| Citation: |
Jiaojiao Zhou, Fei Luo, Jinkai Jiang, Yundong Li. STRANGE NONCHAOTIC ATTRACTORS AND COMPLICATED QUASI-PERIODIC CASCADES INDUCED BY BIFURCATIONS IN A DRY-FRICTION OSCILLATOR UNDER QUASI-PERIODIC EXCITATION[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 963-997. doi: 10.11948/20260165
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STRANGE NONCHAOTIC ATTRACTORS AND COMPLICATED QUASI-PERIODIC CASCADES INDUCED BY BIFURCATIONS IN A DRY-FRICTION OSCILLATOR UNDER QUASI-PERIODIC EXCITATION
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Abstract
This paper explores the crossing bifurcation, switching-sliding and grazing-sliding bifurcations in a dry-friction oscillator on a moving belt under quasi-periodic excitation with two frequencies. We present a concrete method to approximate the largest Lyapunov exponent. By applying the method of some numerical experiments, we find that as bifurcation parameter varies, the system shows quite rich dynamical behaviors, including: (1) For crossing and switching-sliding bifurcations, there exists an alternating sequence of chaotic attractors, quasi-periodic toruses and strange nonchaotic attractors (SNAs), in which SNA exists between chaotic attractor and quasi-periodic torus via an evolution route of torus → SNA → chaos. (2) Grazing-sliding bifurcation triggers a direct transition from a pre-grazing three-torus quasi-periodic orbit to an SNA. Furthermore, we also observe torus-adding cascades and weak torus-adding cascades in quasi-periodic attractors. The unique properties of the observed SNAs are comprehensively validated by the largest Lyapunov exponent, rational approximation, phase sensitivity property, singular continuous spectrum analysis and separation of nearby trajectories.
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References
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Jiaojiao Zhou, Fei Luo, Jinkai Jiang, Yundong Li. STRANGE NONCHAOTIC ATTRACTORS AND COMPLICATED QUASI-PERIODIC CASCADES INDUCED BY BIFURCATIONS IN A DRY-FRICTION OSCILLATOR UNDER QUASI-PERIODIC EXCITATION[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 963-997. doi: 10.11948/20260165
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Figure 1.
A quasi-periodically excited nonlinear dry-friction oscillator
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Figure 2.
Four cases of sliding bifurcation: (a): Crossing-sliding. (b): Switching-sliding. (c): Grazing-sliding. (d): Adding-sliding. In all cases,
$ \Sigma $ $ F_1 $ $ F_2 $ $ \bar{\Sigma} $ $ F_1 $ $ F_2 $ -
Figure 3.
Bifurcation diagram in the
$ (f_{n}, x) $ $ \lambda_{\max} $ $ a_1=3.6, a_2=0.1, d_1=2.2, d_2=0.1, v=1, e_1=0.1, e_2=2, \omega_1=6, \omega_2=\sqrt{10} $ $ f_n\in[3, 4.2] $ -
Figure 4.
The phase diagram of system (2.4) with
$ a_1=3.6, a_2=0.1, d_1=2.2, d_2=0.1, v=1, e_1=0.1, e_2=2, \omega_1=6, \omega_2=\sqrt{10} $ $ f_{n}=3 $ $ (x,y)\in (-2.5, 2.5)\times(-5, 5) $ -
Figure 5.
The five-torus (
$ 5T $ $ f_{n}=3.081 $ $ (\theta, x) $ $ (x, y) $ -
Figure 6.
The seventeen-torus (
$ 17T $ $ f_{n}=3.2264 $ $ (\theta, x) $ $ (x, y) $ -
Figure 7.
The twelve-torus (
$ 12T $ $ f_{n}=3.27956 $ $ (\theta, x) $ $ (x, y) $ -
Figure 8.
SNA attractor at
$ f_{n}=3.333 $ $ (\theta, x) $ $ (x, y) $ -
Figure 9.
The seven-torus (
$ 7T $ $ f_{n}=3.421 $ $ (\theta, x) $ $ (x, y) $ -
Figure 10.
Chaotic attractor at
$ f_{n}=3.8507 $ $ (\theta, x) $ $ (x, y) $ -
Figure 11.
The nine-torus (
$ 9T $ $ f_{n}=3.634 $ $ (\theta, x) $ $ (x, y) $ -
Figure 12.
The two-torus (
$ 2T $ $ f_{n}=4.2 $ $ (\theta, x) $ $ (x, y) $ -
Figure 13.
Bifurcation diagram in the
$ (a_1, x) $ $ \lambda_{\max} $ $ a_2=1.23, d_1=2.5, d_2=0.4, f_{n}=1.2, v=0.5, e_1=0.1, e_2=1.9, \omega_1=2, \omega_2=\sqrt{10} $ $ a_{1}\in[3.5, 3.62] $ -
Figure 14.
The phase diagram of system (2.4) with
$ a_2=1.23, d_1=2.5, d_2=0.4, f_{n}=1.2, v=0.5, e_1=0.1, e_2=1.9, \omega_1=2, \omega_2=\sqrt{10} $ $ a_{1}=3.5 $ $ (x,y)\in (-1.5, 1.5)\times(-3, 3) $ -
Figure 15.
The three-torus (
$ 3T $ $ a_{1}=3.5 $ $ (\theta, x) $ $ (x, y) $ -
Figure 16.
The eight-torus (
$ 8T $ $ a_{1}=3.51 $ $ (\theta, x) $ $ (x, y) $ -
Figure 17.
The five-torus (
$ 5T $ $ a_{1}=3.54 $ $ (\theta, x) $ $ (x, y) $ -
Figure 18.
SNA attractor at
$ a_{1}=3.548 $ $ (\theta, x) $ $ (x, y) $ -
Figure 19.
The seven-torus (
$ 7T $ $ a_{1}=3.56 $ $ (\theta, x) $ $ (x, y) $ -
Figure 20.
The nine-torus (
$ 9T $ $ a_{1}=3.575 $ $ (\theta, x) $ $ (x, y) $ -
Figure 21.
The eleven-torus (
$ 11T $ $ a_{1}=3.585 $ $ (\theta, x) $ $ (x, y) $ -
Figure 22.
Chaotic attractor at
$ a_{1}=3.61 $ $ (\theta, x) $ $ (x, y) $ -
Figure 23.
The two-torus (
$ 2T $ $ a_{1}=3.62 $ $ (\theta, x) $ $ (x, y) $ -
Figure 24.
The three-torus (
$ 3T $ $ a_{1}=2.1, a_{2}=3.5, d_{1}=2.5, d_{2}=0.8, f_{n}=1.5, e_{1}=0.285323, e_{2}=2.5, \omega_{1}=3, \omega_{2}=\sqrt{10} $ $ v\approx 2.42228001 $ $ (x, y) $ $ P_{s} $ $ (\theta, x) $ $ P_{s} $ $ (x, y) $ -
Figure 25.
Bifurcation diagram in the
$ (e_1, x) $ $ \lambda_{\max} $ $ a_{1}=2.1, a_{2}=3.5, d_{1}=2.5, d_{2}=0.8, f_{n}=1.5, v\approx 2.42228001,e_{2}=2.5, \omega_{1}=3, \omega_{2}=\sqrt{10} $ $ e_{1}\in[0.2852, 0.2856] $ -
Figure 26.
SNA attractor at
$ e_{1}=0.28535 $ $ (\theta, x) $ $ (x, y) $ -
Figure 27.
The phase diagram of map
$ P_{s} $ $ (x, y) $ -
Figure 28.
Rational approximation of the SNA for
$ \mathcal{P}=\mathcal{P}_{1} $ $ f_{n}=3.333 $ -
Figure 29.
Rational approximation of the SNA for
$ \mathcal{P}=\mathcal{P}_{2} $ $ a_1=3.548 $ -
Figure 30.
Rational approximation of the SNA for
$ \mathcal{P}=\mathcal{P}_{3} $ $ e_1=0.28535 $ -
Figure 31.
Phase sensitivity of attractors of system (2.4) at
$ \mathcal{P}=\mathcal{P}_{1} $ $ f_{n}=3.081 $ $ f_{n}=3.333 $ -
Figure 32.
Phase sensitivity of attractors of system (2.4) at
$ \mathcal{P}=\mathcal{P}_{2} $ $ a_{1}=3.5 $ $ a_{1}=3.548 $ -
Figure 33.
Phase sensitivity of attractors of system (2.4) at
$ \mathcal{P}=\mathcal{P}_{3} $ $ e_{1}=0.285323 $ $ e_{1}=0.28535 $ -
Figure 34.
SNA for
$ \mathcal{P}=\mathcal{P}_{1} $ $ f_{n}=3.333 $ $ P:=P(\omega) $ $ \log_{10} |X|^2:=\log_{10} |X(\omega,N)|^2 $ $ \log_{10} N $ $ ({\rm Re}(X),{\rm Im}(X)) $ -
Figure 35.
SNA for
$ \mathcal{P}=\mathcal{P}_{2} $ $ a_{1}=3.548 $ $ P:=P(\omega) $ $ \log_{10} |X|^2:=\log_{10} |X(\omega,N)|^2 $ $ \log_{10} N $ $ ({\rm Re}(X),{\rm Im}(X)) $ -
Figure 36.
SNA for
$ \mathcal{P}=\mathcal{P}_{3} $ $ e_{1}=0.28535 $ $ P:=P(\omega) $ $ \log_{10} |X|^2:=\log_{10} |X(\omega,N)|^2 $ $ \log_{10} N $ $ ({\rm Re}(X),{\rm Im}(X)) $ -
Figure 37.
Separation of two nearby trajectories
$ (x_i(t), y_i(t)) $ $ (x_j(t), y_j(t)) $ $ \Delta:=\Delta(t)=\sqrt{(x_i-x_j)^2+(y_i-y_j)^2} $ $ \mathcal{P}=\mathcal{P}_1 $ -
Figure 38.
Separation of two nearby trajectories
$ (x_i(t), y_i(t)) $ $ (x_j(t), y_j(t)) $ $ \Delta:=\Delta(t)=\sqrt{(x_i-x_j)^2+(y_i-y_j)^2} $ $ \mathcal{P}=\mathcal{P}_2 $ -
Figure 39.
Separation of two nearby trajectories
$ (x_i(t), y_i(t)) $ $ (x_j(t), y_j(t)) $ $ \Delta:=\Delta(t)=\sqrt{(x_i-x_j)^2+(y_i-y_j)^2} $ $ \mathcal{P}=\mathcal{P}_3 $ $ e_{1}=0.28535 $