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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.

Show MeSH
System configuration B in normal form (a), Jordan form (b), and first and second equivalent.
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Related In: Results  -  Collection


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fig2: System configuration B in normal form (a), Jordan form (b), and first and second equivalent.

Mentions: The three-dimensional perspective views on the chaotic state space attractors together with plane projections for each dynamical system mentioned above and obtained by using Mathcad with build-in fourth-order Runge-Kutta numerical integration method are shown in Figures 1 and 2, respectively. For these simulations final time equals tmax⁡ = 1000 with time step tstep = 0.01. The initial conditions were set: x0A = (0.1,0, 0)T and x0B = (0.1,0, 0)T. Note that LTC operation is demonstrated by means of Figure 3.


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

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

System configuration B in normal form (a), Jordan form (b), and first and second equivalent.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

fig2: System configuration B in normal form (a), Jordan form (b), and first and second equivalent.
Mentions: The three-dimensional perspective views on the chaotic state space attractors together with plane projections for each dynamical system mentioned above and obtained by using Mathcad with build-in fourth-order Runge-Kutta numerical integration method are shown in Figures 1 and 2, respectively. For these simulations final time equals tmax⁡ = 1000 with time step tstep = 0.01. The initial conditions were set: x0A = (0.1,0, 0)T and x0B = (0.1,0, 0)T. Note that LTC operation is demonstrated by means of Figure 3.

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