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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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Several examples of the laboratory measurements (for case A): double-scroll attractor (a,b), single-scroll (c), AV curve of nonlinear resistor including saturation (d), limit cycle (e), and period doubling (f).
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fig11: Several examples of the laboratory measurements (for case A): double-scroll attractor (a,b), single-scroll (c), AV curve of nonlinear resistor including saturation (d), limit cycle (e), and period doubling (f).

Mentions: The experimental verification (measurements) of this chaotic oscillator is provided by means of Figure 11. Note that proposed circuitry represents only case A dynamical system (2). Variant B can be modeled by using slightly modified nonlinear resistor. Transformation from normal form into voltages across capacitors is represented by square regular matrix:(19)Txyz=uc1uc2uc3,  T=−1c2(r1+r2)c2c3r1r21000c2r10.


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

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

Several examples of the laboratory measurements (for case A): double-scroll attractor (a,b), single-scroll (c), AV curve of nonlinear resistor including saturation (d), limit cycle (e), and period doubling (f).
© Copyright Policy - open-access
Related In: Results  -  Collection

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

fig11: Several examples of the laboratory measurements (for case A): double-scroll attractor (a,b), single-scroll (c), AV curve of nonlinear resistor including saturation (d), limit cycle (e), and period doubling (f).
Mentions: The experimental verification (measurements) of this chaotic oscillator is provided by means of Figure 11. Note that proposed circuitry represents only case A dynamical system (2). Variant B can be modeled by using slightly modified nonlinear resistor. Transformation from normal form into voltages across capacitors is represented by square regular matrix:(19)Txyz=uc1uc2uc3,  T=−1c2(r1+r2)c2c3r1r21000c2r10.

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