What are autonomous and non-autonomous systems and how are they different from each other. What are the differences between the types of systems they are describing? Do both autonomous and non-autonomous system describe dynamical systems or are dynamical systems something different?

  • $\begingroup$ have you tried googling ? $\endgroup$
    – Our
    Mar 5, 2019 at 16:40
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    $\begingroup$ @onurcanbektas Yes I have...but none of the answers seem to tell me the difference between them. Mathematically I know how they are..that is their general form with non-autonomous systems being explicitly dependent on the independent variable. But how does that make it different than autonomous systems (which are still implicitly dependent on the independent-variable. They are called invaarient with time(the independent-variable) but not independent of time ) $\endgroup$
    Mar 5, 2019 at 16:44
  • $\begingroup$ Your question is rather too broad. Do you mean nonautonomous vs. autonomous systems of differential equations? Then an autonomous system of differential equations gives rise to a (true) dynamical system. A nonautonomous system of differential equations gives rise to a more general object (sometimes it is called a nonautonomous dynamical system, however the terminology is not fixed). Or perhaps you have some other systems in mind? $\endgroup$
    – user539887
    Mar 6, 2019 at 8:08
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    $\begingroup$ For differential equations (ordinary, partial, with deviating argument, etc.), as well as for some integral equations, a rule of thumb is the following: the equation is called autonomous when for any of its solutions any (admissible) time translate of that solution is its solution, too. $\endgroup$
    – user539887
    Mar 6, 2019 at 8:12
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    $\begingroup$ Not at all: the state variables of the system change with time, but the law governing their change does not change with time. For instance, in radioactive decay, the number of nuclei (state of the system) decreases, but according to the law (represented as decay constant, or half-life) that is constant in time. $\endgroup$
    – user539887
    Mar 7, 2019 at 7:21

1 Answer 1


I assume you refer to dynamical systems; that is, differential equations of the form $$ \dot x = f(x,t,u).$$

These are classified according to which terms appear in $f(x,t,u)$:

  • Time invariant if $f(x,t,u)=f(x,u)$ is independent of time,
  • Autonomous if $f(x,t,u) = f(x)$ is time invariant and independent of the input.

(These definitions come from Khalil 2001)

What is important is that the evolution of an autonomous system cannot be influenced using an external input and only depends on the initial condition whereas an appropriate controller can change the behavior of non-autonomous systems.

  • $\begingroup$ Are you sure, that autonomous does include dependence on time? $\endgroup$ Mar 5, 2019 at 20:52
  • $\begingroup$ I guess it depends on the definition; I have changed my answer to reflect what is in Khalil 2001. $\endgroup$ Mar 6, 2019 at 7:26
  • $\begingroup$ @RiccardoSvenRisuleo Are you sure Time invariant means independent of time? I am asking because from what I understand time independent means a static system. Time invariant systems are dynamical system too in which the system in implicitly dependent on time. These are dynamical time invariant systems. $\endgroup$
    Mar 6, 2019 at 9:23
  • $\begingroup$ Time invariant means that if $x_1(t)$ is the response of the system started at $x_1(0)$ and $x_2(t)$ is the response of the system started at $x_1(T)$ for some $T>0$, then $x_2(t)=x_1(t)$ for all $t>T$. In other words: the response looks the same irrespective of when the system is started. $\endgroup$ Mar 6, 2019 at 9:47
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    $\begingroup$ @GRANZER nono! The differential equation is independent of time, not the state variables! For instance $\dot x = -ax$ for some constant $a$ is independent of time and autonomous; the solution started at $x(0) = x_0$ is $x(t) = x_0 e^{-a t}$, which is a function of time. $\endgroup$ Mar 7, 2019 at 8:17

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