Differential equations – Nonlinear differential equations

Nonlinear differential equations are much more difficiult to solve than linear differential equations. There exists almost no theory about solving nonlinear differential equations. However, sometimes one can acquire qualitative information about the solution(s) without solving the differential equation(s) explicitly (which would lead to quantative information). This is similar to the qualitative information, that can be obtained from the so-called phase portret of a first-order differential equation of the form

\[\frac{dy}{dt}=F(t,y),\]

where \(F(t,y)\) denotes any expression in terms of the independent variable \(t\) and the dependent variable \(y\) (\(y=y(t)\)). For each point \((t,y)\) in the \((t,y)\)-plane one obtains the slope of the tangent line to the graph of a solution (a so-called trajectory). This leads to a so-called direction field in the phase plane. This is already treated during the course calculus.

In the case of a homogeneous linear system

\[\mathbf{x}'(t)=A\mathbf{x}(t)\quad\text{with}\quad\mathbf{x}(t)=\begin{pmatrix}x_1(t)\\x_2(t)\end{pmatrix}\quad\text{and}\quad A\quad\text{a \(2\times2\) matrix,}\]

we can draw the trajectories in the \((x_1,x_2)\)-plane (the phase plane). This situation is quite similar to the case of a so-called autonomous system \(\displaystyle\frac{dy}{dt}=F(y)\), where \(F\) only depends on \(y\) and not explicitly on \(t\). The values of \(y\) for whichr \(F(y)=0\) are called critical or stationary points. Then we have that \(\displaystyle\frac{dy}{dt}=0\); that's why these are also called equilibrium solutions.

Now consider: \(\mathbf{x}'(t)=A\mathbf{x}(t)\) with \(A\) a \(2\times2\) matrix. Then we knwo: if \(\mathbf{x}(t)=\mathbf{v}e^{r t}\), then \(\mathbf{v}\neq\mathbf{0}\) is an eigenvector of \(A\) corresponding to the eigenvalue \(r\). Suppose that \(r_1\) and \(r_2\) are the eigenvalues of \(A\), then we distinguish between the following possiblities:

1. \(r_1,r_2\in\mathbb{R}\) with \(r_1\neq r_2\) and \(r_1r_2 > 0\): the origin \(O\) is called a node;


2. \(r_1,r_2\in\mathbb{R}\) with \(r_1\neq r_2\) and \(r_1r_2 < 0\): the origin \(O\) is called a saddle point;


3. \(r_1,r_2\in\mathbb{R}\) with \(r_1=r_2=r\) (algebraic multiplicity \(2\)):
   
   
   1. geometric multiplicity \(2\): the origin \(O\) is called a proper node;
   
   
   2. geometric multiplicity \(1\): the origin \(O\) is called an improper node;
4. \(r_1,r_2\notin\mathbb{R}\) with \(r_{1,2}=\lambda\pm i\mu\), \(\lambda,\mu\in\mathbb{R}\) and \(\lambda\neq0\): the origin \(O\) is called a spiral point;


5. \(r_1,r_2\notin\mathbb{R}\) with \(r_{1,2}=\lambda\pm i\mu\), \(\lambda,\mu\in\mathbb{R}\) and \(\lambda=0\): the origin \(O\) is called a center point.

Stability

Consider the behaviour of the solutions for \(t\to\infty\): the system is called

1. unstable if there exist solutions which tend to infinity;


2. stable if all solution stay bounded;


3. asymptotically stable if all solutions tend to the origin.

In the case of autonomous systems, which are locally linear, we can locally (near the critical points) use linearizations to find the behaviour of the solutions of the nonlinear system.

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Last modified on August 16, 2021