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gradient system (Definition)

A gradient system in $\mathbb{R}^n$ is an autonomous ordinary differential equation \begin{equation} \dot{x}=-\operatorname{grad}V(x)\label{eq} \end{equation}defined by the gradient of $V$ where $V:\mathbb{R}^n\to \mathbb{R}$ and $V\in C^\infty$ . The following results can be deduced from the definition of a gradient system.
Properties:

References

HSD
Hirsch, W. Morris, Smale, Stephen, Devaney, L. Robert: Differential Equations, Dynamical Systems & An Introduction to Chaos. Elsevier Academic Press, New York, 2004.




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Cross-references: periodic solutions, level curves, perpendicular, function, strictly decreasing, solution, asymptotically stable, isolated, real, equilibrium point, linearization, eigenvalues, gradient, ordinary differential equation, autonomous
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This is version 4 of gradient system, born on 2005-05-05, modified 2006-09-20.
Object id is 7014, canonical name is GradientSystem.
Accessed 5553 times total.

Classification:
AMS MSC34A34 (Ordinary differential equations :: General theory :: Nonlinear equations and systems, general)

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