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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 circuitry implementation of chaotic oscillator for case A.
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fig10: The circuitry implementation of chaotic oscillator for case A.

Mentions: In order to evaluate geometrical structural stability of equations a new circuit (not presented so far for (1)) was assembled and measured. The circuit synthesis methods dedicated to modeling the nonlinear dynamical systems are well-known and commonly used [28, 29]. Assume canonical (in the sense of minimum circuit components) network shown in Figure 10 which represents a parallel connection of the third-order linear admittance and two-segment piecewise-linear resistor. The straightforward analysis gives us admittance function in Laplace transform, namely,(15)Ys=s3+φ1s2φ2s≈c1c2c3r1r2s3+c1c2(r1+r2)s2+(c1+c2)s,where resistors and capacitors have normalized values so far. To obtain the real passive element values let consider impedance normalization factor ξI and frequency norm ξF. By comparing the individual coefficients (15) with (1) we get following simple relationships(16)C1=C2=φ22ξIξF,  C3=φ22φ12ξIξF,  R1=R2=2φ1ξIφ22.


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

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

The circuitry implementation of chaotic oscillator for case A.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

fig10: The circuitry implementation of chaotic oscillator for case A.
Mentions: In order to evaluate geometrical structural stability of equations a new circuit (not presented so far for (1)) was assembled and measured. The circuit synthesis methods dedicated to modeling the nonlinear dynamical systems are well-known and commonly used [28, 29]. Assume canonical (in the sense of minimum circuit components) network shown in Figure 10 which represents a parallel connection of the third-order linear admittance and two-segment piecewise-linear resistor. The straightforward analysis gives us admittance function in Laplace transform, namely,(15)Ys=s3+φ1s2φ2s≈c1c2c3r1r2s3+c1c2(r1+r2)s2+(c1+c2)s,where resistors and capacitors have normalized values so far. To obtain the real passive element values let consider impedance normalization factor ξI and frequency norm ξF. By comparing the individual coefficients (15) with (1) we get following simple relationships(16)C1=C2=φ22ξIξF,  C3=φ22φ12ξIξF,  R1=R2=2φ1ξIφ22.

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