June, : WSPC - Proceedings Trim Size: in x in SPT-broer Survey of strong normal-internal k : l resonances in quasi-periodically driven oscillators for l =,,. H.W. BROER and R. VAN DIJK Institute for mathematics and computing sciences, University of Groningen, The Netherlands E-mail: broer@math.rug.nl http://www.math.rug.nl/ broer R. VITOLO Dipartimento di Matematica e Informatica, Università di Camerino, Italy E-mail: renato.vitolo@unicam.it Recently, semi-global results have been reported by Wagener for the k : l resonance where l =,. In this work we add the l = strong resonance case and give an overview for l =,,. For an introduction to the topic, as well as results on the non-resonant and weakly-resonant cases, see Refs.,,. Keywords: Quasi-periodically driven oscillators; Duffing - Van der Pol oscillator; Stoker s response problem ; strong resonance.. Introduction to the problem We consider the forced Duffing - Van der Pol oscillator, given by ẋ j = ω j, j =,...,m, ẏ = y, ẏ = (a + cy)y by dy + εf(x,...,x m,y,y,a,b,c,d,ε), which is a vector field defined on T m R = {(x,...,x m ),(y,y )}. Here f is a smooth function which is π-periodic in its first m-arguments. We take σ = (a,b) as parameter, ε as perturbation parameter and (c,d) as coefficients. The internal frequency vector ω = (ω,...,ω m ) is to be nonresonant, or quasi-periodic. The search is for quasi-periodic response solutions, i.e. invariant m-tori of (), with frequency vector ω, that can be represented as graphs y = y(x) over T = T m {}. For large values of a we can use a contraction argument. However, if a this approach fails due to small divisors. Braaksma and Broer use KAM theory to find ()
June, : WSPC - Proceedings Trim Size: in x in SPT-broer invariant tori for small a. This requires Diophantine non-resonance conditions on the frequency vector ω and the normal frequencies. The case a involves a Hopf bifurcation of the corresponding free oscillator (ε = ). We generalise the above setup by considering the family of vector fields X σ,ǫ (x,y) = ω x + (A(σ) + B(y,c,d) + εf(x,y,σ,c,d,ε)) y, () where again (x,y) T m R, and σ, ε, (c,d) are taken as before. The normal linear part of the unperturbed invariant torus T is given by N σ (x,y) = ω x + A(σ)y y. Let λ = µ(σ) ± iα(σ) denote the eigenvalues of A(σ) where α(σ) is called the normal frequency. We assume that the map σ (µ(σ),α(σ)) is a submersion, which allows to take (µ,α) as parameters instead of σ. A normalinternal k : l resonance of T occurs if k,ω + lα(σ) = () for k Z m \{} and l Z\{}. The smallest integer l for which () holds, is called the order of the resonance. Resonances up to order are called strong. The non-resonant and weakly resonant cases have been investigated in Ref.. Here we consider resonances such that () holds for l =, and. Therefore we assume that at σ = σ the normal part N σ of X versally unfolds a Hopf bifurcation at y = and that torus T is at strong normalinternal resonance. To study (), an averaging procedure is followed to push time dependence to higher order terms. Next a Van der Pol transformation is applied to bring the normal frequencies close to zero. Using a suitable complex variable z, the vector field X transforms to X λ,θ,δ (x,z) = ω x + δ R(λz + e iθ z z + z l ) z + δ Z(x,z). For details see Wagener. Here θ is a generic constant, λ = µ + iα a parameter and δ = ε l is a rescaled perturbation parameter. All nonintegrable terms are truncated, yielding the so-called principal part : Z = R(λz + e iθ z z + z l ) z. () In the next section we describe the bifurcation diagrams of Z for l =, and summarising results from Refs.,,, for a periodic analogue see Gambaudo. Bifurcations of equilibria of Z correspond to bifurcations of quasi-periodic m-tori of the integrable vector field ω x +Z. This interpretation remains valid if we add integrable perturbations. However, adding non-integrable (δ ) terms breaks the torus symmetry and quasi-periodic bifurcation theory has to be invoked. For persistence results see Ref..
June, : WSPC - Proceedings Trim Size: in x in SPT-broer Table. Notation used for the codimension one bifurcation curves (leftmost two columns) and codimension two points (rightmost two columns) for all figures in Sec.. Notation Name Notation Name HE Heteroclinic BT Bogdanov-Takens L Homoclinic bifurcation DL Degenerate homoclinic H Hopf bifurcation H Degenerate Hopf PF Pitchfork PF Degenerate Pitchfork SN Saddle Node (Fold) SN Cusp SNLC Saddle node of limit cycles L Homoclinic at saddle node SDZ Symmetric double zero. Results In the previous section a normal form truncation () was introduced for the general family of vector fields () at normal-internal resonance. Here we present the bifurcation diagram of () in the (µ,α)-plane (we recall that λ = µ + iα) for l =,,. However, in certain cases the coefficient θ is sometimes activated as an extra parameter. For the k : resonance, this gives rise to a codimension three bifurcation point of nilpotent elliptic type, which can be seen as an organising centre. Auxiliary variables ζ, ψ R and ρ C are introduced by setting ζ = z, z = ζe iψ, ρ = ρ + iρ = e iθ λ, see Refs.,. The bifurcation diagrams are reported at the end of the paper and the notation used there is summarized in Table. Local bifurcations are obtained analytically, while global bifurcations are found and continuated by standard numerical tools such as AUTO. Proofs of the cases l =, are given in Ref., whereas the case l = is treated in Ref.... The k : resonance For l = the family () rewrites to ż = λz + e iθ z z +. () System () is symmetric with respect to the group generated by (t,z,λ,θ ) (t, z, λ, θ ) and (t,z,λ,θ ) ( t, z, λ,θ + π). Hence, we can restrict θ to θ π. Two types of local codimension one bifurcations occur in system (): () a saddle node bifurcation SN, given by ρ ρ + ρ ρ + ρ + ρ + ρ ρ + = ;
June, : WSPC - Proceedings Trim Size: in x in SPT-broer () a Hopf bifurcation (with two branches H a and H b ), given by µ µ α cos θ sin θ + µα cos θ + 8cos θ =, (α µtan θ ) (µsec θ ) >. Three types of local codimension two bifurcations occur for (): () two cusp points SN + and SN, depending on θ, given by SN ± : µ = cos θ ± sinθ, α = sin θ cos θ ; () two Bogdanov-Takens points BT + and BT, given by BT ± : µ = cos θ, α = sin θ + ; ( ± sin θ ) ( ± sin θ ) () a degenerate Hopf-bifurcation point H given by H : µ = cos θ, α =, π < θ < π. In the restricted parameter space (µ,α,θ ) there exists a unique codimension three singularity of nilpotent elliptic type. In Fig. the bifurcation diagram is presented along with the codimension two bifurcation points... The k : resonance For l = the family () rewrites to ż = λz + e iθ z z + z. () System () is symmetric with respect to the group generated by (z,λ,θ) ( z, λ, θ) and (z,λ,θ) ( z,λ,θ). Hence, we can restrict θ to θ π. We have depicted the bifurcation diagram in Fig. Three types of local codimension one bifurcations occur in system (): () a curve of pitchfork bifurcations PF, given by ρ = ; () two curves of saddle-node bifurcations SN, given by ρ =, ρ < ; () three Hopf bifurcation curves H, two are given by R(e iθ ρ) =, ρ >, and the third by (ρ + ρ tan θ ) + ρ =, ρ + tan θ ρ ρ <, ρ tan θ ρ >. Moreover, the following codimension two bifurcations take place:
June, : WSPC - Proceedings Trim Size: in x in SPT-broer () Two degenerate pitchfork bifurcation points PF at ρ = ±i. () Two symmetric double-zero bifurcation points SDZ at λ = ±i. () A Bogdanov-Takens bifurcation point BT at ρ = tan θ ( + icot θ )... The k : resonance For l = the family () rewrites to ż = (λz + e iθ z z + z ). () System () is symmetric with respect to the group generated by (t,z,λ,θ) (t, z, λ, θ) and (t,z,λ,θ) ( t, z, λ, θ + π). Hence, we can restrict θ to θ π. Two types of local codimension one bifurcations, depending on θ, occur in system (): () a saddle-node bifurcation curve SN, given by ρ = ρ + ; () a Hopf curve H, given by R(ρe iθ + ζe iθ ) =, ζ + ρ >. If θ π there is no Hopf bifurcation of a non-central equilibrium, while the central equilibrium always undergoes Hopf bifurcation for µ =. Furthermore, there are two Bogdanov-Takens points BT given by BT ± : ρ = (tan θ ± tan θ ). We consider two scenarios: the first corresponds to θ π. Here, the non-central equilibria will not have Hopf bifurcations and, as a result, no Bogdanov-Takens bifurcations occur, see Fig.. The second scenario, depicted in Fig., corresponds to θ > π. Here there are Hopf as well as Bogdanov-Takens bifurcations and the latter involve the occurrence of additional phenomena of global nature. References. B.L.J. Braaksma and H.W. Broer, Ann. Inst. Henri Poincaré, Analyse non linéare () (8), -8.. H.W. Broer, V. Naudot, R. Roussarie, K. Saleh and F.O.O. Wagener, IJAMAS (), to appear.. R. van Dijk,master thesis, Universiteit Groningen, ().. E.J. Doedel et al., AUTO,http//indy.cs.concordia.ca.. J.M. Gambaudo, J. Diff. Eqn. (8), -.. F.O.O. Wagener, Global Analysis of Dynamical Systems, Festschrift dedicated to Floris Takens for his th birthday, Bristol and Philadelphia IOP (), -.
June, : WSPC - Proceedings Trim Size: in x in SPT-broer H b α L b L c H DL 8 - SNLC SN SN + - BT L c L b BT + SN c SN b L b L a - - SN b L a H a -. - -. (a) µ α H b SN L b BT L b H b SN a SN c - SN b H a - BT + L a SN + - L a H a - -. - -. µ (b) Fig.. Two-dimensional bifurcation diagram of the principal part () for the k : resonance () for two values of θ ; (a) : θ = π/ and (b) : θ = π/. Phase portraits for the noted regions are given in Fig., for the notation we refer to Table.
June, : WSPC - Proceedings Trim Size: in x in SPT-broer y... -. -. - -. - - - - - -. -. - -... - - - - x Region Region Region Region -. - -.... -. - -... -. - -. -. - -... - - - - -. Region Region Region -. - -.... -. - -. - - -. - -.. - - Region Region Region - - - -.. -. - - -. - -... -. -. -. -. -.......8 Region 8.8. -. -. -. -.8 - -. -. -..8.....8 -. -.....8.....8 Region Region Region Fig.. Phase portraits of the principal part for the k : resonance (). The labels Region... correspond to the regions in the bifurcation diagram in Fig..
.. -. - -... -. - -... -. - -... -. - -... -. - -... -. - -... -. - -... -. - -. -. -... -. - -... -.. -. - -... -. - -... -. - -... -. - -. June, : WSPC - Proceedings Trim Size: in x in SPT-broer 8 y - - -. - -... - - - - -. - -... - -. - -... - -. - -... x (b) Region (c) Region (d) Region (e) Region (f) Region - -. - -... - PF H SDZ V(a) H PF 8 - -. - -... - (g) Region 8 PF L L - SDZ BT α SNLC SN - -. - -... - - HE HE SN (h) Region - H - - -. - -... - (i) Region -. - -.... (a) µ - -. - -... - (j) Region - -. - -... - - -. - -... - - -. - -... - - -. - -... - - -. - -... - (k) Region (l) Region (m) Region (n) Region (o) Region Fig.. (a) Two-dimensional bifurcation diagram of the principal part for the k : resonance (); (b)-(o) Phase portraits for the indicated regions.
June, : WSPC - Proceedings Trim Size: in x in SPT-broer -.. -. - - - -. -.. - -.. (b) Region (c) Region (d) Region α H. HE -. - - -.. (e) Region SN (a) µ - - -.. (f) Region. -. - - - -.. - -.. (g) Region (h) Region Fig.. (a) Two-dimensional bifurcation diagram of the principal part for the k : resonance () for θ < π ; (b)-(h) Phase portraits for the indicated regions.
June, : WSPC - Proceedings Trim Size: in x in SPT-broer H SNLC L BT HE SN (c) Blowup of indicated region α H HE BT HE 8 (b) Blowup of indicated region µ - - - (a) Global bifurcation diagram Fig.. (a) Two-dimensional bifurcation diagram of the principal part for the k : resonance () for θ > π ; (b)-(c) Blowups of the indicated rectangles. Phase portraits for the regions labelled,..., are given in Fig..
June, : WSPC - Proceedings Trim Size: in x in SPT-broer.. -. y - y - -. -. - -... - - - (a) Region (b) Region - - - - - - - - (c) Region (d) Region - - - - - -.. -. - -. - - -. - -... (e) Region (f) Region - - - Magnification - - - (g) Region - - - - - - - - - - - - (h) Region 8 (i) Region Fig.. (a)-(i) Phase portraits of the principal part for the k : resonance (). The labels Region... correspond to the regions in the bifurcation diagram in Fig.. Note that two limit cycles coexist in region, see the magnification.