The beauty of the geometry of dynamical systems is due to the existence and uniqueness theorems for ODEs. These results give us the conditions under which ODEs have solutions, and when those solutions are unique. Once we have these results, we can make some very powerful observations about any system described by them.
Preliminary Result
Before we get into the general theorems, let us recall the solution to first-order linear ODEs with constant coefficients. These are equations of the form
which always have solutions
where the constant \(A\) is determined by the initial condition \(x(0) = x_0\).
Existence Theorem
Consider the general initial value problem
The Existence Theorem states: if the function \(f(t,x)\) is continuous in a region around \((t_0, x_0)\), then there exists at least one solution \(x(t)\) defined in some interval containing \(t_0\).
This guarantees that continuity of the vector field is enough to ensure that trajectories actually exist.
Uniqueness Theorem
Continuity alone does not prevent multiple solutions. To guarantee uniqueness, we require more. Suppose \(f(t,x)\) satisfies a Lipschitz condition in the variable \(x\); that is, there exists \(L > 0\) such that
for all \(x_1, x_2\) in the region of interest. Then the initial value problem has exactly one solution.
Geometrically, this means that in the phase space, no two trajectories can cross, and each point belongs to one and only one solution curve.
Failure of Uniqueness
If the Lipschitz condition fails, uniqueness may break down. For instance, consider
Both \(x(t) = 0\) and \(x(t) = \tfrac{1}{4}(t-c)^2\) (for any \(c \geq 0\)) are valid solutions. Thus the system is not deterministic, and multiple solution curves can pass through the same initial point.