Nonlinear dynamics of a classical rotating pendulum system with multiple excitations

(a) The physical model of the classical rotating pendulum with multiple excitations, (b) the simple pendulum with rotation feature, (c) the mass-spring oscillator with bi-stable characteristics (SD oscillator).

When f₀ = 0 and ξ = 0: (a) the equilibrium bifurcation set (λ,q), (b) the nonlinear restoring force for (λ,q) ∈ I, (c) the nonlinear restoring force for $(\lambda,q)\in {\mathcal B} $, (d)–(f) three types of the nonlinear restoring forces for (λ,q) ∈ II, where ${\rm{II}}={{\rm{II}}}_{1}\displaystyle \cup {\mathcal L} \displaystyle \cup {{\rm{II}}}_{2}$.

(a) Phase portrait with the pendulum-type homoclinic orbits for (λ,q)=(2,3) ∈ I, (b) phase portrait with the coupling of the pendulum-type and Duffing-type homoclinic orbits for (λ,q) = (1.2,3) ∈ II.

(a) Cylindrical phase portrait with the pendulum-type homoclinic orbits for (λ,q) = (2,3) ∈ I, (b) cylindrical phase portrait with the coupling of the pendulum-type and Duffing-type homoclinic orbits for (λ,q) = (1.2,3) ∈ II.

Dynamic behavior of system (3) for (λ,q) ∈ I: (a) the equilibrium bifurcation diagram for f₀ versus x, (b)–(e) the corresponding phase portraits and the nonlinear restoring forces marked by dashed lines.

Dynamic behavior of system (3) for (λ,q) ∈ II₁: (a) the equilibrium bifurcation diagram for f₀ versus x, (b)–(i) the corresponding phase portraits and the nonlinear restoring forces marked by dashed lines.

Dynamic behavior of system (3) for $(\lambda,q)\in {\mathcal L} $: (a) the equilibrium bifurcation diagram for f₀ versus x, (b)–(f) the corresponding phase portraits and the nonlinear restoring forces marked by dashed lines.

Dynamic behavior of system (3) for (λ,q) ∈ II₂: (a) the equilibrium bifurcation diagram for f₀ versus x, (b)–(i) the corresponding phase portraits and the nonlinear restoring forces marked by dashed lines.

Dynamic behaviors of the pendulum system with viscous damping and constant excitation for (λ,q) ∈ I: (a) none of limit cycle surrounding the equilibrium for 2 π f₀ – a₁ξ < 0, (b) the bifurcation of the upper homoclinic orbit for 2πf₀ – a₁ξ = 0, (c) a rotating limit cycle for 2πf₀ – a₁ξ > 0.

Dynamic behaviors of the pendulum system with viscous damping and constant excitation for (λ,q) ∈ II: (a) none of limit cycle surrounding the equilibrium for 2πf₀ – a₁ξ < 0, (b) the bifurcation of upper homoclinic orbit for 2πf₀ – a₁ξ = 0, (c) a rotating limit cycle for 2πf₀ – a₁ξ > 0.

Cylindrical phase portraits of the pendulum system with viscous damping and constant excitation for q = 0, ξ = 0.2 and a₁ = 8: (a) no limit cycle for f₀ = 0.0001, (b) the bifurcation of the upper homoclinic orbit for f₀ ≈ 0.2527, (c) a rotating limit cycle for f₀ = 0.3.

The chaotic boundaries detected by the Melnikov analysis for the Duffing-type and pendulum-type homoclinic orbits at q = 3 and λ = 1.5: (a) σ = 0.5 (${R}_{{\rm{\hom }}1}^{\pm }(1.8)=0.7556$, ${R}_{{\rm{\hom }}2}^{+}(1.8)=4.1508$, ${R}_{{\rm{\hom }}2}^{-}(1.8)=6.5314$), (b) σ = 1 (${R}_{{\rm{\hom }}1}^{\pm }(0.5)=0.8541$, ${R}_{{\rm{\hom }}2}^{+}(0.5)=1.3500$, ${R}_{{\rm{\hom }}2}^{-}(0.5)=3.5212$), (c) σ = 2.

When q = 3, λ = 1.5, ξ = 0.06, f₀ = 0.03 and ω = 1.8: (a) bifurcation diagram for f₁ versus x with the threshold values ${f}_{1}^{{\rm{\hom }}1}=0.0427$ (solid line), ${f}_{1}^{{{\rm{\hom }}2}^{+}}=0.0675$ (dashed line) and ${f}_{1}^{{{\rm{\hom }}2}^{-}}=0.1761$ (dotted line), (b) the corresponding largest Lyapunov exponents under the initial condition (1,0); (c) the phase portraits of a pair of period-1 solutions for f₁ = 0.1 and their time histories (d); (e) the phase portrait of period-3 solution for f₁ = 0.65 and the corresponding time history (f); (g) the phase portrait of period-13 solution for f₁ = 1.5 and the corresponding time history (h); (i)–(k) three chaotic attractors for f₁ = 0.92, 1.7, 1.9, respectively.

When q = 3, λ = 1.5, ξ = 0.05, f₀ = 0.05 and ω = 0.5: (a) bifurcation diagram for f₁ versus x with the threshold values ${f}_{1}^{{\rm{\hom }}1}=0.0453$ (solid line), ${f}_{1}^{{{\rm{\hom }}2}^{+}}=0.2490$ (dashed line) and ${f}_{1}^{{{\rm{\hom }}2}^{-}}=0.3919$ (dotted line); (b) the local enlargements for f₁ ∈ [1.5,1.8]; (c)–(e) three types of chaotic motions for f₁ = 0.8, 1.55, 1.8 and the corresponding time histories (f)–(h); (i)–(k) three periodic motions for f₁ = 1.575, 1.616, 1.78, respectively.

(a) A two-dimensional parameter space plot for q = 3, λ = 1.5, f₀ = 0.1, ξ = 0.2 in the range of f₁ ∈ [0,2] and ω ∈ [1,4] and theoretical boundary derived by the Melnikov analysis, (b) the bifurcation diagram of f₁ with the threshold f₁ = 0.13 and (c) the corresponding Lyapunov exponents for ξ = 1.515 (the gray region corresponds to the chaotic region, the white region represents non-chaotic region and the black curve ${\bar{R}}_{{\rm{\hom }}}(\omega )=1$ is the theoretical boundary of chaos).

Cylindrical phase portraits of oscillations starting initial condition (x,y) = (1,0): (a) SD-type chaotic motion, (b) Duffing-type chaotic motion; chaotic cylindrical phase portraits of the coupling of oscillations and rotations (c) and (d).