Autonomous (meaning independent of time variable) equations are equations of the form

\begin{equation} \label{EqAuto.1} y' = f(y) \qquad\mbox{or}\qquad \frac{{\text d}y}{{\text d}t} = f(y) , \end{equation}
where slope function f(y) is a function of only dependent variable and does not involve independent variable explicitly. This equation encompasses, for example, all one-dimensional motions subject to conservative forces, but also many one-dimensional physical systems. This equation \eqref{EqAuto.1} occurs in cosmology, fluid mechanics, glaciology, hydrology, oceanography, and seismology.

We can make some observations regarding solutions to autonomous equations because their solutions have very limited types of behavior. Their certain mathematical properties are translated into physical properties relevant for natural phenomena. The integral curves (solutions) are invariant under translation along independent variable line, i.e., a horizontal shift of a solution is another solution. For example, consider an autonomous equation

\[ y' = 4 - y^2 . \]
We can plot a family of solutions using Mathematica:
sola = DSolve[{y'[x] == (4 - (y[x])^2)}, y[x], x];
g[x_, c_] = sola[[1, 1, 2]] /. C[1] -> c;
Plot[Table[g[x, c], {c, -5, 5}], {x, -2, 2}, PlotStyle -> Thick, PlotRange -> {-3, 3}, ImageSize -> 250]
StreamDensityPlot[{1, 4 - y^2}, {x, 0, 5}, {y, -6, 6}]
line = Graphics[Line[{{0, -2}, {0, 2}}], Epilog -> Inset[Text[Style["-2", Black, Medium]], {-0.2, -1}]];
p1 = Graphics[{PointSize[Large], Red, Point[{0, -1}]}];
p2 = Graphics[{PointSize[Large], Red, Point[{0, 1}]}];
a1 = Graphics[{Arrowheads[0.1], Arrow[{{0, -1.2}, {0, -2.2}}]}];
a2 = Graphics[{Arrowheads[0.1], Arrow[{{0, -1}, {0, 0.5}}]}];
a3 = Graphics[{Arrowheads[0.1], Arrow[{{0, 2.3}, {0, 1.3}}]}];
Show[line, a1, a2, a3, p1, p2]
    Family of general solutions.
      Phase portrait.
   Phase line for slope function 1-y².

 

All autonomous equations can be solved at least implicitly by separating variables:
\[ \frac{{\text d}y}{f(y)} = {\text d}x \qquad \Longrightarrow \qquad \int \frac{{\text d}y}{f(y)} = x+C, \quad f(y) \ne 0, \]
where C is an arbitrary constant. Since the above formula involves the reciprocal of the slope function, we have to exclude from our consideration all points where f vanishes---we have analyses this situation separately.
A point y = y* where \( f\left( y^{\ast} \right) = 0 \) is called a critical point or equilibrium solution for the given differential equation \( y' = f(y) . \)

Equilibrium solutions can be classified as follows.

  • Stable: The equilibrium solution y(t) = y* is stable if all solutions with initial condition y(t0) = y0 "near" the critical point y* approach it as t → ∞. This point is also called a sink.
  • Unstable: The equilibrium solution y(t) = y* is unstable if all solutions with initial conditions "near" the critical point y* do not approach it as t → ∞. This point is also called a source.
  • Semi-stable: The equilibrium solution y(t) = y* is semistable if initial conditions y(t0) = y0 "near" the critical point on one side of it lead to solutions y(t) that approach y* as t → ∞, while initial conditions on the other side of the critical point do not approach it. This point is also called a node.
A critical point is called globally stable if every solution approaches the equilibrium solution independently of the initial conditions.

If the slope function f(x) is continuous, the behavior of solutions of the autonomous equation can be determined from the slope lines along the vertical axis. This leads to construction of what is called a phase line for the differential equation.

 

Every critical point is a solution of the autonomous differential equation \( y' = f(y) , \) but solutions cannot cross it. If a solution touches the equilibrium solution, then this stationary solution is a singular solution. According to the existence theorem, if f(y) and ∂f(y)/ ∂y are continuous in some domain, then in this domain we are guaranteed that no other solution can intersect an equilibrium solution.

The nature of autonomous equations makes spotting constant solutions and interpreting the general behavior of solutions fairly straightforward by analyzing the slope function.

  • If the slope function is increasing below and decreasing above the critical point    ⇒   Stable
  • If the slope function is decreasing below and increasing above the critical point    ⇒   Unstable
  • If the slope function is decreasing below and above OR increasing below and above    ⇒   Semi-stable
Theorem 1: Suppose that the slope function f(y) of the autonomous differential equation \( {\text d}y/{\text d}x = f(y) \) has a continuous derivative in some interval containing an equilibrium point y*, that is, f(y*) = 0.
  • If the derivative of f(y) of the slope function evaluated at the critical point is negative, i.e., \( f'\left( y^{\ast} \right) < 0, \) then the equilibrium solution y = y* is stable and the equilibrium solution y* is an attractor. This is a situation when neighbouring solutions that start above with y > y* decrease toward it, but those neighbouring solutions that start below it with y < y* increase towards the constant solution y*.
  • If this derivative is positive, that is, \( f'\left( y^{\ast} \right) > 0, \) then the equilibrium solution y = y* is unstable and the critical point is a repellor. This is a situation when neighbouring solutions that start above with y > y* depart it, but those neighbouring solutions that start below it with y < y* decrease away from the constant solution y*.
When the derivative of the slope function evaluated at the critical point is zero, \( f'\left( y^{\ast} \right) = 0, \) then the above test is inconclusive. For example, the autonomous differential equation
\[ \dot{y} = f(y) = (y-1)\left( y-2 \right)^2 \]
has two critical points y = 1 (unstable and y = 2 semistable. The derivatives at these points are \( f' (1) = 1 > 0 \) and \( f' (2) = 0 . \)
p1 = Graphics[{PointSize[Large], Red, Point[{0, 1}]}];
p2 = Graphics[{PointSize[Large], Red, Point[{0, 2}]}];
txt = Graphics[{Text[Style["1", Black, Medium], {-0.2, 1.01}],
Text[Style["unstable", Black, Medium], {2, 1.01}],
Text[Style["2", Black, Medium], {-0.2, 3.1}],
Text[Style["semi-stable", Black, Medium], {2.1, 3.1}]}];
line = Graphics[Line[{{0, 0}, {0, 3}}], Epilog -> Inset[txt]];
a1 = Graphics[{Arrowheads[0.1], Arrow[{{0, 0.8}, {0, -0.2}}]}];
a2 = Graphics[{Arrowheads[0.1], Arrow[{{0, 1.2}, {0, 1.9}}]}];
a3 = Graphics[{Arrowheads[0.1], Arrow[{{0, 2.2}, {0, 3.2}}]}];
Show[line, p1, p2, a1, a2, a3]
StreamPlot[{1, (y - 1)*(y - 2)^2}, {x, -3, 3}, {y, -1, 4}, StreamStyle -> Yellow, FrameStyle -> LightGray, Background -> Black]
    Phase portrait for (y-1)(y-2)².
   Phase line for slope function (y-1)(y-2)².

On the other hand, the differential equation \( \dot{y} = f(y) = (y-1)^3 \) has one unstable critical point y = 1.

Theorem 2: Suppose that the slope function f(y) of the autonomous differential equation \( {\text d}y/{\text d}x = f(y) \) has two continuous derivatives in some interval containing an equilibrium point y*, at which the slope function has a doublke root: \( f\left( y^{\ast} \right) = f'\left( y^{\ast} \right) = 0 , \quad\mbox{but} \quad f''\left( y^{\ast} \right) \ne 0. \)
  • If \( f''\left( y^{\ast} \right) < 0, \) then the function f(y) has a maximum at y* and \( y' = f(y) < 0 \) in neighbourhood of y*. If neighbouring solutions that start below it y < y*, they are running away from it, but neighbouring solutions that start above it, approach it in phase space. The equilibrium solution is semi-stable.
  • If \( f''\left( y^{\ast} \right) > 0, \) then the function f(y) has a minimum at y* and \( y' = f(y) > 0 \) in neighbourhood of y*. The reagion of stability and instability are reversed with respect to the previous case. The equilibrium solution is still semi-stable.

A critical point y = y* of the autonomous differential equation \( y' = f(y) \) is referred to as being hyperbolic if the derivative of the slope function is not zero at the stationary point: \( f' \left( y^{\ast} \right) \ne 0 . \) Correspondingly, when \( f' \left( y^{\ast} \right) = 0 \) the case is called nonhyperbolic.
When you observe a higher order root of the slope function f(y), you must look at the lowest order derivative \( f^{(n)} (y) \) that does not vanish at the critical point y*.
  • If n is odd, the sign of \( f^{(n)} \left( y^{\ast} \right) \) determines stability. So if \( f^{(n)} \left( y^{\ast} \right) < 0,\) it is stable, and if \( f^{(n)} \left( y^{\ast} \right) > 0,\) it is unstable.
  • If n is even, there is stability on one side and instability on the other side of the constant solution y = y*.
Theorem 3: Let y(x) be a solution of the autonomous differential equation \( {\text d}y/{\text d}x = f(y) . \) Suppose that the slope function has a continuous derivative in some interval containing an equilibrium point y* and that at some point x0 the solution y(x) enters this interval.
  • If f(y) > 0 for y(x0) ≤ y < y* or f(y) < 0 for y* < y ≤ y(x0), then \( \lim_{x\to \infty} y(x) = y^{\ast} . \)
  • If f(y) < 0 for y(x0) ≤ y < y* or f(y) > 0 for y* < y ≤ y(x0), then \( \lim_{x\to -\infty} y(x) = y^{\ast} . \)

Here are three basic examples for illustration of each type of stability:
\[ \frac{{\text d}y}{{\text d}t} = \dot{y} = 1- y , \qquad \mbox{critical point $y=1$ is stable}. \]
\[ \dot{y} = y-1 , \qquad \mbox{critical point $y=1$ is unstable}. \]
\[ \dot{y} = \left( 1- y \right)^2 , \qquad \mbox{critical point $y=1$ is semi-stable}. \]
Theorem 4: Let f( y) be a continuous function on the closed interval [𝑎,b] that has only one null y* ∈ (𝑎,b) and f( y) ≠ 0 for all other points y ∈ ([𝑎,b). If the integral
\[ \int_{y}^{y^\ast} \frac{{\text d}y}{f(y)} \]
diverges, then the initial value problem for the autonomous differential equation \( y' = f(y), \quad y(t_0 ) = y^{\ast} \) has the unique solution y( t) = y*. If the above integral converges, then the initial value problem has multiple solutions.