Dynamics $delta x(t)=delta x(0) e^{lambda t}$ of Henon Attractor
$begingroup$
Recall the question I asked before:
Linearized perturbation dynamics of Henon Attractor
So, I have the following separation dynamics for Henon attractor ($a = 1.4. b = 0.3$) $$delta x(t)=delta x(0) e^{lambda t},$$
where $lambda = 0.42$ is a Lyapunov exponent. Since $e^{lambda}>1$, so the difference of separation will increase exponentially.
However, the following is what I do in MATLAB, I choose two initial conditions: $$x(0) = begin{bmatrix} 0 \ 0 end{bmatrix}, x'(0) = begin{bmatrix} 0.1 \ 0.1 end{bmatrix}.$$ It is easily calculated (by the Henon dynamical system) that $$x(1) = begin{bmatrix} 1 \ 0 end{bmatrix}, x'(1) = begin{bmatrix} 1.086 \ 0.03 end{bmatrix}.$$
It seems that the difference of separation (distance) decreases.
The following picture is simulated in MATLAB.
My question is why the simulated result is different from the theoretical prediction $delta x(t)=delta x(0) e^{lambda t}$. Do I misunderstand something?
dynamical-systems chaos-theory stability-theory
asked Dec 1 '18 at 21:42
sleeve chensleeve chen
3,07141852
$endgroup$
add a comment |
$begingroup$
Recall the question I asked before:
Linearized perturbation dynamics of Henon Attractor
So, I have the following separation dynamics for Henon attractor ($a = 1.4. b = 0.3$) $$delta x(t)=delta x(0) e^{lambda t},$$
where $lambda = 0.42$ is a Lyapunov exponent. Since $e^{lambda}>1$, so the difference of separation will increase exponentially.
However, the following is what I do in MATLAB, I choose two initial conditions: $$x(0) = begin{bmatrix} 0 \ 0 end{bmatrix}, x'(0) = begin{bmatrix} 0.1 \ 0.1 end{bmatrix}.$$ It is easily calculated (by the Henon dynamical system) that $$x(1) = begin{bmatrix} 1 \ 0 end{bmatrix}, x'(1) = begin{bmatrix} 1.086 \ 0.03 end{bmatrix}.$$
It seems that the difference of separation (distance) decreases.
The following picture is simulated in MATLAB.
My question is why the simulated result is different from the theoretical prediction $delta x(t)=delta x(0) e^{lambda t}$. Do I misunderstand something?
dynamical-systems chaos-theory stability-theory
asked Dec 1 '18 at 21:42
sleeve chensleeve chen
3,07141852
$endgroup$
add a comment |
$begingroup$
Recall the question I asked before:
Linearized perturbation dynamics of Henon Attractor
So, I have the following separation dynamics for Henon attractor ($a = 1.4. b = 0.3$) $$delta x(t)=delta x(0) e^{lambda t},$$
where $lambda = 0.42$ is a Lyapunov exponent. Since $e^{lambda}>1$, so the difference of separation will increase exponentially.
However, the following is what I do in MATLAB, I choose two initial conditions: $$x(0) = begin{bmatrix} 0 \ 0 end{bmatrix}, x'(0) = begin{bmatrix} 0.1 \ 0.1 end{bmatrix}.$$ It is easily calculated (by the Henon dynamical system) that $$x(1) = begin{bmatrix} 1 \ 0 end{bmatrix}, x'(1) = begin{bmatrix} 1.086 \ 0.03 end{bmatrix}.$$
It seems that the difference of separation (distance) decreases.
The following picture is simulated in MATLAB.
My question is why the simulated result is different from the theoretical prediction $delta x(t)=delta x(0) e^{lambda t}$. Do I misunderstand something?
dynamical-systems chaos-theory stability-theory
asked Dec 1 '18 at 21:42
sleeve chensleeve chen
3,07141852
$endgroup$
Recall the question I asked before:
Linearized perturbation dynamics of Henon Attractor
So, I have the following separation dynamics for Henon attractor ($a = 1.4. b = 0.3$) $$delta x(t)=delta x(0) e^{lambda t},$$
where $lambda = 0.42$ is a Lyapunov exponent. Since $e^{lambda}>1$, so the difference of separation will increase exponentially.
However, the following is what I do in MATLAB, I choose two initial conditions: $$x(0) = begin{bmatrix} 0 \ 0 end{bmatrix}, x'(0) = begin{bmatrix} 0.1 \ 0.1 end{bmatrix}.$$ It is easily calculated (by the Henon dynamical system) that $$x(1) = begin{bmatrix} 1 \ 0 end{bmatrix}, x'(1) = begin{bmatrix} 1.086 \ 0.03 end{bmatrix}.$$
It seems that the difference of separation (distance) decreases.
The following picture is simulated in MATLAB.
My question is why the simulated result is different from the theoretical prediction $delta x(t)=delta x(0) e^{lambda t}$. Do I misunderstand something?
dynamical-systems chaos-theory stability-theory
dynamical-systems chaos-theory stability-theory
asked Dec 1 '18 at 21:42
sleeve chensleeve chen
3,07141852
asked Dec 1 '18 at 21:42
sleeve chensleeve chen
3,07141852
edited Dec 6 '18 at 6:45
sleeve chen
asked Dec 1 '18 at 21:42
sleeve chensleeve chen
3,07141852
asked Dec 1 '18 at 21:42
sleeve chensleeve chen
3,07141852
asked Dec 1 '18 at 21:42
sleeve chensleeve chen
3,07141852
3,07141852
add a comment |
add a comment |
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