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

Petržela 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.

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The graphical illustration of the sensitivity to initial conditions. Where the total number of random generations is 10 · 103 with standard deviation σ = 0.01 around initial point , total time Tend = 100, and integration step h = 0.01. On the left there is case A and on the right there is case B. The green points represent initial conditions, red color is a final point, and gray is original attractor.
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fig9: The graphical illustration of the sensitivity to initial conditions. Where the total number of random generations is 10 · 103 with standard deviation σ = 0.01 around initial point , total time Tend = 100, and integration step h = 0.01. On the left there is case A and on the right there is case B. The green points represent initial conditions, red color is a final point, and gray is original attractor.

Mentions: The graphical illustration of the sensitivity to initial conditions can be seen in Figure 9. The total number of random generations is 10 · 103 with standard deviation σ = 0.01 around initial point x0AB = (0,0, 0)T, total time Tend = 100, and integration step h = 0.01.


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

Petržela J, Gotthans T, Guzan M - ScientificWorldJournal (2014)

The graphical illustration of the sensitivity to initial conditions. Where the total number of random generations is 10 · 103 with standard deviation σ = 0.01 around initial point , total time Tend = 100, and integration step h = 0.01. On the left there is case A and on the right there is case B. The green points represent initial conditions, red color is a final point, and gray is original attractor.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

fig9: The graphical illustration of the sensitivity to initial conditions. Where the total number of random generations is 10 · 103 with standard deviation σ = 0.01 around initial point , total time Tend = 100, and integration step h = 0.01. On the left there is case A and on the right there is case B. The green points represent initial conditions, red color is a final point, and gray is original attractor.
Mentions: The graphical illustration of the sensitivity to initial conditions can be seen in Figure 9. The total number of random generations is 10 · 103 with standard deviation σ = 0.01 around initial point x0AB = (0,0, 0)T, total time Tend = 100, and integration step h = 0.01.

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