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A versatile class of prototype dynamical systems for complex bifurcation cascades of limit cycles.

Sándor B, Gros C - Sci Rep (2015)

Bottom Line: The prototype systems are adaptive, with friction forces f(V(x)) being functionally dependent exclusively on the mechanical potential V(x), characterized in turn by a finite number of local minima.We discuss several low-dimensional systems, with friction forces f(V) which are linear, quadratic or cubic polynomials in the potential V.We point out that the zeros of f(V) regulate both the relative importance of energy uptake and dissipation respectively, serving at the same time as bifurcation parameters, hence allowing for an intuitive interpretation of the overall dynamical behavior.

View Article: PubMed Central - PubMed

Affiliation: Institute for Theoretical Physics, Goethe University Frankfurt, Frankfurt am Main, 60438, Germany.

ABSTRACT
A general class of prototype dynamical systems is introduced, which allows to study the generation of complex bifurcation cascades of limit cycles, including bifurcations breaking spontaneously a symmetry of the system, period doubling and homoclinic bifurcations, and transitions to chaos induced by sequences of limit cycle bifurcations. The prototype systems are adaptive, with friction forces f(V(x)) being functionally dependent exclusively on the mechanical potential V(x), characterized in turn by a finite number of local minima. We discuss several low-dimensional systems, with friction forces f(V) which are linear, quadratic or cubic polynomials in the potential V. We point out that the zeros of f(V) regulate both the relative importance of energy uptake and dissipation respectively, serving at the same time as bifurcation parameters, hence allowing for an intuitive interpretation of the overall dynamical behavior. Starting from simple Hopf- and homoclinic bifurcations, complex sequences of limit cycle bifurcations are observed when the energy uptake gains progressively in importance.

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The logarithmic growth rate 〈ln(Δr)〉 averaged for 100 random initial conditions as a function of time for three qualitatively different types of dynamics: spiraling into a fixpoint (μ1 = −0.05), limit cycle oscillations (μ1 = 0.2) and chaotic behavior (μ1 = 0.3). Brown lines correspond to the best linear regression. In the first and last case, the line is fitted only to the first part of the trajectory. The dashed line indicates that the distance of the point pairs has reached the maximal accuracy of the integrator.
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f7: The logarithmic growth rate 〈ln(Δr)〉 averaged for 100 random initial conditions as a function of time for three qualitatively different types of dynamics: spiraling into a fixpoint (μ1 = −0.05), limit cycle oscillations (μ1 = 0.2) and chaotic behavior (μ1 = 0.3). Brown lines correspond to the best linear regression. In the first and last case, the line is fitted only to the first part of the trajectory. The dashed line indicates that the distance of the point pairs has reached the maximal accuracy of the integrator.

Mentions: Our prototype system (4) is not generically dissipative. We have evaluated the average contraction rate σ, as defined by (22) in the Methods section, and presented the results in Fig. 5. Phase space contracts trivially along the attracting limit cycles, but also, on average, in the chaotic region, where the average Lyapunov exponent becomes positive. is negative for μ1 < 0, when only stable fixpoints are present, vanishing for intermediate values of μ1, when stable limit cycles are present. The later is due to the fact, see Fig. 7 and the corresponding Methods section, that two initially close trajectories will generally flow to the same limit cycle with the relative distance becoming constant.


A versatile class of prototype dynamical systems for complex bifurcation cascades of limit cycles.

Sándor B, Gros C - Sci Rep (2015)

The logarithmic growth rate 〈ln(Δr)〉 averaged for 100 random initial conditions as a function of time for three qualitatively different types of dynamics: spiraling into a fixpoint (μ1 = −0.05), limit cycle oscillations (μ1 = 0.2) and chaotic behavior (μ1 = 0.3). Brown lines correspond to the best linear regression. In the first and last case, the line is fitted only to the first part of the trajectory. The dashed line indicates that the distance of the point pairs has reached the maximal accuracy of the integrator.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4510578&req=5

f7: The logarithmic growth rate 〈ln(Δr)〉 averaged for 100 random initial conditions as a function of time for three qualitatively different types of dynamics: spiraling into a fixpoint (μ1 = −0.05), limit cycle oscillations (μ1 = 0.2) and chaotic behavior (μ1 = 0.3). Brown lines correspond to the best linear regression. In the first and last case, the line is fitted only to the first part of the trajectory. The dashed line indicates that the distance of the point pairs has reached the maximal accuracy of the integrator.
Mentions: Our prototype system (4) is not generically dissipative. We have evaluated the average contraction rate σ, as defined by (22) in the Methods section, and presented the results in Fig. 5. Phase space contracts trivially along the attracting limit cycles, but also, on average, in the chaotic region, where the average Lyapunov exponent becomes positive. is negative for μ1 < 0, when only stable fixpoints are present, vanishing for intermediate values of μ1, when stable limit cycles are present. The later is due to the fact, see Fig. 7 and the corresponding Methods section, that two initially close trajectories will generally flow to the same limit cycle with the relative distance becoming constant.

Bottom Line: The prototype systems are adaptive, with friction forces f(V(x)) being functionally dependent exclusively on the mechanical potential V(x), characterized in turn by a finite number of local minima.We discuss several low-dimensional systems, with friction forces f(V) which are linear, quadratic or cubic polynomials in the potential V.We point out that the zeros of f(V) regulate both the relative importance of energy uptake and dissipation respectively, serving at the same time as bifurcation parameters, hence allowing for an intuitive interpretation of the overall dynamical behavior.

View Article: PubMed Central - PubMed

Affiliation: Institute for Theoretical Physics, Goethe University Frankfurt, Frankfurt am Main, 60438, Germany.

ABSTRACT
A general class of prototype dynamical systems is introduced, which allows to study the generation of complex bifurcation cascades of limit cycles, including bifurcations breaking spontaneously a symmetry of the system, period doubling and homoclinic bifurcations, and transitions to chaos induced by sequences of limit cycle bifurcations. The prototype systems are adaptive, with friction forces f(V(x)) being functionally dependent exclusively on the mechanical potential V(x), characterized in turn by a finite number of local minima. We discuss several low-dimensional systems, with friction forces f(V) which are linear, quadratic or cubic polynomials in the potential V. We point out that the zeros of f(V) regulate both the relative importance of energy uptake and dissipation respectively, serving at the same time as bifurcation parameters, hence allowing for an intuitive interpretation of the overall dynamical behavior. Starting from simple Hopf- and homoclinic bifurcations, complex sequences of limit cycle bifurcations are observed when the energy uptake gains progressively in importance.

No MeSH data available.


Related in: MedlinePlus