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A non-autonomous system is a dynamic equation on a smooth fiber bundle over . For instance, this is the case of non-autonomous mechanics . An r -order differential equation on a fiber bundle Q → R {\displaystyle Q\to \mathbb {R} } is represented by a closed subbundle of a jet bundle J r Q {\displaystyle J^{r}Q} of Q → R {\displaystyle Q\to ...
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 .
Consider the autonomous Itō stochastic differential equation: = + with initial condition =, where denotes the Wiener process, and suppose that we wish to solve this SDE on some interval of time [,]. Then the Milstein approximation to the true solution X {\displaystyle X} is the Markov chain Y {\displaystyle Y} defined as follows:
The system of autonomous differential equations for ~ is equivalent to that of non-autonomous ones for , and (,) is a bijection between the sets of integral curves of and ~, respectively. Flow [ edit ]
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation , for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature ...
In mathematics, a phase line is a diagram that shows the qualitative behaviour of an autonomous ordinary differential equation in a single variable, = (). The phase line is the 1-dimensional form of the general n {\displaystyle n} -dimensional phase space , and can be readily analyzed.
Differential equations are prominent in many scientific areas. Nonlinear ones are of particular interest for their commonality in describing real-world systems and how much more difficult they are to solve compared to linear differential equations.
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).