The nonlinear Rayleigh damping term that is introduced to the classical parametrically excited pendulum makes the parametrically excited pendulum more complex and interesting. The effect of the nonlinear damping term on the new excitable systems is investigated based on analytical techniques such as Melnikov theory. The threshold conditions for the occurrence of Smale-horseshoe chaos of this deterministic system are obtained. Compared with the existing conclusion, i.e. the smaller the damping term is, the easier the chaotic motions become when the damping term is linear, our analysis, however, finds that the smaller or the larger the damping term is, the easier the Smale-horseshoe heteroclinic chaotic motions become. Moreover, the bifurcation diagram and the patterns of attractors in Poincaré map are studied carefully. The results demonstrate the new system exhibits rich dynamical phenomena: periodic motions, quasi-periodic motions and even chaotic motions. Importantly, according to the property of transitive as well as the fractal layers for a chaotic attractor, we can verify whether a attractor is a quasi-periodic one or a chaotic one when the maximum lyapunov exponent method is difficult to distinguish. Numerical simulations confirm the analytical predictions and show that the transition from regular to chaotic motion.
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Figure 1. The corresponding curves of the potential function and the phase spaces portraits when $ \gamma = 0 $ and $ \gamma = 0.05 $. (a) The potential function $ V(x) $ for $ \gamma = 0 $. (b) The phase spaces portraits when $ \gamma = 0 $. The closed trajectories represent oscillation motion around the hanging position, while the opened trajectories $ R_{1} $, $ R_{2} $ with unchanging sign of velocity visualize the rotating motion of the pendulum in the anti-clockwise or clockwise direction, respectively. (c) The potential function $ V(x) $ for $ \gamma = 0.05 $. (d) The phase spaces portraits when $ \gamma = 0.05 $. Starting from each hyperbolic saddle, there exists an asymmetric homoclinic orbit located at the right-hand of each saddle, which means oscillation motions from one peak point can not reach to the next right peak point but turn downward.
Figure 3. The critical heteroclinic bifurcation curves are ploted when $ \gamma = 0 $. (a) $ F = 2 $.The region (I) below the heteroclinic bifurcation curve represents $ M_{-1,1}(t) $ has simple zeros, which means the upper heteroclinic orbit will break and then the chaotic behavior may appear. Similarly, the region (II) over the heteroclinic bifurcation curve represents $ M_{1,-1}(t) $ has simple zeros, which means the lower heteroclinic orbit will break and then the chaotic behavior may appear. (b) $ F = 2 $ and $ F = 3 $.The larger the exciting term coefficient $ F $ is, the easier the chaotic motions become.
Figure 4. The critical homoclinic bifurcation curves are plotted when $ \gamma = 0.05 $, $ F = 2 $, and $ \gamma = 0.05 $, $ F = 3 $, respectively. The possible chaotic motions will be getting popular in system with decrease of $ \mu $ and the increase of $ F $. It is also found that for a fixed value of $ F $, there are three critical values under which homoclinic bifurcation may occur.
Figure 5. The stable and unstable manifolds of the fixed points of equation $ \ddot{x}-\mu(1-\dot{x}^2)\dot{x}+\sin x+\gamma+F\sin x\cos(\omega{t}) = 0 $ (with $ \gamma = 0, F = 2, \omega = 1.3 $, $ \mu = 0.12 $ in (a) and $ \mu = 0.83 $ in (b)). (a) Fixed points, stable and unstable manifolds of the fixed point (approximately $ (3.067,-0.622)) $. In the upper left window, the fixed points are marked by crosses. In the upper right window, a part of stable manifold (blue online) of the fixed point (approximately $ (3.067,-0.622) $) is superimposed. In the lower left window, a part of unstable manifold (red online) of the fixed point (approximately $ (3.067,-0.622) $) is superimposed. The lower right window contains a superposition of the pictures of the upper left, upper right, lower left windows. The intersection of the stable and unstable manifolds implies the existence of chaos. (b) A part of the stable (blue online) and unstable manifold (red online) of the fixed point (approximately $ (3.063,-0.021) $). The intersection of the stable and unstable manifolds implies the existence of chaos.
Figure 6. The stable and unstable manifolds of the fixed points of equation $ \ddot{x}-\mu(1-\dot{x}^2)\dot{x}+\sin x+\gamma+F\sin x\cos(\omega{t}) = 0 $ (with $ \gamma = 0.05, F = 3, \omega = 2.4 $, $ \mu = 0.23 $ in (a) and $ \mu = 0.72 $ in (b)). (a) Fixed points, stable and unstable manifolds of the fixed point (approximately $ (-3.052,-0.018)) $. In the upper left window, the fixed points are marked by crosses. In the upper right window, a part of stable manifold (blue online) of the fixed point (approximately $ (-3.052,-0.018)) $ is superimposed. In the lower left window, a part of unstable manifold (red online) of the fixed point (approximately $ (-3.052,-0.018)) $ is superimposed. The lower right window contains a superposition of the pictures of the upper left, upper right, lower left windows. The intersection of the stable and unstable manifolds implies the existence of chaos. (b) A part of stable (blue online) and unstable (red online) manifolds of the fixed point (approximately $ (-3.064,-0.032) $). The separation of the stable and unstable manifolds means that chaos does not exist.
Figure 7. Bifurcation diagrams and the corresponding maximal Lyapunov exponents of equation $ \ddot{x}-\mu(1-\dot{x}^2)\dot{x}+\sin x+\gamma+F\sin x\cos(\omega{t}) = 0 $, (a) and (b) with $ \gamma = 0, F = 2, \omega = 1.3 $, (c) and (d) with $ \gamma = 0.05, F = 3, \omega = 2.4 $, respectively, where $ \mu $ represents the bifurcation parameter and $ L_{max} $ denotes the maximum Lyapunov exponents.
Figure 8. A chaotic attractor and consecutive blow-ups in $ (x,y) $ plane. (a) shows a chaotic attractor with the parameter values $ \mu = 0.1 $, $ \gamma = 0, F = 2, \omega = 1.3 $. The coordinates of the (b) are the coordinates of the rectangle in (a). Similarly, the coordinates of (c) are the coordinates of the rectangle in (b).
Figure 13. (a) Five different attractors when $ \mu = 0.01, \gamma = 0.05, F = 3, \omega = 5.1 $. Limit cycle (as shown in the upper left window); Period-$ 2 $ points (as shown in the upper right window); Period-$ 6 $ points (as shown in the lower left and right windows). (b) The corresponding attractive basins of the five attractors.
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The corresponding curves of the potential function and the phase spaces portraits when
(a) Two heteroclinic orbits, two hyperbolic saddles
The critical heteroclinic bifurcation curves are ploted when
The critical homoclinic bifurcation curves are plotted when
The stable and unstable manifolds of the fixed points of equation
The stable and unstable manifolds of the fixed points of equation
Bifurcation diagrams and the corresponding maximal Lyapunov exponents of equation
A chaotic attractor and consecutive blow-ups in
(a) A quasi-periodic attractor with the parameter values
(a) Two attractors of fixed points
Period-8 points when parameters
(a) Bifurcation diagrams in
(a) Five different attractors when