Parameters Estimation of Chaotic System for Single-Objective Optimization

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Chaotic System Instruction

1. Loren

$$ \dot x_1(t)=a[x_2(t)-x_1(t)]\\ \dot x_2(t)=bx_1(t)-x_1(t)x_3(t)-x_2(t)\\ \dot x_3(t)=x_1(t)x_2(t)-cx_3(t) $$

2. Chen

$$ \dot x_1(t)=a(x_2(t)-x_1(t))\\ \dot x_2(t)=dx_1(t)-x_1(t)x_3(t)+cx_2(t)\\ \dot x_3(t)=x_1(t)x_2(t)-bx_3(t) $$

3. Newton-Leipnik

$$ \dot x_1(t)=-ax_1(t)+x_2(t)+10x_2(t)x_3(t)\\ \dot x_2(t)=-x_1(t)-0.4x_2(t)+5x_1(t)x_3(t)\\ \dot x_3(t)=bx_3(t)-5x_1(t)x_2(t) $$

4. Rossler

$$ \dot x_1(t)=-x_2(t)-x_3(t)\\ \dot x_2(t)=x_1(t)+ax_2(t)\\ \dot x_3(t)=bx_1(t)-cx_3(t)+x_1(t)x_3(t) $$

5. Volta

$$ \dot x_1(t)=-x_1(t)-ax_2(t)-x_2(t)x_3(t)\\ \dot x_2(t)=-x_2(t)-bx_1(t)-x_1(t)x_3(t)\\ \dot x_3(t)=cx_3(t)+x_1(t)x_2(t)+1 $$

Usage:

f = chaotic_system(x,func_num)

• x: the parameters to estimate
• func_num: function number

Example:

f = chaotic_system([9,27.5,2.6],1) # Calculate objective function of Loren chaotic system