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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.
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Figure 2.
(a) Two heteroclinic orbits, two hyperbolic saddles $ x_{1} $, $ x_{-1} $ and a turning point $ x_{TP} $. (b) A homoclinic orbit, a hyperbolic saddle $ x_{-1} $ and a turning point $ x_{TP} $.
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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.
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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.
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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.
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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.
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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.
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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).
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Figure 9.
(a) A quasi-periodic attractor with the parameter values $ \mu = 0.41 $, $ \gamma = 0.05, F = 3, \omega = 2.4 $. (b) A blow-up of (a). The coordinates of the (b) are the coordinates of the rectangle in (a).
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Figure 10.
(a) Two attractors of fixed points $ (-1.5964,1.7136) $ and $ (1.5964,-1.7136) $ in $ (x,y) $ plane with the parameters are $ \mu = 0.8, $ $ \gamma = 0, $ $ F = 2 $ and $ \omega = 1.3 $. (b) Basins of attraction for the two fixed points in $ (x,y) $ plane,
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Figure 11.
Period-8 points when parameters $ \mu = 0.52, \gamma = 0.05, F = 3, \omega = 2.4 $.
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Figure 12.
(a) Bifurcation diagrams in $ (\mu, x) $ plane with $ \gamma = 0.05, F = 3, \omega = 5.1 $, where $ \mu $ is the bifurcation parameter. (b) The corresponding maximal Lyapunov exponent in $ (\mu, L_{max}) $ plane, where $ L_{max} $ denotes the maximum Lyapunov exponents.
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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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