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In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems .
The stability of fixed points of a system of constant coefficient linear differential equations of first order can be analyzed using the eigenvalues of the corresponding matrix. An autonomous system ′ =, where x(t) ∈ R n and A is an n×n matrix with real entries, has a constant solution =
That is to say, by evaluating the Jacobian matrix at each of the equilibrium points of the system, and then finding the resulting eigenvalues, the equilibria can be categorized. Then the behavior of the system in the neighborhood of each equilibrium point can be qualitatively determined, (or even quantitatively determined, in some instances ...
A Lyapunov function for an autonomous dynamical system {: ˙ = ()with an equilibrium point at = is a scalar function: that is continuous, has continuous first derivatives, is strictly positive for , and for which the time derivative ˙ = is non positive (these conditions are required on some region containing the origin).
Ergodic dynamical systems, that is, systems exhibiting randomness, exhibit flows as well. The most celebrated of these is perhaps the Bernoulli flow . The Ornstein isomorphism theorem states that, for any given entropy H , there exists a flow φ ( x , t ) , called the Bernoulli flow, such that the flow at time t = 1 , i.e. φ ( x , 1) , is a ...
A line, usually vertical, represents an interval of the domain of the derivative.The critical points (i.e., roots of the derivative , points such that () =) are indicated, and the intervals between the critical points have their signs indicated with arrows: an interval over which the derivative is positive has an arrow pointing in the positive direction along the line (up or right), and an ...
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In mathematics, the Bendixson–Dulac theorem on dynamical systems states that if there exists a function (,) (called the Dulac function) such that the expression . According to Dulac theorem any 2D autonomous system with a periodic orbit has a region with positive and a region with negative divergence inside such orbit.