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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
Monte Carlo analysis (for case A) for various standard deviations (Gaussian distributed) around nominal values (2) for resistors σR and capacitors σC. Graphs can be understood as histograms in %, where the sum of values in histogram represents 100%. Dark magenta denotes fixed point solution, pink color is a limit cycle, yellow is chaotic motion, and brown color marks unbounded trajectories.
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fig12: Monte Carlo analysis (for case A) for various standard deviations (Gaussian distributed) around nominal values (2) for resistors σR and capacitors σC. Graphs can be understood as histograms in %, where the sum of values in histogram represents 100%. Dark magenta denotes fixed point solution, pink color is a limit cycle, yellow is chaotic motion, and brown color marks unbounded trajectories.

Mentions: Figure 12 give an idea if uncertainties in values of the passive components are capable to destruct chaotic nature of circuit. It is obvious that 1% tolerances cannot cause such changes, but 10% tolerances can lead to death of double-scroll in about 8% of cases. The total number of turns for each histogram is 10000 (for case A) with initial conditions ic = (0.1,0, 0)T.


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

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

Monte Carlo analysis (for case A) for various standard deviations (Gaussian distributed) around nominal values (2) for resistors σR and capacitors σC. Graphs can be understood as histograms in %, where the sum of values in histogram represents 100%. Dark magenta denotes fixed point solution, pink color is a limit cycle, yellow is chaotic motion, and brown color marks unbounded trajectories.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

fig12: Monte Carlo analysis (for case A) for various standard deviations (Gaussian distributed) around nominal values (2) for resistors σR and capacitors σC. Graphs can be understood as histograms in %, where the sum of values in histogram represents 100%. Dark magenta denotes fixed point solution, pink color is a limit cycle, yellow is chaotic motion, and brown color marks unbounded trajectories.
Mentions: Figure 12 give an idea if uncertainties in values of the passive components are capable to destruct chaotic nature of circuit. It is obvious that 1% tolerances cannot cause such changes, but 10% tolerances can lead to death of double-scroll in about 8% of cases. The total number of turns for each histogram is 10000 (for case A) with initial conditions ic = (0.1,0, 0)T.

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