Dynamics of weakly coupled parametrically forced oscillators

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Dynamics of weakly coupled parametrically forced oscillators

P. Salgado Sánchez, J. Porter, I. Tinao, and A. Laverón-Simavilla

Phys. Rev. E 94, 022216 – Published 29 August 2016

Abstract

The dynamics of two weakly coupled parametric oscillators are studied in the neighborhood of the primary subharmonic instability. The nature of both primary and secondary instabilities depends in a critical way on the permutation symmetries, if any, that remain after coupling is considered, and this depends on the relative phases of the parametric forcing terms. Detailed bifurcation sets, revealing a complex series of transitions organized in part by Bogdanov-Takens points, are calculated for representative sets of parameters. In the particular case of out-of-phase forcing the predictions of the coupled oscillator model are compared with direct numerical simulations and with recent experiments on modulated cross waves. Both the initial Hopf bifurcation and the subsequent saddle-node heteroclinic bifurcation are confirmed.

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Received 27 November 2015
Revised 16 June 2016

DOI:https://doi.org/10.1103/PhysRevE.94.022216

Physics Subject Headings (PhySH)

Coupled oscillators

Chaos & nonlinear dynamics

Nonlinear Dynamics

Authors & Affiliations

P. Salgado Sánchez, J. Porter^*, I. Tinao, and A. Laverón-Simavilla

Escuela Técnica Superior de Ingeniería Aeronáutica y del Espacio, Universidad Politécnica de Madrid, Plaza de Cardenal Cisneros 3, 28040 Madrid, Spain

^*jeff.porter@upm.es

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Vol. 94, Iss. 2 — August 2016

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Images

Figure 1
Pitchfork (P, thick curve) and saddle-node (SN, thin curve) bifurcation sets for a single parametrically forced oscillator in the case of conservative nonlinearity $a = 0$ .
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Figure 2
Contours of the normal form (first Lyapunov) coefficient $l_{1}$ (see [29]) in the $(ν, a)$ plane. This coefficient is always negative, showing that the Hopf bifurcation, which occurs for $ν < (a^{2} - 1) / (2 a)$ , is supercritical.
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Figure 3
Bifurcation sets in the case of $a = - 0.3$ : pitchfork (P, thick solid curve), saddle-node (SN, thin solid line), Hopf (H, lower dashed curve), saddle-node of periodic orbits (SN Per, upper dashed curve), homoclinic gluing (Hom Gluing, thin solid curve above pitchfork), homoclinic (Hom, lower thin solid curve below pitchfork), heteroclinic (Het, upper thin solid curve below pitchfork), saddle-node heteroclinic (SN Het), and Bogdanov-Takens point (BT, solid dot).
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Figure 4
Bifurcation diagrams for $a = - 0.3$ and (a) $ν = - 0.5$ and (b) $ν = 1$ , showing the (time-averaged) norm of the nontrivial steady states ( $A_{s}$ ), periodic modulated solutions ( $P_{m}$ ), and periodic rotating solution ( $P_{r}$ ). Solid (dashed) curves denote stable (unstable) solutions. With increasing $ν$ , the primary bifurcation transitions from supercritical to subcritical and the gluing bifurcation splits into distinct homoclinic and heteroclinic bifurcations. A selection of periodic orbits at particular points labeled $a - h$ in these diagrams are shown in (c) and (d), respectively. $a, e$ , rotating orbit at $f = 0$ ; $b$ , limit point at $f = 2.747$ ; $c$ , symmetric homoclinic orbit at $f = 2.678$ ; $d$ , small modulated orbits at $f = 2.3$ ; $f$ , heteroclinic orbit at $f = 1.275$ ; $g$ , homoclinic orbits at $f = 1.071$ ; $h$ , small modulated orbits at $f = 0.9$ .
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Figure 5
Bifurcation sets for $ϕ = 0, a = 0, μ = 0.05 + 0.2 i$ : primary pitchfork in $Σ_{\pm}$ subspaces ( $P_{\pm}$ , thick solid curves), saddle-node in $Σ_{\pm}$ ( ${SN}_{\pm}$ , thin solid lines), symmetry-breaking (pitchfork) from $Σ_{\pm}$ to mixed mode state ( ${SB}_{\pm}$ , dash-dotted curves), saddle-node on mixed mode state ( ${SN}_{M}$ , thin solid curves). In the larger regions the types of solutions that exist there are indicated in parentheses.
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Figure 6
Bifurcation diagram for $ν = - 1.5, ϕ = 0, a = 0, μ = 0.05 + 0.2 i$ showing $| A + B |$ (projection onto $Σ_{+}$ ) and $| A - B |$ (projection onto $Σ_{-}$ ). Solid (dashed) curves denote stable (unstable) solutions. Solutions in the $Σ_{\pm}$ subspaces are labeled with $S_{\pm}$ and asymmetric mixed modes with $M$ .
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Figure 7
Phase space portraits corresponding to the case of Fig. 6 with $f = 2.5$ . There are two stable steady states $S_{+}$ (solid dots) in $Σ_{+}$ , two stable steady states $S_{-}$ (asterisks) in $Σ_{-}$ and four unstable asymmetric steady states $M$ (open circles); one representative solution on each group orbit is labeled. Panels (a) and (b) show projections onto $A = A_{r} + i A_{i}$ and $B = B_{r} + i B_{i}$ , respectively, while the projections in (c) and (d) illustrate the interchange symmetry (21). The trivial state is not shown.
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Figure 8
Bifurcation diagram for $ν = 1.5, ϕ = 0, a = 0, μ = 0.05 + 0.2 i$ showing $| A + B |$ and $| A - B |$ . Solid (dashed) curves denote stable (unstable) solutions. Solutions in the $Σ_{\pm}$ subspaces are labeled with $S_{\pm}$ and asymmetric mixed modes with $M$ .
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Figure 9
Bifurcation sets for $ϕ = π, a = 0, μ = 0.2 i$ : Hopf (H, dashed curves) and saddle-node of periodic orbits (SN Per, dashed curve), saddle-node (SN), heteroclinic (Het), saddle-node heteroclinic (SN Het), homoclinic (Hom), saddle-node homoclinic (SN Hom), Bogdanov-Takens points (BT). In the larger regions, the types of solutions that exist there are indicated in parentheses. In (a) the regions of one and two $C_{4}$ -symmetric periodic orbits ( $P_{C_{4}}$ ) are further marked with “1” and “2”, respectively. In (b) a close-up of the complex region between two BT points near the cusp of SN bifurcations is shown, as is, in the inset, a further magnification showing the cusp region.
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Figure 10
Bifurcation diagrams for $ϕ = π, a = 0, μ = 0.2 i$ showing the norm $| A | = | (A, B) |$ of steady states ( $S$ ), $C_{4}$ -symmetric periodic orbits ( $P_{C_{4}}$ ), and asymmetric periodic orbits ( $P_{a}$ ): (a) $ν = - 0.5$ , (b) $ν = 0.5$ , (c) $ν = 1.4$ , (d) $ν = 2$ . For $ν = 1.4$ , there is a secondary supercritical Hopf bifurcation on the isola of steady states that generates stable asymmetric periodic orbits, shown in the inset. Stable (unstable) solutions are shown with solid (dashed) lines.
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Figure 11
Phase space portraits corresponding to the case of Fig. 10 with $f = 1.5$ . There are four stable steady states $S_{s}$ (solid dots) and four unstable steady states $S_{u}$ (open circles) related by the symmetry (22); one representative solution on each group orbit is labeled. The $C_{4}$ -symmetric periodic orbit, $P_{C_{4}}$ , is also shown. Panels (a) and (b) show projections onto $A = A_{r} + i A_{i}$ and $B = B_{r} + i B_{i}$ , respectively, while the projections in (c) and (d) illustrate the $C_{4}$ symmetry. The trivial state is not shown.
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Figure 12
Bifurcation sets for $ϕ = π, a = 0, μ = 0.2 + 0.04 i$ : Hopf (H, dashed curve) and saddle-node of periodic orbits (SN Per, dashed curve), saddle-node (SN), heteroclinic (Het), saddle-node heteroclinic (SN Het), and double-zero point. In the larger regions, the types of solutions that exist there are indicated in parentheses.
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Figure 13
(a) Close-up of the double-zero bifurcation point and (b) cusp region of Fig. 12 showing bifurcation sets: Hopf (H, dashed curve), saddle-node of periodic orbits (SN Per, dashed curve), saddle-node (SN), saddle-node heteroclinic (SN Het), and heteroclinic (Het). The heteroclinic bifurcation set near the cusp does not collide there, but with the upper branch, although this cannot be distinguished on the scale of the plot. In the larger regions, the types of solutions that exist there are indicated in parentheses.
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Figure 14
Contours showing the transition from supercritical to subcritical Hopf bifurcation for coupling values with $μ_{r} = 0, 0.1, 0.2, 0.3, 0.4, 0.5$ . The transition is independent of $μ_{i}$ .
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Figure 15
Contour map showing the location of the $Z_{2}$ -symmetric Bogdanov-Takens point with $ν > 0$ in the conservative and purely dissipative cases, as given by Eqs. (41) and (47) . Curves for $ν < 0$ are obtained by reflection. Curves are shown for constant $μ$ , in steps of $0.2 i$ ( $μ \in Im$ ) and $0.2$ ( $μ \in Re$ ), and for constant $ϕ$ (radial lines), in steps of $π / 6$ . The central curve intersecting $(ν, f) = (0, 1)$ corresponds to the uncoupled case $μ = 0$ .
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Figure 16
Primary Hopf (H) and pitchfork (P) bifurcation sets for (a) $μ = 0.2 i$ , (b) $μ = 0.04 + 0.2 i$ , (c) $μ = 0.2$ , and (d) $μ = 0.2 + 0.04 i$ . $Z_{2}$ -symmetric Bogdanov-Takens points (not marked) lie on the intersection.
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Figure 17
(a) Bifurcation sets for $ϕ = π / 2, a = 0, μ = 0.2 i$ : Hopf (H, dashed curves), saddle-node of periodic orbits (SN Per, dashed curve), saddle-node (SN), heteroclinic (Het), saddle-node heteroclinic (SN Het), homoclinic (Hom), Bogdanov-Takens points (BT). Some bifurcation sets (SN Per, homoclinic, and gluing bifurcations, for example) are hard to see on the scale of the plot and are not labeled. Panel (b) shows a close-up of the complex transitions for positive detuning. In the larger regions, the types of solutions that exist there are indicated in parentheses.
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Figure 18
Close-up of the (a) left and (b) right transition, between pitchfork and Hopf bifurcation of Fig. 17 showing bifurcation sets: pitchfork (P), Hopf (H, dashed curves), saddle-node (SN), saddle-node heteroclinic (SN Het), heteroclinic (Het), homoclinic gluing (Hom Gluing), and Bogdanov-Takens points (BT). Some bifurcation sets [homoclinic and heteroclinic, near the lower, asymmetric BT point in panel (b)] are not visible on the scale of the plot and are not labeled. In the larger regions, the types of solutions that exist there are indicated in parentheses.
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Figure 19
Bifurcation diagrams for $ϕ = π / 2, a = 0, μ = 0.2 i$ showing the norm $| A | = | (A, B) |$ of steady states ( $S$ ), $Z_{2}$ -symmetric periodic orbits ( $P_{Z_{2}}$ ), and asymmetric periodic orbits ( $P_{a}$ ): (a) $ν = - 1.5$ , (b) $ν = 0$ , (c) $ν = 1$ , (d) $ν = 2$ . For $ν = 1$ there is a secondary supercritical Hopf bifurcation on the isola of steady states that generates stable asymmetric periodic orbits, shown in the inset. Stable (unstable) solutions are shown with solid (dashed) lines.
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Figure 20
Phase space portraits corresponding to the case of Fig. 19 with $f = 1.374$ at the saddle-node heteroclinic bifurcation. There are two stable steady states (upper branch denoted by solid dots, and the lower branch by asterisks) and two (larger amplitude) unstable steady states (upper branch denoted by open circles, and the lower branch by open squares). The $Z_{2}$ -symmetric periodic orbit, which encloses the origin (trivial state not shown), is colliding with two smaller-amplitude unstable steady states (open triangles). Projections onto (a) $(A_{r}, A_{i})$ , (b) $(B_{r}, B_{i})$ , (c) $(A_{r}, B_{i})$ , and (d) $(A_{i}, B_{r})$ .
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Figure 21
Modulated (quasiperiodic) solution found experimentally in a 20-cm square container with 5 cSt oil driven at 13.95 Hz. The crosswise mode number is $N = 11$ and the modulation period is about 45.6 s. The snapshots (a)–(d) are separated by approximately $1 / 8$ of this period, or 5.7 s. Reproduced from [20].
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Figure 22
Modulation (Hopf) frequency (in Hz) of the solution shown in Fig. 21 over an interval of forcing frequencies $ω / π$ (in Hz) around the resonance frequency of $13.52 Hz$ . The experimentally measured Hopf frequency $Ω$ is compared with that predicted by the model obtained in [20] with the derived boundary conditions (solid curve) and with Dirichlet boundary conditions (dashed curve); remaining parameters for these calculations are viscosity, $5 cSt$ ; surface tension, $19.7 dyne / cm$ ; density, $0.913 g / {cm}^{3}$ ; container depth, $5 cm$ ; and width, $9 cm$ . Reproduced from [20].
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Figure 23
Primary Hopf bifurcation for $ϕ = π$ and $μ (ν)$ given by Eq. (48) with $μ_{0} = 0.06, σ = 0.4, θ_{0} = 7 π / 18, Ω_{ν} = 0.76 π$ . (a) Critical forcing $f (ν)$ (solid curve) compared to the pitchfork bifurcation of a single oscillator (dashed curve). (b) Hopf frequency $Ω (ν)$ ; this should be compared with Fig. 22.
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Figure 24
Selected bifurcation sets for $ϕ = π$ and $μ (ν)$ given by Eq. (48) with $μ_{0} = 0.06, σ = 0.4, θ_{0} = 7 π / 18, Ω_{ν} = 0.76 π$ : Hopf (H, dashed curve), saddle-node heteroclinic (SN Het), saddle-node of periodic orbits (SN Per, solid curves for $ν ≳ 2.15$ ). Dotted vertical lines mark $ν$ values where the Hopf frequency vanishes. Additional bifurcation sets near these double-zero points, which may include saddle-node, pitchfork, asymmetric Hopf, homoclinic, and heteroclinic bifurcations [38], are not shown in detail.
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Figure 25
Harmonic (black curve) and subharmonic (gray curve) components as forcing (in units of the gravitational acceleration g) is increased for frequencies of (a) 30 Hz, (b) 40 Hz, (c) 50 Hz, and (d) 60 Hz. A transient has been removed from each simulation. Note the modulation of the subharmonic waves in all but (d) the 60-Hz case.
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Figure 26
Snapshot of the change in fluid surface height (in cm) for the 50-Hz simulation at $0.9088 g$ . Above onset the subharmonic waves are dominant throughout the domain.
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Figure 27
(a) Close-up of the Hopf bifurcation to modulated subharmonic waves in the case of 50 Hz in Fig. 25. Peak modulation amplitude (maximum over one modulation period) of the filtered subharmonic signal is shown for increasing forcing (open circles, up arrow) and decreasing forcing (open squares, down arrow). (b) Linear fit of the growth rate (in $s^{- 1}$ ) of perturbations at fixed values of the forcing predicts an onset of $f = 0.804$ g.
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Figure 28
Time series showing (signed) modulation amplitude for points equidistant from the left (solid curves) and right (dashed curves) boundaries for the case of 50 Hz forcing. The transition from sinusoidal modulation near the Hopf bifurcation to slower irregular modulation near the global bifurcation can be seen with increasing forcing values (in units of g): (a) 1, (b) 1.2, (c) 1.25, (d) 1.27. A 30-s transient has been removed from each simulation.
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Figure 29
The increase in modulation period ( $T_{m}$ ) as the global bifurcation is approached. The data from the simulation (open circles) fit well with the scaling expected for a saddle-node heteroclinic bifurcation: The solid line shows the fitting function, $0.059 + 0.835 {(1.296 - f)}^{- 1 / 2}$ .
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Figure 30
Harmonic (black curve) and subharmonic (gray curve) components as forcing (in units of g) is increased with 60.4-Hz vibration.
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Figure 31
(a) Estimated location of onset (triangles) and saddle-node heteroclinic bifurcation (circles); the parameters shown are applied forcing $f$ (in units of $g$ ) and applied frequency $ω / π$ (in Hz). (b) Estimated modulation frequency $Ω$ (in Hz) near onset.
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