On the Detection and Dynamical Consequences of Orbits Homoclinic to Hyperbolic Periodic Orbits and Normally Hyperbolic Invariant Tori in a Class of Ordinary Differential Equations

1988 ◽  
Vol 48 (2) ◽  
pp. 262-285 ◽  
Author(s):  
Stephen Wiggins
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Orbits ◽  
Invariant Tori

scholarly journals On Using Curvature to Demonstrate Stability

2008 ◽  
Vol 2008 ◽  
pp. 1-7
Author(s):  
C. Connell McCluskey
Keyword(s):  
Differential Equations ◽  
Global Stability ◽  
Ordinary Differential Equations ◽  
Periodic Orbits ◽  
Large Amplitude ◽  
Minimum Distance ◽  
New Approach ◽  
Sudden Appearance ◽  
Globally Stable ◽  
Rule Out

A new approach for demonstrating the global stability of ordinary differential equations is given. It is shown that if the curvature of solutions is bounded on some set, then any nonconstant orbits that remain in the set, must contain points that lie some minimum distance apart from each other. This is used to establish a negative-criterion for periodic orbits. This is extended to give a method of proving an equilibrium to be globally stable. The approach can also be used to rule out the sudden appearance of large-amplitude periodic orbits.

SUFFICIENT CONDITIONS FOR THE EXISTENCE OF INVARIANT TORI IN MULTIDIMENSIONAL DYNAMICAL SYSTEMS

1991 ◽  
Vol 01 (03) ◽  
pp. 681-689
Author(s):  
V. S. AFRAIMOVICH ◽  
A. L. ZHELEZNYAK ◽  
I. L. ZHELEZNYAK
Keyword(s):  
Mathematical Model ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Invariant Tori ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Strange Attractors

A method for analyzing the existence of a multidimensional torus for certain systems of ordinary differential equations is proposed. Using this method, the existence of a multidimensional torus in one of such systems is analyzed. This system is a mathematical model of the dynamics of interacting structures within the drift hydrodynamical systems. The behavior of trajectories on the multidimensional torus is numerically investigated. The existence of two- and three-dimensional tori as well as strange attractors are considered.

Existence of periodic orbits of autonomous ordinary differential equations

1980 ◽  
Vol 85 (1-2) ◽  
pp. 153-172 ◽  
Author(s):  
Russell A. Smith
Keyword(s):  
Feedback Control ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Orbits ◽  
Autonomous Systems ◽  
Higher Order ◽  
Control Equation ◽  
Recurrent Orbits

SynopsisThe Poincaré-Bendixson theorem, concerning the existence of periodic orbits of plane autonomous systems, is extended to higher order systems under certain conditions. Under similar conditions, a complementary theorem on the existence of recurrent orbits is also proved. For the feedback control equation, these conditions are reduced to a form which can be easily verified in practice.

scholarly journals Persistence of periodic orbits in periodically forced impact systems

2014 ◽  
Vol 64 (1) ◽  
Author(s):  
Michal Fečkan ◽  
Michal Pospíšil
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Periodic Orbits

AbstractThis paper is devoted to the study of persistence of forced periodic solutions for impact systems from single periodic solutions of unperturbed impact equations. An example of planar discontinuous ordinary differential equations is given to illustrate the theory.

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