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Nonlinear Dynamics
Published in Bogdan M. Wilamowski, J. David Irwin, Control and Mechatronics, 2018
The CTM is applied for describing the system. Invariant manifold is a curve (trajectory) in plane (N = 2) (Figure 1.15), or curve or surface in space (N = 3) (Figure 1.16), or in general a subspace (hypersurface) of the state space (N > 3). The manifolds are always associated with saddle point denoted here by x⋆. Any initial condition in the manifold results in movement of the operation point in the manifold under the action of the relevant differential equations. There are two kinds of manifolds: stable manifold denoted by Ws and unstable manifold denoted by Wu. If the initial points are on Ws or on Wu, the operation points remain on Ws or Wu forever, but the points on Ws are attracted by x⋆ and the points on Wu are repelled from x⋆. By considering t → –∞, every movement along the manifolds is reversed.
Dynamical Systems and Linear Algebra
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
Fritz Colonius, Wolfgang Kliemann
The local stable (and unstable) manifolds can be extended to global invariant manifolds by following the trajectories, i.e., Ws(p)=∪t≥0Φ(−t,Wlocs(p))andWu(p)=∪t≥0Φ(t,Wlocs(p)).
Periodic Orbits and Poincaré Return Maps
Published in Eric R. Westervelt, Jessy W. Grizzle, Christine Chevallereau, Jun Ho Choi, Benjamin Morris, Feedback Control of Dynamic Bipedal Robot Locomotion, 2018
Eric R. Westervelt, Jessy W. Grizzle, Christine Chevallereau, Jun Ho Choi, Benjamin Morris
The underlying idea is the following: The robot models addressed in this book are underactuated in one or more phases. The unactuated degrees of freedom in these models must be controlled indirectly through the actuated degrees of freedom. A good feedback design typically results in relatively higher bandwidth—that is, faster rates of convergence—for variables that are closer2 to the actuators. Also, with feedback, it is often possible to create invariant manifolds—that is, lower-dimensional surfaces with the property that if the system is initialized on the surface, its evolution remains on the surface. It is often quite advantageous to exploit timescale and invariance properties in stability analysis.
Understanding the geometry of dynamics: the stable manifold of the Lorenz system
Published in Journal of the Royal Society of New Zealand, 2018
The theory of dynamical systems describes such ‘universal’ geometric features of behaviour, which comprise a language that has proven extraordinarily useful in the analysis of dynamical models across the sciences: it effectively tells one what to expect and what to look for in a given system. The theory is inherently geometric in nature, in that it requires one to understand the overall organisation of phase space by (hyper)surfaces known as global invariant manifolds, which are associated with objects such as saddle equilibria (rest-state behaviour) and periodic orbits (oscillatory behaviour). In short, the interactions of invariant manifolds determine the overall behaviour, and they are the key to understanding the geometry of the observed dynamics. This point of view was introduced by Poincaré to demonstrate instabilities of planetary motion (Barrow–Green 1997). Similar geometric arguments were used in the 1970s to show that the famous Lorenz system features chaotic dynamics and unpredictability (Lorenz 1963; Guckenheimer and Williams 1979). This geometric theory has benefited enormously from the progress in scientific computing and the recent development of numerical methods that allow one to find invariant manifolds, and to investigate their interactions and bifurcations as parameters are changed (Krauskopf and Osinga 2007; Guckenheimer et al. 2015). Numerical approximations have yielded insights into the overall organisation of phase space in the transition to chaos (Doedel et al. 2011), near certain types of global bifurcations and singularities (Aguirre et al. 2014), and as mechanisms for so-called mixed-mode oscillations (MMOs) (Desroches et al. 2012).