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Local Bifurcations
Published in LM Pismen, Working with Dynamical Systems, 2020
In 2D phase space, there are only five possibilities. The stationary state is either a stable or an unstable node when both eigenvalues are real and, respectively, either negative or positive. The stationary state is either a stable or an unstable focus when both eigenvalues are complex with, respectively, either negative or positive real parts. For a stable node or focus, the dimension of the stable manifold is two and the dimension of the unstable manifold is zero, and for an unstable node or focus, the other way around. The remaining possibility is a saddle, realized when the eigenvalues are real and have opposite signs; then the dimension of both stable and unstable manifolds is one. Typical trajectories in the vicinity of a stationary point for all these cases are sketched in Fig. 2.3. The dashed lines, which are, generally, not mutually perpendicular, show the direction of real eigenvectors. Arrows showing the direction of motion are missing, and therefore the pictures for stable and unstable nodes and foci, which can be converted one into the other by reversing time, are identical. Note that trajectories are, generally, curvilinear even in the vicinity of a node, since evolution is faster in the direction of the eigenvector with the eigenvalue largest by its absolute value.
Saturation of generalized partially hyperbolic attractors
Published in Dynamical Systems, 2019
In what follows, stands for the Lebesgue measure on and . In this section an example of singular hyperbolic attractor with positive volume is given. The idea is to construct a fat horseshoe1H on the Poincaré section Σ of a geometric Lorenz attractor Λ such that . Let us assume that we have constructed the fat horseshoe H. Then is a subset of Λ and . Therefore the geometric Lorenz attractor Λ has positive volume for its associated vector filed and a priori for the time one map diffeomorphism. Note that the attractor Λ is horseshoe-like, i.e., it dose not contain any local stable manifold, but it dose contain an unstable manifold2.
Convergence rate of stationary solutions to outflow problem for full Navier–Stokes equations
Published in Applicable Analysis, 2019
Yazhou Chen, Hakho Hong, Xiaoding Shi
By (2.13), (1.6), , and (1.12), we have . Therefore, applying the center manifold theory, by the same lines as in Section 2.1 of [14], there exist a local center manifold and a local stable manifold such that if the data satisfies
Topological pressure for conservative C 1-diffeomorphisms with no dominated splitting
Published in Dynamical Systems, 2023
If Λ is a hyperbolic set for a diffeomorphism f, and , the stable manifold of x is given by . Similarly, we can define the unstable manifold of x as .