A Diagrammatic Representation of Phase Portraits and Bifurcation Diagrams of Two-Dimensional Dynamical Systems
JJune 29, 2018 0:36 main
A DIAGRAMMATIC REPRESENTATION OF PHASEPORTRAITS AND BIFURCATION DIAGRAMS OFTWO-DIMENSIONAL DYNAMICAL SYSTEMS
JAVIER ROULET
Physics Department, Princeton UniversityPrinceton, NJ 08544, [email protected]
GABRIEL B. MINDLIN
Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,Buenos Aires, C1053ABJ, Argentinaand IFIBA, CONICETCiudad Universitaria, 1428 Buenos Aires, [email protected]
We treat the problem of characterizing in a systematic way the qualitative features of two-dimensional dynamical systems. To that end, we construct a representation of the topologicalfeatures of phase portraits by means of diagrams that discard their quantitative information.All codimension 1 bifurcations are naturally embodied in the possible ways of transitioningsmoothly between diagrams. We introduce a representation of bifurcation curves in parameterspace that guides the proposition of bifurcation diagrams compatible with partial informationabout the system.
Keywords : two-dimensional systems, bifurcations, nonlinear dynamics
1. Introduction
By understanding the dynamics displayed by a nonlinear system we typically refer to the capacity tolist all the qualitatively different phase space portraits that the system can display for different valuesof its control parameters. Even when the equations ruling the system are known, it is often a very hardproblem. Yet, there are algorithmic ways to proceed. One computes some key invariant sets, analyzes theirstability, finds the normal forms that allow mapping the problem onto a (hopefully) studied one close to abifurcation. . . until consistent bifurcation diagrams are sketched for all the parameters of interest [Wiggins,2003; Guckenheimer & Holmes, 1983]. Eventually, the educated intuition of a dynamicist allows filling agap, so that every single change in the phase portrait, as the parameters are changed, can be explainedby either a local or global bifurcation. The program becomes much more difficult when the equations arenot known, for example, if one explores the problem experimentally [Green et al. , 1990; D’Angelo et al. ,1992; Valling et al. , 2007; Ondar¸cuhu et al. , 1993; Ondar¸cuhu et al. , 1994; Mindlin et al. , 1994; Berry et al. ,1996]. In that case, one starts with some sets of attractors, obtained for different parameter values, which apriori are not “close” in any way. Actually, a similar situation is faced when a system (whose equations areknown) is explored numerically. Is it possible to algorithmically list and classify the dynamical possibilitiescompatible with sparse information of this sort? a r X i v : . [ n li n . C D ] J un une 29, 2018 0:36 main J. Roulet & G. B. Mindlin
In this work we explore this question for planar systems. These systems are near and dear to thehearts of dynamicists, since two is the minimal dimensionality in which we can embed nontrivial, recurrentdynamics. Moreover, it is typical to study the different behaviors that these bidimensional models candisplay when two parameters are varied, since this allows us to consider cases in which the linear part ofthe vector fields are doubly degenerate. Yet, even these modest models can present a significant puzzlefor a natural scientist designing the set of experiments (or numerical simulations) necessary to unveilthe structure of his/her problem’s bifurcation diagram. The tools we present here provide an algorithmicmeans for generating and classifying all phase portraits compatible with a given, limited information abouta dynamical system. This could be for instance the knowledge from experiments or simulations about whatthe attractors of the system are.The work is organized as follows. In Section 2 we introduce a way of representing phase portraits byusing diagrams that capture important qualitative information of the system’s dynamics. Specifically, theyencode what the limit sets of the system are, their stability and their distribution in phase space.Section 3 discusses how smooth modifications of these diagrams give rise naturally to bifurcations, inwhich phase portraits change qualitatively. All codimension 1 bifurcations are obtained in this way. Weintroduce a representation of bifurcation curves by means of “dressed” lines, that encode the direction inwhich new limit sets are created and their type and stability. The possible ways of connecting differentbifurcation curves in a higher codimension bifurcation can be constrained by simple rules concerning theirdressings.In Section 4 we show, as an instructive example, how the theoretical framework developed here can beapplied to the Wilson-Cowan oscillator. We present our conclusions in Section 5.
2. Diagram Representation of Phase Portraits
We now describe a representation of phase portraits by diagrams that discard all the quantitative informa-tion that portraits convey (i.e., the specific trajectories in phase space), while preserving the qualitativefeatures. Unlike the phase portrait, the resulting diagram is robust under changes in the system’s param-eters as long as these do not reach a bifurcation, in which the qualitative features of the system change.We will restrict ourselves to two-dimensional, structurally stable dynamical systems. Further, we willassume that the region of interest of phase space can be enclosed by a closed transversal curve (i.e., a closedcurve along which the velocity vector is neither tangent nor zero, so it always traverses the curve from oneof the sides to the other). This is, we assume that the flow traverses the whole boundary of the region ofinterest either inward or outward. We note that the structural stability hypothesis is usually generic exceptfor systems that have some kind of symmetry or conserved quantity, to which this method does not applyin a straightforward manner.
Construction of the Diagrams
The diagrams encode information about the limit sets of the system, which in two dimensions can onlybe stable or unstable nodes (or foci, which are topologically equivalent), stable or unstable limit cycles, orsaddle points. We will represent those with the symbols of Table 1.In Table 1 we have introduced two quantities associated to limit sets: the index and the repulsion.These can be computed easily from the diagrams: the index is the number of arrows above its symbolminus the arrows below, and the repulsion is the number of outgoing arrows minus the incoming ones.Their interpretation will be given in § § − A Diagrammatic Representation of Phase Portraits and Bifurcation Diagrams of Two-Dimensional Dynamical Systems − − − − − n saddles Additive If a system has several saddle points, we shall represent them with a compound symbol as shown inthe last two entries of Table 1 (unless they are separated by a limit cycle so that one is inside the limitcycle and the other outside, in which case we treat them individually). In those symbols, the undirectedlines should be replaced with incoming or outgoing arrows, and one of the squares should be crossed foreach extra incoming arrow (see Fig. 1c for an example). As before, it is forbidden to have all the n + 2arrows outgoing or all incoming.Every well-formed diagram consists of any number of the symbols in Table 1, connected by arrows sothat there is only one arrow unmatched, at the top of the diagram. Each connecting arrow should have adefinite direction (upward or downward), so two symbols can only be connected if the directions of theirarrows match. For example, some well-formed diagrams are shown in Fig. 1.Suppose we know from experiments or simulations what the stable limit sets of a two-dimensionaldynamic system are. Then it is possible to obtain all the phase portraits compatible with these stablesolutions by adding unstable sets (i.e. unstable nodes, unstable cycles or saddle points) in a way such thatthe resulting diagrams are well-formed. Each diagram can then be interpreted as a specific class of phaseportraits. Interpretation of the Diagrams as Phase Portraits
The interpretation of the diagrams is as follows: each arrow in the diagram represents a family of closedtransversal curves in the phase plane, that enclose all the limit sets in the branch of the diagram belowthat arrow (see insets in Fig. 1). An arrow pointing downwards means that the flow enters the regiondelimited by the associated closed transversal, and an arrow pointing upwards means that it exits it. Givenune 29, 2018 0:36 main J. Roulet & G. B. Mindlin (a)(b) (c)
Fig. 1. Some examples of well-formed diagrams and their interpretation as phase portraits. In the portraits, filled dotsrepresent stable nodes, empty dots unstable nodes, squares saddle points, and closed curves limit cycles. Closed transversalsare shown in blue, with an arrowhead indicating the inward or outward direction of the flow. Each closed transversal isassociated to an arrow in the diagram. Diagram (c) admits multiple topologically different phase portraits, that would beseparated by heteroclinic bifurcations in parameter space. a diagram, we draw a closed curve in phase space for each arrow, keeping track of its direction by addingan inward or outward arrowhead to the curve. We draw each curve inside the one associated to the arrowimmediately above in the diagram.Notice that every symbol in Table 1 has one arrow on top and a number of arrows at the bottom.Accordingly, the transversal curves in phase space will delimit regions with an outer boundary and anumber of holes inside. In each of these regions we draw the limit sets associated with the correspondingsymbol in the diagram. Regions with a limit cycle will have a single hole, and the limit cycle should bedrawn around it (see Fig. 1b).To complete the phase portrait, trajectories that cross all the transversals in the direction they defineshould be sketched. Of these, the ones along the saddles’ invariant manifolds are the most interesting.Two begin in each saddle and two end, constituting its unstable and stable manifold respectively. Eachpair approaches or leaves the saddle in opposite directions, so the incoming and outgoing trajectoriesalternate around it. For a single saddle point, there is a unique qualitative (topologically equivalent) wayof connecting its manifolds to the transversals bounding its region, that depends on its repulsion as shownin Fig. 2. This follows since these trajectories must each traverse one of the bounding transversals in thedirection it induces, without intersecting the other trajectories. For several saddle points, the ways ofconnecting their invariant manifolds to the transversals are no longer unique. However, all of them areequivalent up to topological equivalence and heteroclinic bifurcations. An example is shown in Fig. 1c.In the case of multiple saddle points, we do not attempt to assign repulsions to them individually,because it cannot always be done without ambiguity (see § A Diagrammatic Representation of Phase Portraits and Bifurcation Diagrams of Two-Dimensional Dynamical Systems
Obtaining the Diagram from the Phase Portrait
Given a phase portrait, the diagram that represents it can be constructed following algorithmic steps. Weneed to identify the closed transversals on the phase portrait and associate an arrow in the diagram toeach of them. The arrows then define the limit set symbols according to Table 1.For every stable or unstable node, a sufficiently small transversal can always be found that encirclesit, we draw one of these with the appropriate orientation for each node present (inward flow for stable,outward for unstable). Similarly, for every stable or unstable limit cycle a pair of closed transversals canalways be found sufficiently close to the cycle, one on the inside and the other on the outside. Finally, byhypothesis the entire region of interest can be enclosed with a closed transversal, which we also draw. Allof these transversals can – and should – be chosen so that they do not intersect. As a result, the closedtransversals will define regions with an outer boundary and a number of holes. It could be the case that aregion has exactly one hole and the two bounding transversals are oriented both inward or both outward.Such regions cannot have any limit set inside, we eliminate them by discarding either of the two boundingtransversals.After this process, the phase space will be divided by the transversals in regions enclosing a single node,a single cycle or a nonzero number of saddle points. That is, matching one of the entries of Table 1. The fulldiagram can now be constructed following the hierarchy induced by the distribution of these regions. Thesymbol of a region lying inside a hole of another should be located below the outer region’s symbol, andconnected with an arrow in the direction induced by the orientation of the transversal separating them.Thus, we have an algorithmic way of identifying any phase portrait with a specific diagram, and anydiagram with a specific class of phase portraits. The diagrams may be used to consistently generate andclassify phase portraits in a highly qualitative approach, and restrictions about the system’s limit sets canbe naturally applied to them.
Index and Repulsion
We can now give an interpretation to the index and repulsion introduced in Table 1. In planar dynamicalsystems, the index of a closed curve is defined as the amount of counterclockwise revolutions that thevector field does as one travels counterclockwise once around the curve. In particular, the index of a closedtransversal is always 1. To compute the index of a limit set, we extend that definition in the following way:we first choose a region of phase space that contains the limit set we are interested in and no other, andthat is bounded by closed curves. The index of the limit set is the sum of the indexes of all the boundingcurves, with the following proviso: the direction for moving around the bounding curves should be with theregion to the left, i.e. counterclockwise for “outer” boundaries but clockwise for inner, “hole” boundaries,which gives the opposite sign. For example, the simplest region containing a limit cycle and no otherlimit sets has a ring shape with an outer boundary and an inner hole, which can be chosen transversalto the flow. The outer boundary should be traveled counterclockwise, giving an index of 1, and the holeclockwise yielding index −
1. The index of limit cycles is thus 0. Since the index of a curve is invariantunder continuous deformations of it that do not traverse a fixed point, the choice of a boundary transversalto the flow is not necessary. In general, the index of a region bounded by transversal curves is the numberune 29, 2018 0:36 main J. Roulet & G. B. Mindlin of outer boundaries minus the number of holes, which justifies the prescription of obtaining it from thediagram by subtracting the number of arrows below the symbol to the arrows above.We define the repulsion only for regions bounded by closed transversals, and it is directly the numberof transversals through which the flow leaves the region minus the number through which it enters it. Therepulsion of a limit set is the repulsion of a region bounded by closed transversals that contains it and noother limit sets, where we are using “limit set” to actually refer to any of the entries of Table 1. The subtletyis that if a system has multiple saddle points, a repulsion cannot always be assigned unambiguously toindividual saddle points. For instance, in the lower inset of Fig. 1c, we could enclose both stable nodes andeither of the two saddles with a new closed transversal, which naively would give that saddle a repulsion of+1, and − −
2, motivating the convention for the symbolsof Table 1.Both quantities are additive: the index of the union of two disjoint regions is the sum of their indexes,and similarly for the repulsion. The interesting case is when two symbols adjacent in the diagram areconsidered together: then the arrow that connects them is not counted when computing these quantities.However, the arrow must have been above one of the symbols and below the other, so the contributionto the index of one of the limit sets cancels the contribution to the other, yielding the same total index.Similarly, the connecting arrow is necessarily outgoing for one of the symbols and incoming for the other,so it does not contribute to the total repulsion either.An important observation is that for the class of dynamical systems we are considering, i.e. whoserelevant phase diagram can be bounded by a single closed transversal, the total index is 1 and the totalrepulsion is −
3. Bifurcations3.1.
Continuous Transitions between Diagrams
Varying the system’s parameters continuously may change its behavior qualitatively, a process known asbifurcation. Then, in crossing a bifurcation the diagram describing the system should change, and doso somehow “continuously”. Interestingly, the diagram formalism allows a natural interpretation of allbifurcations as the continuous ways to change a diagram into another.By changing a diagram continuously we mean either shrinking the length of a connecting arrow untilit disappears, or alternatively creating a new zero-length arrow and enlarging it. Since the arrows implythe limit set symbols according to Table 1, these must be updated as the arrow configuration changes.Recall that each arrow represents a family of closed transversals that separate the limit sets associatedto the symbols in its ends. An arrow length approaching zero is interpreted as the two involved limit setsapproaching each other in phase space, so that a transversal curve should be finely tuned to separate them.In the bifurcation, the limit sets collide and no separating curves can be found any more.Depending on the type of limit sets that are connected to the arrow involved, different kinds ofbifurcation can occur. Table 2 shows all the possible connections between symbols that can be madeusing the entries of Table 1, and the transition to a different diagram that takes place when the involvedarrow shrinks to zero-length. Each of these transitions has an interpretation as a bifurcation. The (partial)une 29, 2018 0:36 main
A Diagrammatic Representation of Phase Portraits and Bifurcation Diagrams of Two-Dimensional Dynamical Systems diagrams of Table 2 display only the limit sets involved in the bifurcation. They must be completed withother limit sets, which would not participate in the bifurcation, in order to represent a full phase portrait.For instance, at the bottom of the first diagram there is a downward arrow left unmatched, that could becompleted by adding a stable node at both sides of the transition. Table 2. All possible continuous transitions between dif-ferent diagrams, each involves an arrow whose length ap-proaches zero and represents a codimension 1 bifurca-tion. Conversely, all codimension 1 bifurcations in two-di-mensional systems that change the limit sets can beexpressed as a continuous transition between diagrams.Bifurcation Transition between diagramsSaddle-NodeHopfSaddle-Node onInvariant CycleHomoclinicSaddle-Node ofLimit Cycles
Notice that the amount and orientation of all the arrows external to each partial diagram remainsunchanged in the bifurcations. This yields conservation laws for the index and repulsion of the system,une 29, 2018 0:36 main J. Roulet & G. B. Mindlin
Table 3. Representation of codimension 1 bifurcationsin a two-dimensional parameter space. The bifurcationsare represented by dressed curves, the dressing symbolsindicate what limit sets are created or destroyed in thebifurcation.Bifurcation Representation in parameter spaceSaddle-NodeHopfSaddle-Node onInvariant CycleHomoclinicSaddle-Node ofLimit Cycles which must be constant for all parameter values as long as the hypotheses made at the beginning ofSection 2 still hold. Some remarkable consequences follow from this: we can predict a priori from thediagram whether a collision between a saddle and a node would lead to a regular saddle-node bifurcation(if they have opposite repulsions) or to a saddle-node on invariant cycle (if they have the same repulsion).Similarly, from the repulsion of a saddle point we can predict the stability of a limit cycle born from it at ahomoclinic bifurcation (and, conversely, whether a limit cycle of given stability can annihilate against it).Moreover, the specific distribution of arrows around the saddle point symbol determines the topologicallyallowed ways in which the cycle can be created, in particular, whether “big homoclinic loops” can occuror not.
A Representation of Bifurcations in Parameter Space
We will now introduce a representation of codimension 1 bifurcations in parameter space, that explicitswhich limit sets are created or destroyed in crossing the bifurcation. This helps proposing plausible bifur-cation diagrams from partial knowledge of the behavior of the system, in a way analogous to the diagramsintroduced in Section 2, that allowed to systematically construct plausible phase portraits. We will focuson two-dimensional parameter spaces, in which codimension 1 bifurcations are curves.We can keep track of the creation and annihilation of limit sets at a bifurcation by “dressing” itscurve with symbols at its sides. Table 3 shows our convention, in which triangles represent nodes, loopsrepresent cycles and squares represent saddle points, and the filling indicates their repulsion. For example,a filled triangle on the right side of a curve would indicate that a stable node is created when crossing thebifurcation from left to right, or destroyed if crossed from right to left. The symbols associated to eachcurve can be obtained from Table 2 by identifying the limit sets that intervene in the bifurcation. Theyadd to the same index and repulsion at each side of the curve, so that these conserved quantities do notchange when the system undergoes the bifurcation.This notation is well suited for representing codimension 2 bifurcations as junctions of several codi-mension 1 curves. The possible codimension 2 bifurcations are shown in Fig. 3 and can occur genericallyin two-dimensional parameter spaces [Kuznetsov, 2004].A simple, yet powerful constraint we can apply to the set of bifurcation curves of a system is thatwhenever several such curves meet at one point in parameter space, all their dressings must match. Everysymbol that comes “in” through one of the curves must go “out” through another, and on the same sideof the bifurcation line. Otherwise, a closed path in parameter space that went around the junction of thecurves would result in a net creation or destruction of limit sets at each revolution, and the amount andune 29, 2018 0:36 main
A Diagrammatic Representation of Phase Portraits and Bifurcation Diagrams of Two-Dimensional Dynamical Systems type of limit sets of the system would not be uniquely specified by the parameters. μ μ (a) μ μ (b) μ μ μ μ (c) (d) μ μ μ μ (e) (f) Fig. 3. Representation of codimension 2 bifurcations in a two-parameter space ( µ , µ ), at the meeting of codimension 1bifurcation curves. These correspond to (a) cusp, (b) Bogdanov-Takens, (c) saddle-node separatrix loop, (d) generalized Hopf,(e) neutral saddle separatrix loop, (f) cusp of saddle-nodes of limit cycles. In each of the regions, the limit sets that undergobifurcations have been sketched by means of diagrams. In (a), one of the nodes on the inner region has been lined through,to help distinguish them and emphasize that the saddle-node curves that meet in the cusp involve different nodes; idem forthe limit cycles in (f). In each case, reversing the sign of all the arrows (i.e. inverting all the repulsions) also gives a possiblescenario.
4. Examples
In this section we illustrate how the tools we introduced can be used to help guide a reconstruction ofbifurcation diagrams and phase portraits.As an example suppose that, from simulations or experiments, a system is known to have a low-dimensional behavior and that, at three different sets of parameters 1, 2 and 3, it has been observed tobe respectively stationary, or to oscillate, or to have a coexistence of these two attractors. Moreover, largeperturbations of the system tend to die away so that the long term dynamics occurs in a bounded region ofphase space. The aim is to suggest plausible bifurcation diagrams in a two-parameter space, and describethe qualitatively different phase portraits that the system would present.The first step is to make complete phase portraits compatible with these sets of attractors. For thefirst set of parameters, it should have a stable node, for the second, a stable limit cycle, and for the thirdboth, so the diagrams must contain these symbols and eventually other unstable limit sets. Since largeune 29, 2018 0:36 main J. Roulet & G. B. Mindlin
Fig. 4. Simplest diagrams featuring (1) a single stable node, (2) a single stable limit cycle, (3) a stable node and a stablelimit cycle. Two scenarios are shown in the third case, that lead to different bifurcation diagrams. perturbations decay, the phase space can be bounded by a closed transversal with repulsion −
1, and thediagrams should have a downward arrow on top. A single stable node is already a well-formed diagram(since it has exactly one unconnected arrow, at the top) and is the simplest choice for the first case. Forthe second, at least an unstable node must be added below the limit cycle symbol. For the third case, tworelatively simple yet different diagrams can be proposed: by adding an unstable cycle, or a saddle pointand an unstable node, as shown in Fig. 4. More complex diagrams could be proposed for each case byadding more unstable limit sets. Note, however, that the only way to add only unstable limit sets to agiven diagram keeping its index and repulsion constant is by adding pairs of repulsion − x = − x + S ( ρ x + ax − by )˙ y = − y + S ( ρ y + cx − dy ) , (1)une 29, 2018 0:36 main A Diagrammatic Representation of Phase Portraits and Bifurcation Diagrams of Two-Dimensional Dynamical Systems where x, y are the mean activities of the excitatory and inhibitory populations respectively, a, b, c, d arethe couplings between both populations and ρ x , ρ y are the external inputs. S is a sigmoidal function, e.g. S ( ξ ) = (1 + e − ξ ) − . For instance, if we set a = 15, b = 15, c = 12, d = 5, the resulting bifurcation diagramfor the parameters ρ x , ρ y ∈ ( − , −
1) has all possible bifurcations of planar systems, as sketched in Fig. 5.This system can have two qualitatively different stationary states, with both populations active or bothinactive. The node symbol corresponding to the latter has been lined through in the diagrams of Fig. 5 tohelp distinguish them.
5. Conclusions
In this work we have treated the problem of constructing phase portraits and bifurcation diagrams oftwo-dimensional nonlinear systems with a diagrammatic approach.We introduced a class of diagrams to represent the qualitative features of phase portraits of structurallystable, globally attracting (or repelling) two-dimensional dynamical systems. The diagrams emphasize therobust, topological characteristics of the limit sets of the system, and explicitly discard its quantitative,parameter dependent features. There is a one-to-one correspondence between well-formed diagrams andsets of equivalent (up to topological equivalence and heteroclinic bifurcations) phase portraits. Any phaseportrait can be obtained from the diagram and vice versa by following simple algorithmic steps.Smooth transitions between diagrams give rise naturally to all codimension 1 bifurcations of planarsystems (with the exception of heteroclinic connections, which are ignored in our description). We intro-duced the notion of repulsion of a limit set, an additive quantity that is conserved in all bifurcations, andcan be easily computed from a diagram by counting incoming and outgoing arrows. Similarly, the indexis also additive, conserved, and can be computed by counting arrows above and below the symbols ina diagram. The global values that these quantities take (1 for the index, ± J. Roulet & G. B. Mindlin space, by adding dressing symbols to the curves. Apart from describing the type of bifurcation, the dressingmakes explicit the orientation of a bifurcation curve, i.e. to which side of the curve are the involved limitsets destroyed or created. The dressings are particularly useful for studying codimension 2 bifurcations,which can only occur at a meeting of codimension 1 curves if their dressings match properly.
Acknowledgements
This work was supported by CONICET, ANCyT, UBA, and NIH through R01-DC-012859 and R01-DC-006876.
Appendix A Generating sequences of bifurcations
We propose to find sequences of bifurcations that lead from a given phase portrait to another, in away that is similar to finding contributions to the transition amplitude of a scattering process in QuantumField Theory (QFT). Our elements will be the limit sets (the “particles” in QFT), the codimension 1bifurcations will be the “interaction vertices” and the sequences of bifurcations the “Feynman diagrams”[Feynman, 1949]. For this reason we will refer to our diagrammatic representation of bifurcation sequencesas “Feynman-like diagrams” (FL). As with Feynman diagrams in QFT, our representation does not pretendto advance our knowledge of bifurcation theory, but to provide algorithmic tools for keeping track of thepossible bifurcation sequences compatible with finite information about the behavior of the system.The idea is as follows. We can represent the limit sets of planar systems with different types of directedlines, as shown in Table A.1. The direction of the arrow indicates the sign of the repulsion, positive by aleft arrow and negative by a right arrow.
Table A.1. Representationby directed lines of the limit sets possi-ble in two-dimensional dynamical systems.Limit set Directed lineStable nodeUnstable nodeStable cycleUnstable cycleRepulsion − The limits sets can meet in codimension 1 bifurcations, that are represented by vertices, i.e. particularjunctions of lines. These are shown in Table A.2. There, the horizontal axis represents the bifurcationparameter µ , and the vertical axis the spatial distribution of the limit sets. The interpretation is straight-forward: the set of lines is different at both sides of the vertex, which reflects the change in the limitsets of the system at the bifurcation. Note that given an allowed vertex, one can obtain three more byreversing the sign of time (i.e. changing the repulsions, or the direction of the arrows), reversing the signof the bifurcation parameter (interchanging left and right in the vertex), or doing both operations simul-taneously. For example, the four possible Hopf bifurcations (supercritical or subcritical, each of which canbe crossed in either direction) are explicitly shown in Figure A.1. Accepting that these are allowed oper-ations, we can condense all four variants in a single vertex, as in Table A.2. Note that, unlike Feynmanune 29, 2018 0:36 main A Diagrammatic Representation of Phase Portraits and Bifurcation Diagrams of Two-Dimensional Dynamical Systems t (cid:55)→ − t (b), µ (cid:55)→ − µ (c), or both (d). diagrams, single particle lines cannot be moved from one side of the vertex to the other. In the exampleof Fig. A.1, an incoming limit cycle could not yield two outgoing nodes due to the conservation of index.Similar considerations apply to the other bifurcations.A sequence of bifurcations with parameter µ is represented by a FL diagram containing several vertices.The lines present at each value of µ give the succession of limit sets at each stage. The possible sequencesof bifurcations between two phase portraits are given by all FL diagrams whose external lines correspondto the initial and final portraits. Provided that these have the same total repulsion, there will be an infinitenumber of possible sequences that link them. As in QFT, it is reasonable to classify them by the numberof bifurcations (vertices) they involve. The motivation for that is that sequences with few bifurcationsoffer the most parsimonious scenarios and are more limited in number. In perturbative QFT, diagramswith fewer vertices correspond to lower orders in perturbation theory and usually contribute more to thescattering amplitude.We will illustrate the procedure with the transition between regions 1 and 3b in the example consideredin Fig. 4. The first step is to identify the initial and final limit sets, which will be the external lines ofthe FL diagrams. In a diagram representing a transition from region 3 to 1, these would look as shownin Fig. A.2. The rationale we take for the vertical ordering of multiple lines is the same we used for thediagrams introduced in Section 2.1. As we have seen, in these diagrams saddle points emanate “branches”downwards. To keep track of their connectivities, we can label them with dummy symbols α , β , . . . and usethe labels as reference marks, as on the left side of Fig. A.2. There, branches α and β stem from the saddleune 29, 2018 0:36 main J. Roulet & G. B. Mindlin
Fig. A.2. A path in parameter space between two qualitatively different regions defines a series of bifurcations, representedby a FL diagram. Here the blob represents any diagram featuring the depicted external lines. The incoming lines encode thelimit cycles of the initial region, and the outgoing lines, of the final region.Fig. A.3.
Left panels: the three possible FL diagrams linking regions 3 and 1 with the minimal number of vertices (two).
Right panels: sketch of how the bifurcation curves associated with these diagrams would look in a two-parameter space. Theydefine regions, numbered 1-5, in which the phase portraits are given by the diagrams drawn in blue. The node in branch β has been lined through to help distinguish it from the other. point, branch α has a limit cycle on top and branch β only has a node. With this ordering convention,bifurcations can only occur between neighboring lines, including those linked by the reference marks. Thecase of multiple saddle points is incorporated by allowing adjacent saddle-point lines to interchange theirpositions and/or repulsions.The next step is to find, in an orderly manner, well-formed FL diagrams featuring these external lines.As argued above, it is reasonable to look at the diagrams with smaller number of vertices first. In thisparticular case, the initial and final lines cannot be linked by a single bifurcation, a minimum of two mustbe used. There are three different possible diagrams with two vertices, that are shown in Fig. A.3.Note that in case (a) in Fig. A.3, the saddle-node and Hopf bifurcations involve different limit sets,so they may occur in either order. Thus, in the general case these bifurcation curves could intersect inparameter space, as shown in the right panel. Diagram (a) can be interpreted as a path in parameter spacethat goes from region 3 to region 1 crossing these bifurcation curves. If the saddle-node occurs first, itwould traverse an intermediate region with an unstable node surrounded by a stable cycle, which we canune 29, 2018 0:36 main REFERENCES
23 4 15bc a1’ µ µ Fig. A.4. Bifurcation diagram for the second scenario in Fig. 4, describing the possible transitions between regions 1 and3 that involve series of two codimension 1 bifurcations. It features a cusp, Bogdanov-Takens and saddle-node separatrix loopcodimension 2 bifurcations (emphasized with circles). The labeling of the regions 1-5 and paths a-c is consistent with Fig. A.3. recognize as the region 2 introduced in Fig. 4. If the Hopf occurs first, a new region appears, labeled 4.There, two stable fixed points coexist, which is a testable prediction of this bifurcation diagram. A typicalexample of a system exhibiting such coexistence is an “on-off” switch. Note that the intersection of thesetwo curves represents two independent codimension 1 bifurcations rather than a bifurcation of intrinsiccodimension 2.On the contrary, in (b) both bifurcations involve the same node, and the Hopf bifurcation must occurbefore the saddle-node. The path in parameter space would traverse region 4 again but, unlike case (a), thenode that survives is now the one on branch β . Since these nodes represent different states of the system(e.g. “on” or “off”), we label the final region 1 (cid:48) to distinguish it from the former. Notice, however, thatthis is a merely quantitative difference and that it could be possible to transform one state into the othercontinuously.Similarly, in (c) both bifurcations involve the same saddle point and the homoclinic must occur first.The final region is again 1 (cid:48) since the surviving node is the one from branch β .The final step is to combine the information from the three diagrams of Fig. A.3 in a single bifurcationdiagram. We see that region 3 should be adjacent to regions 2, 4 and 5. In turn, regions 2 and 4 limitwith 1, and regions 4 and 5 with 1 (cid:48) . The resulting bifurcation diagram is shown in Fig. A.4, where wehave obtained the junctions of curves (emphasized by gray circles) from the codimension 2 bifurcations ofFig. 3. Note that the dressings of the colliding bifurcation curves matched properly in every case. If thehomoclinic born at the Bogdanov-Takens bifurcation reaches the saddle-node curve separating regions 2an 3, it must collide with it at a saddle-node separatrix loop bifurcation. It cannot cross it, since on theother side lies region 2, that has no saddle point to undergo an homoclinic bifurcation. And indeed, regions2 and 5 can be connected by the resulting saddle node in invariant cycle bifurcation. The three paths a, band c correspond to the three sequences of bifurcations encountered in Fig. A.3. This bifurcation diagramdescribes general transitions between regions 1 and 3 with up to two codimension 1 bifurcations, in thesense that any other such bifurcation diagram will typically be possible to map to a part of this one. Wenote that not all bifurcations shown here must necessarily be accessible to a specific system (for example,the two saddle-node curves need not meet in the cusp, they could extend further). More complex sequencesof bifurcations can be taken into account in a similar manner by drawing FL diagrams with increasingnumber of vertices. References
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