# Autonomous system

of ordinary differential equations

A system of ordinary differential equations which does not explicitly contain the independent variable $t$( time). The general form of a first-order autonomous system in normal form is:

$$\dot{x} _ {j} = f _ {j} ( x _ {1} \dots x _ {n} ) , \ j = 1 \dots n,$$

or, in vector notation,

$$\tag{1 } \dot{x} = f ( x ) .$$

A non-autonomous system $\dot{x} = f(t, x)$ can be reduced to an autonomous one by introducing a new unknown function $x _ {n+1} = t$. Historically, autonomous systems first appeared in descriptions of physical processes with a finite number of degrees of freedom. They are also called dynamical or conservative systems (cf. Dynamical system).

A complex autonomous system of the form (1) is equivalent to a real autonomous system with $2n$ unknown functions

$$\frac{d}{dt} ( \mathop{\rm Re} x ) = \mathop{\rm Re} f ( x ) , \ \frac{d}{dt} ( \mathop{\rm Im} x ) = \mathop{\rm Im} f ( x ).$$

The essential contents of the theory of complex autonomous systems — unlike in the real case — is found in the case of an analytic $f(x)$( cf. Analytic theory of differential equations).

Consider an analytic system with real coefficients and its real solutions. Let $x = \phi (t)$ be an (arbitrary) solution of the analytic system (1), let $\Delta = ( t _ {-} , t _ {+} )$ be the interval in which it is defined, and let $x(t; t _ {0} , x ^ {0} )$ be the solution with initial data $x \mid _ {t = t _ {0} } = x ^ {0}$. Let $G$ be a domain in $\mathbf R ^ {n}$ and $f \in C ^ {1} (G)$. The point $x ^ {0} \in G$ is said to be an equilibrium point, or a point of rest, of the autonomous system (1) if $f ( x ^ {0} ) \equiv 0$. The solution $\phi (t) \equiv x ^ {0}$, $t \in \mathbf R = ( - \infty , + \infty )$, corresponds to such an equilibrium point.

## Local properties of solutions.

1) If $\phi (t)$ is a solution, then $\phi ( t + c )$ is a solution for any $c \in \mathbf R$.

2) Existence: For any $t _ {0} \in \mathbf R , x ^ {0} \in G$, a solution $x(t; t _ {0} , x ^ {0} )$ exists in a certain interval $\Delta \ni t$.

3) Smoothness: If $f \in C ^ {p} (G) , p \geq 1$, then $\phi (t) \in C ^ {p+1} ( \Delta )$.

4) Dependence on parameters: Let $f = f(x, \alpha )$, $\alpha \in G _ \alpha \subset \mathbf R ^ {m}$, where $G _ \alpha$ is a domain; if $f \in C ^ {p} (G \times G _ \alpha )$, $p \geq 1$, then $x(t; t _ {0} , x ^ {0} , \alpha ) \in C ^ {p} ( \Delta \times G _ \alpha )$( for more details see ).

5) Let $x ^ {0}$ be a non-equilibrium point; then there exist neighbourhoods $V, W$ of the points $x ^ {0} , f( x ^ {0} )$, respectively, and a diffeomorphism $y = h(x) : V \rightarrow W$ such that the autonomous system has the form $\dot{y} = \textrm{ const }$ in $W$.

A substitution of variables $x = \phi (y)$ in the autonomous system (1) yields the system

$$\tag{2 } \dot{y} = ( \phi ^ \prime ( y ) ) ^ {-1} f ( \phi ( y ) ),$$

where $\phi ^ \prime (y)$ is the Jacobi matrix.

## Global properties of solutions.

1) Any solution $x = \phi (t)$ of the autonomous system (1) may be extended to an interval $\Delta = (t _ {-} , t _ {+} )$. If $\Delta = \mathbf R$, the solution is said to be unboundedly extendable; if $t _ {+} = + \infty , t _ {-} > - \infty$, the solution is said to be unboundedly extendable forwards in time (and, in a similar manner, backwards in time). If $t _ {+} < + \infty$ then, for any compact set $K \subset \Omega$, $x ^ {0} \in K$, there exists a $\tau = \tau (K) < t _ {+}$ such that the point $x(t; t _ {0} , x ^ {0} )$ is outside $K$ for $t > \tau (K)$( and, analogously, for $t _ {-} > - \infty$; cf. Prolongation of solutions of differential equations).

2) The extension is unique in the sense that any two solutions with common initial data are identical throughout their range of definition.

3) Any solution of an autonomous system belongs to one of the following three types: a) aperiodic, with $\phi (t _ {1} ) \neq \phi ( t _ {2} )$ for all $t _ {1} \neq t _ {2}$, $t _ {j} \in \mathbf R$; b) periodic, non-constant; or c) $\phi (t) \equiv \textrm{ const }$.

## Geometric interpretation of an autonomous system.

To each solution $x = \phi (t)$ is assigned a corresponding curve $\Gamma$: $x = \phi (t)$, $t \in \Delta$, inside the domain $G$. $G$ is then said to be the phase space of the autonomous system, $\Gamma$ is its trajectory in the phase space, and the solution is interpreted as motion along this trajectory in the phase space. The mapping $g ^ {t} : G \rightarrow G$ defined by the formula $g ^ {t} x ^ {0} = x (t; 0, x ^ {0} )$( i.e. each point moves along the phase trajectory during time $t$) is called the phase flow. In its domain of definition the phase flow satisfies the following conditions: 1) $g ^ {t} x$ is continuous in $(t, x)$; and 2) the group property $g ^ {t _ {1} + t _ {2} } x = g ^ {t _ {1} } g ^ {t _ {2} } x$.

The Liouville theorem is valid: Let $D \subset G$ be a domain with a finite volume and let $v _ {t}$ be the volume of the domain $g ^ {t} D \subset G$, then

$$\tag{3 } \left . \frac{d v _ {t} }{dt} \right | _ {t = 0 } = \int\limits _ { D } \mathop{\rm div} f ( x ) dx .$$

For a Hamiltonian system, a consequence of (3) is the conservation of the phase volume by the phase flow. A second variant of (3) is obtained as follows. Let $x = \phi (t, \alpha )$ be a family of solutions of (1), $\alpha = ( \alpha _ {1} \dots \alpha _ {n-1} ) \in G _ \alpha$, let $G _ \alpha$ be a domain and let $\phi \in C ^ {1} ( \Delta \times G _ \alpha )$, then

$$\tag{3'} \frac{d}{dt} \mathop{\rm ln} I ( t , \alpha ) = \mathop{\rm div} f ( x ) ,$$

where $I(t, \alpha ) = \mathop{\rm det} \partial x / \partial (t, \alpha )$.

## Structure of phase trajectories.

1) Any two phase trajectories have either no points in common or coincide.

2) Any phase trajectory belongs to one of the following types: a) a smooth, simple, non-closed Jordan arc; b) a cycle, i.e. a curve diffeomorphic to a circle; or c) a point (an equilibrium point). The local structure of phase trajectories in a small neighbourhood of a point other than an equilibrium point is trivial (cf. local property 5) of the solutions): The family of phase trajectories is diffeomorphic to a family of parallel straight lines. For a linear autonomous system the structure of phase trajectories in a neighbourhood of an equilibrium point is known, since the autonomous system is integrable . For non-linear autonomous systems this problem has not yet been completely solved, even for $n = 2$( cf. Qualitative theory of differential equations). One aspect of this problem is the question of stability of an equilibrium point (cf. Stability theory). A few results will be given below. Let $x ^ {0} , y ^ {0}$ be equilibrium points of the system (1), let

$$\tag{1'} \dot{y} = g ( y )$$

and let $U, V$ be neighbourhoods of the points $x ^ {0} , y ^ {0}$. The systems (1) and (1'}) are said to be equivalent in neighbourhoods of their equilibrium points $x ^ {0} , y ^ {0}$ if there exist neighbourhoods $U, V$ and a bijective mapping $h: U \rightarrow V$ such that $( h \circ f ^ {t} ) x = (g ^ {t} \circ h ) x$( for $x \in U$, $f ^ {t} x \in U$, $( g ^ {t} \circ h)x \in V$), i.e. as a result of the substitution $y = h(x)$ the trajectories of the autonomous system (1) go into trajectories of the autonomous system (1'}). The equivalence is said to be differentiable (topological) if $h$ is a diffeomorphism (homeomorphism). Let $x ^ {0}$ be an equilibrium point of the autonomous system (1), let the matrix $f ^ { \prime } (x ^ {0} )$ be non-degenerate, and let it not possess any pure imaginary eigen values. Then the autonomous system (1) in a neighbourhood of $x ^ {0}$ is topologically equivalent to its linear part $\dot{y} = f ^ { \prime } (x ^ {0} ) y$. An important example is the autonomous system $\dot{x} = Ax , \dot{y} = By$ where $A, B$ are constant matrices with pure imaginary eigen values and $n > 2$; it is not known when these autonomous systems are topologically equivalent. One of the most fundamental problems in the theory of autonomous systems is that of the structure of the entire family of phase trajectories. The most complete results have been obtained for $n = 2$, but even in this case the solution is far from complete.

How to Cite This Entry:
Autonomous system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Autonomous_system&oldid=53284
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article