Elimination of the X Variable in Lorenz’ Chaos Equations

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Abstract

Lorenz’s equations determine x, y and z as functions of time and are used as a classic example of chaotic behavior.

dot x=sigmaleft( y-x right)

dot y= x left( rho-z right) -y
dot z=xy-beta z
These equations yield functions which satisfy the following equation:
y^2+y dot y=left( rho-z right) left( beta z+dot z right)
This suggests that if we define y and z ”energies” varepsilon_y =y^2+y dot y and varepsilon_z =-left( rho-z right) left( beta z+dot z right)  , we see that the total energy varepsilon=varepsilon_y+varepsilon_z is conserved (with value zero) over the time evolution of the system.
This also suggests the definition of a utility function u=uleft( t right)  which represents the common value of y^2+y dot y and left( rho-z right) left( beta z+dot z right)  for each moment in time.
This implies
yleft( t right)=yleft( 0right) expleft(-tright) left(1  + frac{2}{yleft( 0right) ^2}int_0^t expleft(2sright) uleft( s right)  ds right)^{ frac{1}{2}    }=yleft(0right) expleft(-tright) left(1+frac{2}{yleft( 0right) ^2}int_0^t expleft(2sright) left( left( rho-z right) left( beta z+dot z right) right)  ds right)^{ frac{1}{2}    } if yleft( 0 right) neq 0,
yleft( t right)=sqrt{2} expleft(-tright) left( int_0^t expleft(2sright) uleft( s right)  ds right)^{ frac{1}{2}    }=sqrt{2} expleft(-tright) left( int_0^t expleft(2sright) left( left( rho-z right) left( beta z+dot z right) right)   ds right)^{ frac{1}{2}    } if yleft( 0 right) = 0,
where u=uleft( t right)  satisfies
left( rho-z right) left( beta z+dot z right) =u
Then we have
x=frac{u}{yleft( rho-z right) }
and the utility function u=uleft( t right)  then is seen to determine xleft( tright) yleft( tright)  and zleft( tright) .