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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 basin of attraction for chaotic attractor, system case A. These slices are x-z projections of cross-sections with the y-axis. Black region represents initial condition inside basin of attraction. All other trajectories tend to infinity.
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fig7: The graphical illustration of the basin of attraction for chaotic attractor, system case A. These slices are x-z projections of cross-sections with the y-axis. Black region represents initial condition inside basin of attraction. All other trajectories tend to infinity.

Mentions: The remaining question which should be answered is the following: how many attractors are available for the nominal set of the parameters and show the associated basins of attractions. The most straightforward approach to visualize these subspaces is by means of repeated integrations. The grid for the initial condition was a cube with edge lengths δ ∈ (−2,2) with 400 points. Due to the symmetry of the vector field two mirrored strange attractors are highly expected. It eventually turns out that the system case A has only unbounded solution or chaotic attractor; trivial fixed point solution is out of question. These results are visible in Figure 7. For the system case B two chaotic attractors have been found; see Figure 8.


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

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

The graphical illustration of the basin of attraction for chaotic attractor, system case A. These slices are x-z projections of cross-sections with the y-axis. Black region represents initial condition inside basin of attraction. All other trajectories tend to infinity.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

fig7: The graphical illustration of the basin of attraction for chaotic attractor, system case A. These slices are x-z projections of cross-sections with the y-axis. Black region represents initial condition inside basin of attraction. All other trajectories tend to infinity.
Mentions: The remaining question which should be answered is the following: how many attractors are available for the nominal set of the parameters and show the associated basins of attractions. The most straightforward approach to visualize these subspaces is by means of repeated integrations. The grid for the initial condition was a cube with edge lengths δ ∈ (−2,2) with 400 points. Due to the symmetry of the vector field two mirrored strange attractors are highly expected. It eventually turns out that the system case A has only unbounded solution or chaotic attractor; trivial fixed point solution is out of question. These results are visible in Figure 7. For the system case B two chaotic attractors have been found; see Figure 8.

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