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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.

No MeSH data available.


The Bogdanov-Takens system (2) with the potential function V(x) = x3/3 − x2/2 and a friction term x − μ (top row), and its generalization (4) to a friction term μ1 − V(x) (bottom row), compare Eq. (6). Left column: The potential function together with the color-coded regions of energy dissipation and uptake respectively, compare Eq. (3). Right column: The phase planes at the respective homoclinic bifurcation points, with the unstable foci and the saddles denoted by open circles. The green and blue trajectories are the stable and unstable manifolds, while the red trajectory corresponds to the homoclinic loop.
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f1: The Bogdanov-Takens system (2) with the potential function V(x) = x3/3 − x2/2 and a friction term x − μ (top row), and its generalization (4) to a friction term μ1 − V(x) (bottom row), compare Eq. (6). Left column: The potential function together with the color-coded regions of energy dissipation and uptake respectively, compare Eq. (3). Right column: The phase planes at the respective homoclinic bifurcation points, with the unstable foci and the saddles denoted by open circles. The green and blue trajectories are the stable and unstable manifolds, while the red trajectory corresponds to the homoclinic loop.

Mentions: which is often used as a prototype system for homoclinic bifurcations5. Here, the mechanical potential is a third order polynomial, as illustrated in Fig. 1. The friction force is directly proportional to the velocity y, hence fixpoints of (2) correspond to the minima and the maxima of the potential V(x).


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

Sándor B, Gros C - Sci Rep (2015)

The Bogdanov-Takens system (2) with the potential function V(x) = x3/3 − x2/2 and a friction term x − μ (top row), and its generalization (4) to a friction term μ1 − V(x) (bottom row), compare Eq. (6). Left column: The potential function together with the color-coded regions of energy dissipation and uptake respectively, compare Eq. (3). Right column: The phase planes at the respective homoclinic bifurcation points, with the unstable foci and the saddles denoted by open circles. The green and blue trajectories are the stable and unstable manifolds, while the red trajectory corresponds to the homoclinic loop.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f1: The Bogdanov-Takens system (2) with the potential function V(x) = x3/3 − x2/2 and a friction term x − μ (top row), and its generalization (4) to a friction term μ1 − V(x) (bottom row), compare Eq. (6). Left column: The potential function together with the color-coded regions of energy dissipation and uptake respectively, compare Eq. (3). Right column: The phase planes at the respective homoclinic bifurcation points, with the unstable foci and the saddles denoted by open circles. The green and blue trajectories are the stable and unstable manifolds, while the red trajectory corresponds to the homoclinic loop.
Mentions: which is often used as a prototype system for homoclinic bifurcations5. Here, the mechanical potential is a third order polynomial, as illustrated in Fig. 1. The friction force is directly proportional to the velocity y, hence fixpoints of (2) correspond to the minima and the maxima of the potential V(x).

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.