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This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor. Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. [6]
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, 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 ...
The A-stability concept for the solution of differential equations is related to the linear autonomous equation ′ =. Dahlquist (1963) proposed the investigation of stability of numerical schemes when applied to nonlinear systems that satisfy a monotonicity condition.
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:
In autonomous systems, the invariant set theorem can be applied to prove asymptotic stability, but this theorem is not applicable when the dynamics are a function of time. [14] Instead, Barbalat's lemma allows for Lyapunov-like analysis of these non-autonomous systems. The lemma is motivated by the following observations.
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).
The order of the differential equation is the highest order of derivative of the unknown function that appears in the differential equation. For example, an equation containing only first-order derivatives is a first-order differential equation, an equation containing the second-order derivative is a second-order differential equation, and so on.