Melnikov method for parabolic orbits

Casasayas, Josefina; Faísca, Patrícia F. N.; Nunes, Ana

doi:10.1007/s00030-003-1031-4

Cited by 3 publications

(1 citation statement)

References 14 publications

(17 reference statements)

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“…Nevertheless, there has been extensive work in the mathematics literature on non-hyperbolic trajectories and their stable and unstable manifolds, e.g. [65,16,30,17,15,5,10,44,6]. This work should serve as an excellent foundation for developing a theory of 'distinguished saddle-points' and their stable and unstable manifolds in finite time, aperiodically time-dependent velocity fields.…”

Section: Boundary Layer Separation On a Non-slip Boundarymentioning

confidence: 99%

Finite-time Lagrangian transport analysis: stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents

Branicki

Wiggins

2010

Nonlin. Processes Geophys.

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Abstract.We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of transport barriers in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagrangian coherent structures have been the main tools for characterising transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate in a concrete manner the issues and phenomena that arise in the setting of finitetime dynamical systems. Of particular significance for geophysical applications is the notion of flow transition which occurs when finite-time hyperbolicity is lost or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing and important area of dynamical systems theory.Correspondence to: M. Branicki

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Section: Boundary Layer Separation On a Non-slip Boundarymentioning

confidence: 99%

Finite-time Lagrangian transport analysis: stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents

Branicki

Wiggins

2010

Nonlin. Processes Geophys.

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A practical use of the Melnikov homoclinic method

Castilho

Marchesin

2009

Journal of Mathematical Physics

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Knotted Periodic Orbits and Chaotic Behavior of a Class of Three-Dimensional Flows

Chen

2011

Int. J. Bifurcation Chaos

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This paper considers a class of three-dimensional systems constructed by rotating some planar symmetric polynomial vector fields. It shows that this class of systems has infinitely many distinct types of knotted periodic orbits, which lie on a family of invariant torus. For two three-dimensional systems, exact explicit parametric representations of the knotted periodic orbits are given. For their perturbed systems, the chaotic behavior is discussed by using two different methods.

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Melnikov method for parabolic orbits

Cited by 3 publications

References 14 publications

Finite-time Lagrangian transport analysis: stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents

Finite-time Lagrangian transport analysis: stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents

A practical use of the Melnikov homoclinic method

Knotted Periodic Orbits and Chaotic Behavior of a Class of Three-Dimensional Flows

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