

Feigenbaum Constant

Ratio of successive period-doubling intervals:

$\delta \approx 4.669$

Universal across many systems.

Phase Space

Each point $=$ full system state.

Flow: ${\varphi }_t\left(x_0\right)$

Phase portraits visualize trajectories.

Poincaré–Bendixson ($2$D continuous systems):

• Only fixed points and limit cycles possible.

• No chaos in $2$D autonomous flows.

Conservative vs Dissipative Systems

Conservative:

• Preserve phase volume (Liouville’s theorem).

• Example: Hamiltonian mechanics.

Dissipative:

• Contract volume.

• Allow attractors of lower dimension.

Divergence test:

$\nabla \cdot F<0\Rightarrow$ volume contraction

Symbolic Dynamics

Encode trajectories as symbol sequences based on region visits.

Shift map on sequences provides a canonical chaotic model.

Symbolic representations simplify proofs and reveal topological structure.

Applications

• Population dynamics

• Climate models

• Control systems

• Financial markets

• Neural networks

Summary

Dynamical systems describe time evolution via maps or differential equations.

Key structures:

• Fixed points

• Stability via eigenvalues

• Periodic orbits

• Bifurcations

• Chaos

Linearization determines local behavior. Bifurcations mark structural change. Positive Lyapunov exponents signal chaos.

The theory provides a unified language for understanding complex time-dependent behavior across disciplines.