線形動的システムの役割,統計学

この数学の種類の基本を学んでいますが具体的の使い方にかんして少し悩んでいます。
The form of the system is x' = Ax, in its simplest sense. Thus, the time derivative of a state variable must be expressed as a linear function of the state. However, higher order systems like f(x)=x^5+x might not be expressible in this form. 信号処理、電子工学の集積回路、金融、航空工学に使用されているらしいです。
One aspect of linear dynamical systems that is confusing is how eigenvalues of the A matrix govern the behavior of the system. Particularly, how the magnitude of the eigenvector determines the decay of the system and how the imaginary component determines the oscillation of the system is somewhat ambiguous. Aspects of the system which can be revealed include delay, whether it increases or decays, whether or not it oscillates, and period.
動的システムの基本例はx'=ax -> x=Ce^(at). So here, the multiple of x on the right hand side determines the power of the exponential. If a=0, then the system is constant. If a were rewritten as a matrix A and x as a vector, then we have the canonical linear dynamical system. If "a" is less than 0, then the x quantity decays with increase in t. If "a" is greater than 0, then the x quantity increases with t. Thus, if a means to characterize a matrix A in the same fashion as a scalar "a" is classified as positive or negative, this would enable characterization of the system.
一番大事な発明はThe system can be rewritten:
dx/dt = Ax = lambda|1 0||x1| , making the system be decoupled.
|0 1||x2|
The lambda which is a solution to the above is the eigenvalue by definition.
Thus, x2 has no effect on dx1/dt, and x1 has no effect on dx2/dt. Looking at this, if lambda (a scalar) is positive and real, the system will continuously increase (assuming x1 and x2 are initially positive). For example, if at time 0 x1=1, x2 = 1 and lambda=1, dx1/dt =1 and dx2/dt=1. Thus, x1=2 and x2=2 at time 1. Then dx1/dt=2 and dx2/dt=2. Thus, x1=4 and x2=4 at t=2.
Conversely, if lambda is negative and real, the system decays up to the steady state. For example, if lambda=-1, x1=1, and x2=1, then dx1/dt=-1 and dx2/dt=-1. Thus, x1=0 and x2=0 at t=1.
What is more interesting is the case where the eigenvalue contains both real and imaginary components. In this case oscillation occurs which depends on the sign of the real component. If the real component is positive, oscillation is apparently decreasing in amplitude. If the real component is negative, the amplitude is decreasing. If the eigenvalue is imaginary, the oscillation is constant. I have yet to confirm this computationally.
更新：確認が出来ました。I confirmed oscillation with amplitude of [1 1] for eigenvalue of i using the following Matlab code and iterating t:
tv=[1:.1:30];
A=[i 0
0 i];
i=1;
for t=tv
H=expm(A*t);
x=[1,1];
vv=H*x';
vv
j=1;
for v=vv'
v
y(j,i)=v;
j++;
endfor;
i++;
endfor;
y
plot3(tv,real(y(1,:)),real(y(2,:)));
xlabel('time');
ylabel('x1');
zlabel('x2');

二次元と三次元の図を添付しました。


Another area of concern is projection pursuit, which involves determining "interesting" paths in a nonlinear plot of data. How this differs from principal component analysis has yet to be determined.