Take a look at the following system

$$

begin {align}

dot {x} _1 & = x_2 \

dot {x} _2 & = -x_1 + x ^ 3_1 – x_2

end {align}

$$

which has three points of equilibrium (0,0), (1,0) and (-1,0). In the book I am reading, the author asks to define a new system that has the equilibrium point as the origin for (1,0) and (-1,0). The procedure in the book is as follows:

leave $ y = x-x_e $ where $ x_e $ is a balance point and we are going to make the transformation for $ x_e = (1,0) $Thus:

$$

begin {align}

y_1 & = x_1 – (1) = x_1 implies dot {y} _1 = dot {x} _1 \

y_2 & = x_2 – (0) = x_2 implies dot {y} _2 = dot {x} _2

end {align}

$$

The new system is now

$$

begin {align}

dot {y} _1 & = y_2 \

dot {y} _2 & = – (y_1 + 1) + (y_1 + 1) ^ 3 – y_2

end {align}

$$

Now we have to show that the new system $ dot {y} = g (y) $ has a point of equilibrium at the origin (ie, $ dot {y} = g (y) implies 0 = g (y_e) $), we obtain

$$

begin {align}

0 & = y_2 \

0 & = – (y_1 + 1) + (y_1 + 1) ^ 3 + y_2

end {align}

$$

$ y_2 = 0, – (y_1 + 1) + (y_1 + 1) ^ 3 = 0 implies y_1 = 0, -2 $. The new system has two equilibrium points and one of them is not at the origin. How to justify this problem. The real question in the book says: