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Dynamical tangles in third-order oscillator with single jump function.

Petr┼żela J, Gotthans T, Guzan M - ScientificWorldJournal (2014)

Bottom Line: This contribution brings a deep and detailed study of the dynamical behavior associated with nonlinear oscillator described by a single third-order differential equation with scalar jump nonlinearity.The relative primitive geometry of the vector field allows making an exhaustive numerical analysis of its possible solutions, visualizations of the invariant manifolds, and basins of attraction as well as proving the existence of chaotic motion by using the concept of both Shilnikov theorems.The aim of this paper is also to complete, carry out and link the previous works on simple Newtonian dynamics, and answer the question how individual types of the phenomenon evolve with time via understandable notes.

View Article: PubMed Central - PubMed

Affiliation: Department of Radio Electronics, Brno University of Technology, Purkynova 118, 612 00 Brno, Czech Republic.

ABSTRACT
This contribution brings a deep and detailed study of the dynamical behavior associated with nonlinear oscillator described by a single third-order differential equation with scalar jump nonlinearity. The relative primitive geometry of the vector field allows making an exhaustive numerical analysis of its possible solutions, visualizations of the invariant manifolds, and basins of attraction as well as proving the existence of chaotic motion by using the concept of both Shilnikov theorems. The aim of this paper is also to complete, carry out and link the previous works on simple Newtonian dynamics, and answer the question how individual types of the phenomenon evolve with time via understandable notes.

Show MeSH
The geometric structures important for optimization, starting situation, and systems A (a) and B (b), where boundary planes are white, fixed points are black dots, eigenvectors are green, eigenplanes are gray, intersections of eigenvectors and boundary planes are red crosses, and intersections of eigenplanes and boundary plane are blue lines.
© Copyright Policy - open-access
Related In: Results  -  Collection


getmorefigures.php?uid=PMC4269091&req=5

fig4: The geometric structures important for optimization, starting situation, and systems A (a) and B (b), where boundary planes are white, fixed points are black dots, eigenvectors are green, eigenplanes are gray, intersections of eigenvectors and boundary planes are red crosses, and intersections of eigenplanes and boundary plane are blue lines.

Mentions: The goal function value is minimized and penalized if geometry of the vector field or desired property of the system changes. It is preserved by choosing the suitable guess values as well as by the restrictions on the parameter space under inspection. The geometric structures like points, lines, and planes important for optimization are defined in Figure 4.


Dynamical tangles in third-order oscillator with single jump function.

Petr┼żela J, Gotthans T, Guzan M - ScientificWorldJournal (2014)

The geometric structures important for optimization, starting situation, and systems A (a) and B (b), where boundary planes are white, fixed points are black dots, eigenvectors are green, eigenplanes are gray, intersections of eigenvectors and boundary planes are red crosses, and intersections of eigenplanes and boundary plane are blue lines.
© Copyright Policy - open-access
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4269091&req=5

fig4: The geometric structures important for optimization, starting situation, and systems A (a) and B (b), where boundary planes are white, fixed points are black dots, eigenvectors are green, eigenplanes are gray, intersections of eigenvectors and boundary planes are red crosses, and intersections of eigenplanes and boundary plane are blue lines.
Mentions: The goal function value is minimized and penalized if geometry of the vector field or desired property of the system changes. It is preserved by choosing the suitable guess values as well as by the restrictions on the parameter space under inspection. The geometric structures like points, lines, and planes important for optimization are defined in Figure 4.

Bottom Line: This contribution brings a deep and detailed study of the dynamical behavior associated with nonlinear oscillator described by a single third-order differential equation with scalar jump nonlinearity.The relative primitive geometry of the vector field allows making an exhaustive numerical analysis of its possible solutions, visualizations of the invariant manifolds, and basins of attraction as well as proving the existence of chaotic motion by using the concept of both Shilnikov theorems.The aim of this paper is also to complete, carry out and link the previous works on simple Newtonian dynamics, and answer the question how individual types of the phenomenon evolve with time via understandable notes.

View Article: PubMed Central - PubMed

Affiliation: Department of Radio Electronics, Brno University of Technology, Purkynova 118, 612 00 Brno, Czech Republic.

ABSTRACT
This contribution brings a deep and detailed study of the dynamical behavior associated with nonlinear oscillator described by a single third-order differential equation with scalar jump nonlinearity. The relative primitive geometry of the vector field allows making an exhaustive numerical analysis of its possible solutions, visualizations of the invariant manifolds, and basins of attraction as well as proving the existence of chaotic motion by using the concept of both Shilnikov theorems. The aim of this paper is also to complete, carry out and link the previous works on simple Newtonian dynamics, and answer the question how individual types of the phenomenon evolve with time via understandable notes.

Show MeSH