Mathematics Related¶
Qualititative Analysis¶
The vacuum oscillation is determined by autonomous equations. A fixed point of an autonomous system is defined by
which means the so called “velocity” is 0. For vacuum oscillation, we set
Thus we find the fixed points,
If we have only the *i*th function with derivative 0, the line is called the *i*th-nullcline. Thus the fixed points are the interaction points of all the nullclines.
These fixed points are very useful. In general, for a set of autonomous equations,
by definition the fixed point in phase space leads to the result
Thus the equations can be approximated using Taylor expansion near the point , since at the fixed points the derivatives are small.
The equations are simplified to linear equations whose coefficient matrix is simply the Jacobian matrix of the original system at the fixed point . In this example, the coefficient matrix for the linearized system is
As a comparison, the Jacobian matrix for the orginal equations at the fixed point is also the same which quite makes sense because Jacobian itself is telling the first order approximation of the velocity.
This linearization is only valid for hyperbolic fixed points which means that the eigenvalues of Jacobian matrix at fixed point has non-zero real part. Suppose the Jacobian is with eigenvalues are , a hyperbolic fixed point requires that .
For more analysis, checkout Poincare-Lyapunov Theorem.[1]_
Define and then the systems can be categorized into 3 different categories given the case that the fixed point isa hyperbolic one.
[1] | Nonlinear Systems of Ordinary Differential Equations by Massoud Malek, California State University, East Bay. |