Resonance and Fractal Geometry
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Abstract
The phenomenon of resonance will be dealt with from the viewpoint of dynamical systems depending on parameters and their bifurcations. Resonance phenomena are associated to open subsets in the parameter space, while their complement corresponds to quasiperiodicity and chaos. The latter phenomena occur for parameter values in fractal sets of positive measure. We describe a universal phenomenon that plays an important role in modelling. This paper gives a summary of the background theory, veined by examples.
Keywords
Resonance Resonance tongue Subharmonic bifurcation Covering space Cantor set Fractal geometry Devil’s staircase Lyapunov diagram1 What Is Resonance?
A heuristic definition of resonance considers a dynamical system, usually depending on parameters, with several oscillatory subsystems having a rational ratio of frequencies and a resulting combined and compatible motion that may be amplified as well. Often the latter motion is also periodic, but it can be more complicated as will be shown below. We shall take a rather eclectic point of view, discussing several examples first. Later we shall turn to a number of universal cases, these are contextfree models that occur generically in any system of sufficiently highdimensional state and parameter space.
Among the examples are the famous problem of Huygens’s synchronizing clocks and that of the Botafumeiro in the Cathedral of Santiago de Compostela, but also we briefly touch on tidal resonances in the planetary system. As universal models we shall deal with the Hopf–Neĭmark–Sacker bifurcation and the Hopf saddlenode bifurcation for mappings. The latter two examples form ‘next cases’ in the development of generic bifurcation theory. The term universal refers to the context independence of their occurrence: in any with certain generic specifications these bifurcations occur in a persistent way. We witness an increase in complexity in the sense that in the parameter space the resonant phenomena correspond to an open & dense subset union of tongues, while in the complement of this a nowhere dense set of positive measure exists, corresponding to multi or quasiperiodic dynamics. This nowhere dense set has a fractal geometry in a sense that will be explained later. From the above it follows that this global array of resonance tongues and fractal geometry has a universal character. As we shall see, both locally and globally Singularity Theory can give organizing principles. It should be noted at once that next to periodic and quasiperiodic dynamics also forms of chaotic dynamics will show up.
Remark
In many cases the resonant bifurcations are repeated at ever smaller scales inside the tongues, leading to an infinite regress. Then we have to extend the notion of open & dense to residual and that of nowhere dense to meagre. Here a residual set contains a countable intersection of open & dense sets, while a meagre set is a countable union of nowhere dense sets. One sometimes also speaks in terms of G _{ δ } or F _{ σ }sets, respectively [73].
1.1 Periodically Driven Oscillators
Many of the resonant phenomena of interest to us are modelled by periodically driven or coupled oscillators. To fix thoughts we now present two examples, namely the harmonic and the Duffing oscillator subjected to periodic forcing.
1.1.1 The Driven Harmonic Oscillator
1.1.2 The Driven Duffing Oscillator
Remarks

One of the exciting things about resonance concerns the peaks of the amplitude R that can be quite high, even where ε is still moderate. Systems like (1) and (2) form models or metaphors for various resonance phenomena in daily life. In many cases high resonance peaks one needs to ‘detune’ away from the resonance value corresponding to the peak, think of a marching platoon of soldiers that have to go out of pace when crossing a bridge.

In other cases, like when ‘tuning’ the radio receiver to a certain channel, one takes advantage of the peak.

It should be noted that the nonlinear (2) dynamically is far richer than the linear case (1), e.g., see [58] and references therein.
1.1.3 Geometrical Considerations
1.2 Torus Flows and Circle Mappings
1.2.1 The Poincaré Mapping
Consideration of the \(\mathbb{T}^{1}\)dynamics generated by iteration of P gives a lot of information about the original \(\mathbb{T}^{2}\)flow, in particular its asymptotic properties as t→∞. For instance, a fixed point attractor of P corresponds to an attracting periodic orbit of the flow which form a 1:1 torus knot as we saw at the end of Sect. 1.1. Similarly a periodic attractor of P of period q corresponds to an attracting periodic orbit of the flow. In general, periodicity will be related to resonance, but to explain this further we need the notion of rotation number.
1.2.2 Rotation Number
 1.
ϱ(P) depends neither on the choice of the lift \(\tilde {P}\) nor on the choice of φ;
 2.ϱ(P) is invariant under topological conjugation. This means that if \(h: \mathbb{T}^{1} \longrightarrow\mathbb{T}^{1}\) is another orientationpreserving homeomorphism, then$$\varrho\bigl(h P h^{1}\bigr) = \varrho(P);$$
 3.
If P:φ↦φ+2πα is a rigid rotation then ϱ(P)=α mod ℤ.
 4.
ϱ(P)∈ℚ precisely when P has a periodic point. Moreover, ϱ(P)=p/q with p and q relatively prime corresponds to a p:q torus knot.
 5.
If P is of class C ^{2} and ϱ(P)=α for α∈ℝ∖ℚ, then, by a result of Denjoy, the mapping P is topologically conjugated to the rigid rotation φ↦φ+2πα.
Recall that in that case any orbit {P ^{ n }(φ)}_{ n∈ℤ} forms a dense subset of \(\mathbb{T}^{1}\). The corresponding dynamics is called quasiperiodic.
 6.
If P depends continuously on a parameter, then so does ϱ(P).
1.2.3 The Arnold Family of Circle Mappings
Periodicity
From the properties of Sect. 1.2.2 it follows that, for (α,ε)=(p/q,0) with p and q relatively prime, one has \(\varrho(\mathbb{A}_{\alpha ,\varepsilon}) = p/q\). One can show that from each (α,ε)=(p/q,0) an Arnold tongue emanates, in which for all the parameter points (α,ε) one has \(\varrho(\mathbb{A}_{\alpha,\varepsilon}) = p/q\), see Fig. 4. The ‘sharpness’, i.e., the order of contact of the boundaries of the p/qtongue at (α,ε)=(p/q,0) exactly is of order q, see [1, 3, 24].
Quasiperiodicity
In between the tongues the rotation number \(\varrho(\mathbb{A}_{\alpha ,\varepsilon})\) is irrational and by the properties of Sect. 1.2.2 we know that the corresponding iteration dynamics of \(\mathbb{A}\) is quasiperiodic and that each individual orbits densely fills \(\mathbb{T}^{1}\).
Open & Dense Versus Nowhere Dense
In general the (α,ε)plane of parameters contains a catalogue of the circle dynamics. Again fixing ε=ε _{0}>0 small, consider the corresponding horizontal line in the (α,ε)plane of parameters. We witness the following, also see Fig. 5 and compare with [24] and references therein. The periodic case corresponds to an open & dense subset of the line, and the quasiperiodic case to a nowhere dense subset, which in the 1dimensional situation is a Cantor set.
Diophantic Rotation Numbers
Fractal Geometry
The Cantor sets under consideration, since they have positive Lebesgue measure, have Hausdorff dimension equal to 1. Moreover Cantor sets have topological dimension 0, since they are totally disconnected: every point has arbitrarily small neighbourhoods with empty boundary. The fact that the Hausdorff dimension strictly exceeds the topological dimension is a characterisation of fractals, see page 15 of [65]. So our Cantor sets are fractals. They also show a lot of selfsimilarity, a property shared with many other fractals.
Beyond the Arnold Family (10) …
The only point of difference with (10) is formed by the exact ‘sharpness’ of the tongues, which depends on the Fourier coefficients of the function f. In particular a tongue at the tip (α,ε)=(p/q,0) has transverse boundaries if and only if the qth Fourier coefficient does not vanish.
1.2.4 Link with Resonance
Returning to the driven oscillator or the two coupled oscillators we now link periodicity of the Poincaré mapping (8) with resonance. For simplicity we keep ε sufficiently small to ensure this mapping to be a diffeomorphism.
As observed in Sect. 1.2.2 the fact that (α,ε) belongs to the p/qtongue, i.e., that ϱ(P _{ α,ε })=p/q, means that the motion takes place on a p:q torus knot. Generically these periodic orbits come in attracting and repelling pairs and the visible motion takes place on such a periodic attractor. In view of our general ‘definition’ of resonance in that case we say that the oscillators are in p:q resonance, one also speaks of phaselocking or synchronisation. In the case of 1:1 sometimes the term entrainment is being used.
1.3 Conclusions and Examples
The literature on resonance phenomena is immense, apart from the references already given, for instance see [3, 19, 58, 85, 90] and their bibliographies. For even more references see below. The present point of view models resonant systems in terms of dynamical systems depending on parameters, where resonance takes place in a persistent way.
In the parameter space the resonant set is part of the bifurcation set, which forms a catalogue for transitions to various types of dynamics. What we add to the general discussion on this subject is the overall fractal geometry that usually manifests itself in the complement of all the resonances. We now present a couple of examples.
Huygens’s Clocks
This implies that (α,ε) belongs to the 1/1tongue, i.e., that the pendulum clocks are in 1:1 resonance, a situation described before as entrainment which is a form of synchronization. This gives a partial explanation of the phenomena discovered by Huygens [61].
Remarks

Note that the 1:1 resonance of the two clocks could be obtained under quite weak assumptions. If one also wants to understand the phase and antiphase motions, the coupling between the clocks has to be included into the dynamics, compare with [9, 76] and references therein.

For another application of these ideas in terms of circadian rhythms and the response to stimuli see [8]. Here it turns out that next to the 1:1 ‘entrainment’ resonance also certain other resonances have biological significance.

The above ideas can be largely extended to the case of more than two oscillators. For examples in models for the visual neurocortex see [31, 32].
Resonances in the Solar System
From ancient times on resonances have been known to occur in solar system, which are more or less in the spirit of the present section. A wellknown example is the orbital 1:2:4 resonance of Jupiter’s moons Ganymede, Europa, and Io which was studied by De Sitter [51, 52] using the ‘méthodes nouvelles’ of Poincaré [77]. The 2:5 orbital resonance between Jupiter and Saturn is described by Moser et al. [67, 68, 81]. These and other resonances by certain authors are being held responsable for gaps in the rings of Saturn and in the asteroid belt.
Another type of resonance is the spinorbit resonance. As an example thereof, the Moon is ‘captured by’ the Earth in a 1:1 resonance: the lunar day with respect to the Earth is (approximately) equal to one month. Similarly Pluto and Charon have caught each other in such a 1:1 resonance: as an approximately rigid body the two orbit around the Sun. Interestingly, the planet Mercury is captured in a 3:2 spinorbit resonance around the Sun [49].
Remarks

The spin orbit resonances are explained by tidal forces, for instance, the rotation of the Moon has been slowed down to a standstill by tidal friction brought about largely by the reciprocal tidal forces exerted of Earth and Moon. Similarly the rotation of the Earth in the very long run will be put to a stand still by the tidal forces of mainly the Moon. But probably by that time the Sun has already turned into a red giant …

This brings us to the subject of adiabatically changing systems as described and summarized by Arnold [2, 3] and which may be used to model such slow changes.^{2} One may perhaps expect that the 3:2 spin orbit resonance of Mercury in the very long run, and after quite a number of transitions, will evolve towards another 1:1 resonance. This part of nonlinear dynamical systems still is largely unexplored.
2 Periodically Driven Oscillators Revisited
We now return to periodically driven oscillators, showing that under certain circumstances exactly the setup of Sect. 1.2 applies.
2.1 Parametric Resonance
Subharmonics and Covering Spaces
On the tongue boundaries subharmonic bifurcations occur, see [21, 22, 44] where each bifurcation can be understood in terms of a pitchfork bifurcation on a suitable covering space.
Remarks

The geometric complexity of the individual tongues in Fig. 7 can be described by Singularity Theory; in fact it turns out that we are dealing with type \(\mathbb{A}_{2k1}\), see [17, 21].

The parametric 1:2 resonance sometimes also is called the parametric roll. By this mechanism ships have been known to capsize …

In Fig. 8 also invariant circles can be witnessed. kam Theory, as discussed before, in particular an application of Moser’s Twist Theorem [66], shows that the union of such invariant circles carrying quasiperiodic dynamics has positive measure.

In both cases the cloud of points^{5} is formed by just one or two orbits under the iteration of the Poincaré mapping. These clouds are associated to homoclinic orbits related to the upside down unstable periodic solution, which gives rise to horseshoes. Therefore such an orbit is chaotic since it has positive topological entropy, see [19] and references therein. A classical conjecture is that the cloud densely fills a subset of the plane of positive Lebesgue measure on which the Poincaré mapping is ergodic [4].^{6}
2.2 The HillSchrödinger Equation
We like to note that in the corresponding literature usually the value of ε=ε _{0}≠0 is fixed and the intersection of the horizontal line ε=ε _{0} with a tongue is referred to as gap: it is a gap in the spectrum of the Schrödinger operator (15). The approach with tongues and the results of [28] regarding the \(\mathbb{A}_{2k1}\)singularity therefore leads to a generic gap closing theory.
Remarks

In the context of Schrödinger operators the letters are chosen somewhat differently. In particular, instead of x(t) one often considers u(x), which gives this theory a spatial interpretation. Also instead of εf(t) one uses V(x), compare with [69].
 The nonlinear equationwith q quasiperiodic is dealt with in [27]. In comparison with the case of periodic f the averaged, approximating situation, is identical. However, the infinite number of resonances and the Cantorization we saw before leads to an infinite regress of the bifurcation scenarios. For this use was made of equivariant Hamiltonian kam Theory on a suitable covering space [23, 44]. As a consequence the resonant set becomes residual and the quasiperiodic set meagre. Compare this with [7, 45, 46, 47] in the dissipative case.$$\ddot{x} + \bigl(a + \varepsilon f(t)\bigr) \sin x = 0,$$
2.3 Driven and Coupled Van der PolLike Oscillators
The examples of the driven oscillator in Sect. 1.1 were based on approximations of the damped pendulum, the free oscillation of which always tends to the lower equilibrium \(x= 0, \dot{x} = 0\). Our present interest is formed by Van der Pollike oscillators that for x and \(\dot{x}\) sufficiently small have negative damping, for this approach compare with [19, 44]. Such oscillators are known to occur in electronics [80, 88, 89].
Similar results hold for n coupled Van der Pol type oscillators, now with state space \(\mathbb{T}^{n}\), the Cartesian product of n copies of \(\mathbb{T}^{1}\). Next to periodic and quasiperiodic motion, now also chaotic motions occur, see [19] and references therein.
3 Universal Studies
Instead of studying classes of driven or coupled oscillators we now turn to a few universal cases of ‘generic’ bifurcations. The first of these is the HopfNeĭmarkSacker bifurcation for diffeomorphisms, which has occurrence codimension 1. This means that the bifurcation occurs persistently in generic 1parameter families. However, the open & dense occurrence of countably many resonances and the complementary fractal geometry of positive measure in the bifurcation set are only persistent in generic 2parameter families. A second bifurcation we study is the Hopf saddlenode bifurcation for diffeomorphisms where we use 3 parameters for describing the persistent complexity of the bifurcation set.
3.1 The HopfNeĭmarkSacker Bifurcation
3.1.1 The Nondegerate Case
Globally a countable union of such cusps is separated by a nowhere dense set of positive measure, corresponding to invariant circles with Diophantine rotation number. As before, see Fig. 4, the latter set contains the fractal geometry.
Remarks

The above results, summarized from [26, 34, 38, 43], are mainly obtained by ℤ_{ q }equivariant Singularity Theory.

The strong resonances with q=1,2,3 and 4 form a completely different story where the Singularity Theory is far more involved [3, 86]. Still, since the higher order resonances accumulate at the boundaries, there is fractal geometry around, always of positive measure.

Regarding structural stability of unfoldings of P as in (19) under topological conjugation, all hopes had already disappeared since [71].
3.1.2 A Mildly Degenerate Case
To illustrate a mildly degenerate case of the Hopf–Neĭmark–Sacker bifurcation one may well consider the preceding Duffing–Van der Pol–Liénard type driven oscillator (18) where we need all four parameters.
3.1.3 Concluding Remarks
For both cases of the Hopf–Neĭmark–Sacker bifurcation we have a good grip on the part of the bifurcation set that governs the number of periodic points. The full bifurcation set is far more involved and the corresponding dynamics is described only at the level of Poincaré–Takens normalform vector fields [13, 38, 43]. We note that homo and heteroclinic phenomena occur at a flat distance in terms of the bifurcation parameters [15, 18, 75].
3.2 The Hopf SaddleNode Bifurcation for Diffeomorphisms
3.2.1 From Vector Fields to Mappings
3.2.2 In the Product of State Space and Parameter Space
color  Lyapunov exponents  attractor type 

red  ℓ _{1}>0=ℓ _{2}>ℓ _{3}  strange attractor 
yellow  ℓ _{1}>0>ℓ _{2}>ℓ _{3}  strange attractor 
blue  ℓ _{1}=0>ℓ _{2}=ℓ _{3}  invariant circle of focus type 
green  ℓ _{1}=ℓ _{2}=0>ℓ _{3}  invariant 2torus 
black  ℓ _{1}=0>ℓ _{2}>ℓ _{3}  invariant circle of node type 
grey  0>ℓ _{1}>ℓ _{2}=ℓ _{3}  fixed point of focus type 
fuchsia  0>ℓ _{1}=ℓ _{2}≥ℓ _{3}  fixed point of focus type 
pale blue  0>ℓ _{1}>ℓ _{2}>ℓ _{3}  fixed point of node type 
white  no attractor detected 
The Parameter Space
On the righthandside of the figure this method detects an attracting invariant circle of focus type (blue). In the gaps larger resonances are visible, compare with Fig. 4 for a fixed value of ε. Moving to the left, in the neighbourhood of the line indicated by H a quasiperiodic Hopf bifurcation occurs from a circle attractor to a 2torus attractor (green). Also here the parameter space is interspersed with a resonance web of which the larger lines are visible. The remaining features, among other things, indicate invariant tori and strange attractors of various types and also more invariant circles.
The State Space
The upper two figures of Fig. 14 show an invariant circle, once seen from the zdirection and once from some wdirection. The lower two figures indicate how this circle has become a strange attractor, from the same two points of view.
Part of these results can be justified mathematically, as seen from the Perturbation Theory point of view. The invariant circles all have one Lyapunov exponent equal to 0 and these are quasiperiodic, perturbations of closed integral curves of a vector field (averaging) approximation, whence their existence can be proven by kam Theory [5, 23]. A similar statement can be made about the 2tori with two Lyapunov exponents equal to 0. In fact the transition is a quasiperiodic Hopf bifurcation as discussed by Broer et al. [10, 11, 19, 44].
By the same references, this also holds for the quasiperiodic invariant circle in the upper half of Fig. 14. The lower half of this figure is conjectured to show a quasiperiodic Hénonlike attractor, which is the closure of the unstable manifold of an unstable quasiperiodic invariant circle. This is the previous quasiperiodic circle that has become unstable through a quasiperiodic saddlenode bifurcation [11]. For this kind of strange attractor the mathematical background theory largely fails, so the results must remain experimental; for indications in this direction however see [41] and references therein.
For a detailed, computerassisted bifurcation analysis of the 2:5 resonance ‘bubble’ we refer to [35]. Compare with earlier work of Chenciner [45, 46, 47]. We like to note that the family of mappings G forms a concrete model for the Ruelle–Takens scenario regarding the onset of turbulence. In fact it also illustrates how the earlier scenario of Hopf–Landau–Lifschitz is also included: the present multiparameter setup unifies both approaches. For details and background see [19, 44, 59, 60, 63, 64, 70, 78, 79].
Resonance and Fractal Geometry
Interestingly, the blue colors right and left correspond to quasiperiodic circle attractors. The fact that the corresponding regions of the plane look like open sets is misleading. In reality these are meagre sets, dense veined by the residual sets associated to periodicity. These details are just too fine to be detected by the computational precision used.
Particularly in the latter case, in the left half of the diagram, we are dealing with the Arnold resonance web, for a detailed analysis see [36].
4 Conclusions
We discuss a number of consequences of the present paper in terms of modelling of increasing complexity.
4.1 ‘Next Cases’
The Hopf saddlenode bifurcation for maps, see Sect. 3.2, can be viewed as a ‘next case’ in the systematic study of bifurcations as compared to, e.g., [58, 62] and many others. The nowhere dense part of parameter space, since it lacks interior points is somewhat problematic to penetrate by numerical continuation methods. Nevertheless, from the ‘physical’ point of view, this part surely is visible when its measure is positive or, as in the present examples, even close to full measure. Needless to say that this observation already holds for the Hopf–Neĭmark–Sacker bifurcation as described in Sect. 3.1.
Other ‘next cases’ are formed by the quasiperiodic bifurcations which is a joint application of Kolmogorov–Arnold–Moser Theory [5, 11, 19, 30, 44] and Singularity Theory [55, 56, 57, 87]. For overviews see [12, 48, 91]. The quasiperiodic bifurcations are inspired by the classical ones in which equilibria or periodic orbits are replaced by quasiperiodic tori. As an example, in the Hopf saddle node of Sect. 3.2 we met quasiperiodic Hopf bifurcation for mappings from circles to a 2tori in a subordinate way. Here we witness a global geometry inspired by the classical Hopf bifurcation, which concerns the quasiperiodic dynamics associated to the fractal geometry in the parameter space, compare with Fig. 13. The gaps or tongues in between concern the resonances inside, within which we notice a further ‘fractalization’ or ‘Cantorization’.
A similar ‘next case’ in complexity is given by the parametrically forced Lagrange top [29, 30], in which a quasiperiodic Hamiltonian Hopf bifurcation occurs. Indeed, we recall from [50] that in the Lagrange top a Hamiltonian Hopf bifurcation occurs, the geometry of which involves a swallowtail catastrophy. By the periodic forcing this geometry is ‘Cantorized’ yielding countably many tongues with fractal geometry in between.
Remarks

As said before, in cases with infinite regress the fractal complement is a meagre set which has positive measure. Simon [82] describes a similar situation for 1dimensional Schrödinger operators. Also see [6].

It is an interesting property of the real numbers to allow for this kind of dichotomy in measure and topology, compare with Oxtoby [73]. Interestingly, although these properties in the first half of the 20th century were investigated for theoretical reasons, they here naturally show up in the context of resonances and spectra.
4.2 Modelling
We like to note that our investigations on the Hopf saddlenode bifurcation for mappings were inspired by climate models [25, 42, 83], where in about 80dimensional Galerkin projections of PDE models such bifurcations were detected in 3dimensional center manifolds.
Generally speaking there exists a largescale programme of modelling in terms of dynamical systems depending on parameters, with applications varying from climate research to mathematical physics and biological cell systems. These models often are highdimensional and their complexity is partly explained by mechanisms of the present paper, also see [19, 78, 79, 91]. In general such models exhibit the coexistence of periodicity (including resonance), quasiperiodicity and chaos, best observed in the product of state and parameter space.
Footnotes
 1.
For simplicity we take ε<1 which ensures that (10) is a circle diffeomorphism; for ε≥1 the mapping becomes a circle endomorphism and the current approach breaks down.
 2.
Mathematical ideas on adiabatic change were used earlier by Rayleigh and Poincaré and by LandauLifschitz.
 3.
For ‘historical’ reasons we use the letter a instead of α ^{2} as we did earlier.
 4.
The coordinates (ζ,t) sometimes also are called corotating. Also think of the Lagrangean variation of constants.
 5.
Colloquially often referred to as ‘chaotic sea’.
 6.
Also known as the metric entropy conjecture.
Notes
Acknowledgements
The author thanks Konstantinos Efstathiou, Aernout van Enter and Ferdinand Verhulst for their help in the preparation of this paper.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
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