This document discusses using geometric approaches to analyze differential equations. It examines four different differential equations: 1) the damped, forced oscillator equation, 2) the damped, forced pendulum equation, 3) the damped, forced Duffing oscillator equation, and 4) the extended Lorenz-Maxwell-Bloch equation. For each equation, it applies Melnikov's method to show the existence of different types of homoclinic or heteroclinic orbits in the Poincare map. It then asserts properties of the invariant sets that result from these orbits based on homoclinic bifurcation theorems.

1. 1 | P a g e
A Geometric Approach to Differential Equations
Maleafisha Joseph Pekwa Stephen Tladi
Abstract: Most ordinary differential equations occurring in mathematical models of physical,
chemical and biological phenomena cannot be solved analytically. Numerical integrations do not
lead to a desired result without qualitative analysis of the behavior of the equation’s solutions. This
course studies the flows of scalar and planar ordinary differential equations. Stability and bifurcation
are discussed.
1. Consider the damped, forced oscillator or the Korteweg-de Vries (KdV) equation
𝑥̈ − 𝑥 + 𝑥2
= 𝜀(𝛾 cos 𝜔𝑡 − 𝛿𝑥̇).
We apply Melnikov’s method to show that the Poincare map associated with this equation
has transverse homoclinic orbits. We appeal to the Smale-Birkhoff Homoclinic Theorem and
assert the existence of an invariant hyperbolic set which contains a countable infinity of
unstable periodic orbits, a dense orbit and infinitely many homoclinic orbits.
2. Consider the damped, forced pendulum or the sine-Gordon(SG) equation
𝑥̈ + sin 𝑥 = 𝜀(𝛾 cos 𝜔𝑡 − 𝛿𝑥̇).
We apply Melnikov’s method to show that the Poincare map associated with this equation
has transverse heteroclinic orbits. We appeal to the Smale-Birkhoff Homoclinic Theorem
and assert the existence of an invariant hyperbolic set which contains a countable infinity of
unstable periodic orbits, a dense orbit and infinitely many heteroclinic orbits.
3. Consider the damped, forced Duffing oscillator or the nonlinear Shrodinger(NLS) equation
𝑥̈ − 𝑥 + 𝑥3
= 𝜀(𝛾 cos 𝜔𝑡 − 𝛿𝑥̇).
We apply Melnikov’s method to show that the Poincare map associated with this equation
has transverse homoclinic orbits. We appeal to the Smale-Birkhoff Homoclinic Theorem and
assert the existence of an invariant hyperbolic set which contains a countable infinity of
unstable periodic orbits, a dense orbit and infinitely many homoclinic orbits.
4. Consider the extended Lorenz-Maxwell-Bloch equation
𝑥̇ = 𝜎( 𝑦 − 𝑥 − 𝜇𝑤),
𝑦̇ = 𝜀𝑥 + 𝑥 − 𝑦 − 𝑥𝑧,
𝑧̇ = −𝛽𝑧 + 𝑥𝑦,
𝛿𝑤̇ = 𝛼𝑥 − 𝑤.
We apply Melnikov’s method to show that this equation has transverse inclination-flip
homoclinic orbits. We appeal to the Shilnikov-Rychlik Homoclinic Theorem and assert the
existence of Lorenz chaotic attractor.