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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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Trajectory close to the homoclinic orbit for dynamical system case A (a) and case B (b).
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Related In: Results  -  Collection


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fig5: Trajectory close to the homoclinic orbit for dynamical system case A (a) and case B (b).

Mentions: Similarly for dynamical system case B we get φ1 = 0.715499177650, φ2 = 0.577154986868, ξ1 = 1.284935534298, ξ2 = −0.631662627700, initial conditions x0B = (ξ2/ξ1, 0,0)T − (1 · 10−8, 0,0)T, fourth order Runge-Kutta build-in MATLAB function integration with variable step (initial step h = 1 · 10−6) and with maximal step h = 1 · 10−2, and final value of fitness function 7.62 · 10−4. The numerically integrated trajectories are provided in Figure 5.


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

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

Trajectory close to the homoclinic orbit for dynamical system case A (a) and case B (b).
© Copyright Policy - open-access
Related In: Results  -  Collection

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

fig5: Trajectory close to the homoclinic orbit for dynamical system case A (a) and case B (b).
Mentions: Similarly for dynamical system case B we get φ1 = 0.715499177650, φ2 = 0.577154986868, ξ1 = 1.284935534298, ξ2 = −0.631662627700, initial conditions x0B = (ξ2/ξ1, 0,0)T − (1 · 10−8, 0,0)T, fourth order Runge-Kutta build-in MATLAB function integration with variable step (initial step h = 1 · 10−6) and with maximal step h = 1 · 10−2, and final value of fitness function 7.62 · 10−4. The numerically integrated trajectories are provided in Figure 5.

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