## Abstract

The periodically forced KdVB and extended KdVB equations are considered. We investigate the structure of the totality of steady profiles. The existence of profiles that are close to any shuffling of two basic profiles is proved, and hence the existence of spatially chaotic and recurrent solutions. The proofs are based on topological degree theory to analyse chaotic behaviour. These proofs combine ideas suggested by P. Zgliczyński (Zgliczyński 1996 *Topol. Methods Nonlinear Anal*. **8**, 169–177) with the method of topological shadowing. The results are also applicable to the classical problem of a quite general model of a forced nonlinear oscillator with viscous damping.

## 1. Introduction

This paper, the origins of which lie in the experimental observations presented in Chester & Bones (1968), is concerned with the existence of chaotic behaviour of second-order periodically forced ordinary differential equations. These equations are a reduced form of a periodically forced Korteweg–de Vries–Burgers' (KdVB) and extended KdVB (eKdVB) equations when viscous damping is included, and which are given by(1.1)Here, *Δ*, *a*, *b*, *d* and *ν* are real parameters, *R*(*x*) is a periodic function and *R*′ is the derivative of *R*. When *a*=0, equation (1.1) reduces to the forced KdVB equation, which is a well-known model for weakly nonlinear dispersive waves with viscous damping (see Cox & Mortell 1986). When *b*=0, equation (1.1) reduces to the forced eKdVB equation.

Chester (1968) considered the resonantly forced oscillations of a finite tank containing shallow water. He derived his basic steady-state equation (5.17), which governed the periodic oscillations of the water surface, by balancing the effects of frequency dispersion, nonlinearity, dissipation due to the boundary layers and external forcing. From this theory, Chester (1968) predicted that, for wavemaker frequencies about the fundamental resonant frequency, waves produced in the tank are characterized by high peaks separated by low troughs and the response amplitude is an order of magnitude greater than the input amplitude. In reporting their experimental results, Chester & Bones (1968) concluded, ‘There is reasonable agreement between theory and experiment over the range of parameters investigated’.

The ordinary differential equation that we use here, equation (1.5), is a variant of Chester's equation (5.17). We model the dispersion by a higher derivative term instead of Chester's integral term, and we use Burgers' type damping rather than boundary layer damping; this is standard, see Ockendon & Ockendon (1973). With regard to damping, Chester & Bones (1968) remark, ‘The oscillations here are determined essentially by the nonlinear and dispersive properties. Dissipation is of secondary importance.’

The evolution of the resonantly forced oscillations was shown in Cox and Mortell (1983, 1986) to be governed by a periodically forced, damped KdV equation, i.e. *a*=0, *R*=*R*_{0}sin(*ωx*) in equation (1.1), see also Miles (1985). This evolution equation was shown to have multiple periodic solutions involving solitary-like waves superimposed on a periodic wave background and also non-periodic beating solutions, all of which had been observed by Chester & Bones (1968).

Ockendon *et al*. (1986) gave some asymptotic solutions for the periodic problem, essentially solutions of equation (1.1) when *u*=*u*(*x*), *a*=0, and a dimensionless *d* being small. Rigorous proofs of the existence of periodic solutions of the ordinary differential equation with no damping (i.e. equation (1.1) with *u*=*u*(*x*) and *a*=0, *ν*=0), found earlier by numerical and asymptotic methods, were given by Hastings & McLeod (1991).

The question of the existence of chaotic solutions for an equation of the form (1.5) was raised in Grimshaw & Tian (1994), Malkov (1996) and Cao *et al*. (1999). The Melnikov method supported by numerical work was used in the first two, while Malkov (1996) did an analysis to indicate period doubling and a Melnikov sequence of bifurcations leading to chaos. In no case, however, was a rigorous proof of the existence of chaos for specific values of parameters given. That is the purpose of this paper. We give a rigorous proof of the existence of chaotic solutions to the steady-state periodic case governed by the ordinary differential equation (1.5) by using a novel method completely distinct from that of Melnikov. We do not, however, investigate under what conditions an initial state might evolve to a chaotic state. This significantly more difficult problem is part of ongoing work.

Physical problems where the quadratic nonlinearity is small and the cubic nonlinearity cannot be neglected lead to equations such as equation (1.1). Examples include resonant oscillations of a gas in an open tube (here *d*=0; see Seymour & Mortell 1973*a*), fluid suspensions, see Cox *et al*. (2000), magnetohydrodynamics, see Wu (1991), and large amplitude internal ocean waves, see Grimshaw *et al*. (1997). Moreover, the eKdVB equation is also of interest in its own right as a model for undercompressive non-classical shocks, see Jacobs *et al*. (1995). Undercompressive shocks arise when we view equation (1.1), in the absence of dispersion (*d*=0) and damping (*ν*=0), as a scalar conservation law(1.2)where the nonconvex flux function *f* is given by . Shock solutions to this conservation law are then known to be structurally sensitive to regularization through the terms and . This structural sensitivity implies that the shock strength itself is dependent on the ratio of dispersion to dissipation as *d*→0, *ν*→0. As a consequence, while solutions of the eKdV include solitary waves of both positive and negative polarity, there is also for *ad*<0 an undercompressive shock solution, see Perelman *et al*. (1974*a*,*b*).

The extension to a two layer fluid results in equation (1.1), which then describes the forced sloshing of interfacial waves, when *Δ* is the detuning measured from the fundamental frequency of the interfacial wave (see Mackey & Cox 2003).

Eliminating time dependence in equation (1.1), we write the ordinary differential equation(1.3)where *u*=*u*(*x*). This equation describes the steady, time independent profiles for the eKdVB equation (1.1).

We will always assume that *d*≠0. If, additionally, *a*≠0, then performing the scaling as suggested in Mackey & Cox (2003) eliminates the term proportional to *uu*′. Therefore, equation (1.3) can be reduced to the standard form(1.4)where *α* equals either 1 or 2, *γ*_{1}, *γ*_{2}, *γ*_{3} are constants, and *r*(*x*) is a periodic function. Integrating equation (1.4), we arrive at the equation(1.5)where *c* is an integration constant.

Equation (1.5) has a broad spectrum of applications different from those already mentioned. For instance, it can be considered as a generic description of a forced nonlinear oscillator with linear viscous damping. For *α*=2, *γ*_{3}>0 we have motion in a single potential well with potential *V*(*u*) given by(1.6)For the symmetric potential well (*c*=0), a period doubling sequence leading to chaos is predicted in Huberman & Crutchfield (1979), and a forcing amplitude versus frequency diagram involving saddle-node and period-doubling bifurcations is generated in Virgin (1987). Another application of equation (1.4) is in the modelling of the rolling motion of ships in regular seas. In particular the problem of escape from a potential well in response to periodic forcing has been shown to be relevant to loss of stability and eventual capsize in Thompson *et al*. (1990). More recently, equation (1.5) has been shown to arise in the investigation of marginally separated boundary layer fluid flow (see Braun & Kluwick 2004). In this context equation (1.5) can be identified as a time-independent formulation of a forced Fisher equation.

There are also extensive numerical studies for the case of the cubic potential, and Melnikov methods have been used to predict successfully the onset of fractal basin boundaries that lead to escape (see Thompson *et al*. 1990). Melnikov methods have also been used (both classical and adiabatic) to examine the periodic behaviour for a forced eKdV (see Mackey & Cox 2003). In particular, Mackey & Cox (2003) extended the Melnikov approach to the small frequency regime that is directly applicable to the two layer sloshing problem. A motivation for the present paper lies in their results where, for certain parameter states, a Poincaré map exhibits a fractal structure associated with positive Lyapunov exponents.

It was shown in Seymour and Mortell (1973*b*, 1980) that resonant oscillations of a gas in a closed tube are governed by the ‘standard mapping’, which is a fundamental mapping in chaos theory (see Lichtenberg & Lieberman 1992). In this case, the exact equations are hyperbolic as the dispersion parameter is zero. However, chaotic motion does not seem to be possible due to the occurrence of shocks. In a dispersive system shocks do not form, and so the possibility of chaotic motion can arise, especially in the vicinity of the separatrices of the hyperbolic system.

In summary, Melnikov methods have been used to predict parameter domains where chaos may be located, computational solutions indicate the presence of chaos and Lyapunov exponents show sensitivity to initial conditions. What is still required is a rigorous proof of the existence of chaos for equation (1.4) for specific parameter values. This is the aim of the present paper. Technically we will investigate the structure of the totality of bounded solutions of the equation (1.4). In contrast to Grimshaw & Tian (1994) and Mackey & Cox (2003), we are interested in the analysis of this structure at some given values of parameters *γ*_{1}, *γ*_{2}, *γ*_{3} (there will be no ‘small’ or ‘large’ parameters in the paper). The specific values of the parameters and the form of the forcing terms *r*′(*x*) were adapted from the papers Grimshaw & Tian (1994) and Mackey & Cox (2003), as those leading to especially rich behaviour.

### (a) What we are going to achieve

In this subsection we explain the main ideas behind our results at an informal level.

Let *T* be the minimal period of the forcing term *r*′(*x*) in the equation (1.4); thus any periodic solution of that equation must have a period that is a multiple of *T*. Let *n*_{0} be a positive integer and(1.7)be a function with the minimal period *n*_{0}*T*. This function will play the role of an approximate *n*_{0}*T*-periodic solution of the equation (1.4). We will always deal with the situation when the functions *u*_{0}(*x*), , *u*″(*x*) are piece-wise continuous, and the limitsare well defined at the points of discontinuity. In our setting the wording ‘the function *u*_{0}(*x*) can be considered as an approximate *n*_{0}*T*-periodic solution of the equation (1.4)’ means that, firstly, the jumpat any discontinuity point *y* is rather small, and, secondly, the function almost satisfies the equation between successive points of discontinuity (the quantitative bounds for the corresponding discrepancies will be specified below).

Suppose that for a positive integer *n*_{1}<*n*_{0}, the numberis small. Then we can treat the *n*_{1}*T*-periodic function *u*_{1}(*x*), −∞<*x*<∞, which is defined by *u*_{1}(*x*)=*u*_{0}(*x*), 0≤*x*<*n*_{1}*T*, as an approximate *n*_{1}*T*-periodic solution to the equation (1.4). Since *n*_{0}*T* is the least period of *u*_{0}(*x*) and *n*_{1}<*n*_{0}, we define *r* by(1.8)Figure 1 illustrates the situation for the case *n*_{0}=2, *n*_{1}=1: the functions *u*_{0} and *u*_{1} are graphed by a solid and a dashed line, respectively.

We can use as approximate patterns for possible wave profiles not only the functions *u*_{0}(*x*) and *u*_{1}(*x*) themselves, but also all possible ‘shufflings of these functions’. To explain what this means we need additional notation and definitions. Let *Ω* be the totality of all bi-infinite binary sequences with *ω*_{i}∈{0,1} for *i*=0, ±1, ±2, …. To each sequence *ω*∈*Ω* there corresponds the sequence of numbers *x*_{i} to be defined by *x*_{0}=0,That is, the real line is partitioned into the intervals where the length of the interval *I*_{i} equals . Figure 2 illustrates this idea: here(1.9)and the element *ω*_{0} is in bold.

Corresponding to any *ω*∈*Ω*, the function *u*_{ω}(*x*) is defined by , for *x*_{i}≤*x*<*x*_{i+1}. Figure 3 graphs a fragment of the function *u*_{ω} for the case when *ω* satisfies equation (1.9).

We will use the notation .

*Any two functions* *are different if ω*^{(1)}≠*ω*^{(2)}. *Moreover*,(1.10)

The proof of the proposition is straightforward and so is omitted.

*The value*(1.11)*is strictly positive*.

We call *r*(**U**_{Ω}) the *separation threshold* for the set **U**_{Ω}. If the sequence *ω* is *p*-periodic in the sense that *ω*_{i}=ω_{i+p} for *i*=0, ±1, ±2, …, then the function *u*_{ω}(*x*) is *T*_{ω}-periodic, where(1.12)The number *T*_{ω} in equation (1.12) is the minimal period of the function *u*_{ω}, provided that *p* is the minimal period of the sequence *ω*. Thus, among the elements of the set **U**_{Ω} there are periodic functions with arbitrary large minimal periods. However, the main part of the set **U**_{Ω} consists of functions that do not follow any particular pattern.

Let *ϵ* be a positive number satisfying(1.13)We will say that the equation (1.4) is (*u*_{0}, *u*_{1}, *ϵ*)—*compatible* if the following two conditions hold.

For any function

*u*_{ω}(*x*),*ω*∈*Ω*, there exists a continuum of solutions*u*(*x*) of the equation (1.4) satisfying the uniform estimate(1.14)Additionally, if the sequence

*ω*is*p*-periodic, then there exists a continuum of*T*_{ω}-periodic solutions*u*(*x*) of the equation (1.4) satisfying the uniform estimate (1.14), where*T*_{ω}is given by equation (1.12).

Figure 4 graphs a function *u*(*x*) that satisfies equation (1.14), where *ω* satisfies equation (1.9).

For small *ϵ*, functions *u*(*x*) satisfying equation (1.14) follow closely the ‘guidance’ of a corresponding function *u*_{ω}(*x*). In the terminology of hyperbolicity theory, a function *u*(.) is an *ϵ*-*shadow* of *u*_{ω}(.). Since *ϵ* is strictly less than half of the separation threshold *r*(**U**_{Ω}), any solution *u*(*x*) can be considered as an *ϵ*-shadow of at most one guide *u*_{ω}. Thus, the meaning of (*u*_{0}, *u*_{1}, *ϵ*)-compatibility is that for any pattern given by a function *u*_{ω}∈**U**_{Ω} we can find plenty of wave profiles each of which is an *ϵ*-shadow of *u*_{ω}.

The principal purpose of this paper is to present (for particular values of the parameters) functions *u*_{0}, *u*_{1} and a positive *ϵ*<*r*(**U**_{Ω})/2, such that the equation (1.4) is (*u*_{0}, *u*_{1}, *ϵ*)-compatible.

### (b) Why it is interesting and important

The (*u*_{0}, *u*_{1}, *ϵ*)-compatibility implies, in particular, that *the totality of bounded wave profiles is spatially chaotic in a topological sense*. We recall, as an aside, that the three main aspects of chaotic behaviour are equally important: the *measure theory aspect*, such as the existence of the Sinai–Ruelle–Bowen invariant measures (Ruelle 1989); the ‘*sensitivity*’ *aspect*, such as positivity of Lyapunov exponents (Katok & Hasselblatt 1995); and the *topological aspect* describing irregular mixing properties in terms of symbolic dynamics (Katok & Hasselblatt 1995).

At an informal level the topological aspect of the spatially chaotic behaviour can be described as follows. Suppose that we register the forms of the waves within a certain precision *δ*, where *δ* is greater than *ϵ*, but significantly less than the separation threshold (see the relevant numerical information in the next section). Then a ‘shadow-solution’ *u*(*x*), satisfying equation (1.14), is indistinguishable from its guide *u*_{ω}(*x*), although different guides are quite distinguishable. Thus any *chaotic shuffling u*_{ω} of two basic patterns *u*_{0}, *u*_{1} provides a possible steady profile (within the accuracy *δ*).

The next level of formalization requires ideas of symbolic dynamics. Denote by *n* the least common multiple of the numbers *n*_{0}, *n*_{1}. Also, let *y*∈[0,*n*_{0}*T*) be a fixed number such that |*u*_{0}(*y*)−*u*_{1}(*y*)|>2*ϵ* (such an *y* exists by equation (1.13)). Firstly, we characterize a function *u*(*x*), −∞<*x*<∞, by the bi-infinite sequence of its values(1.15)That is, we measure the values of *u*(*x*) ‘stroboscopically’ with the uniform interval *nT* between the measurement positions. Further, encode the function *u*(*x*) by a binary sequence *ω*∈*Ω*, which is constructed as follows. LetThe intervals *I*_{0} and *I*_{1} do not intersect since |*u*_{0}(*y*)−*u*_{1}(*y*)|>2*ϵ*. If for any *i* the value *u*_{i} belongs either to *I*_{0} or to *I*_{1}, then we define the code sequence *ω*=ω(*u*(.)) by setting *ω*_{i}=0 in the first instance, and *ω*_{i}=1 in the second. In other words, we indicate the measurement positions *inT*, *i*=0, ±1, ±2, …, where the function *u*(*x*) goes through a window *I*_{0} or *I*_{1} (see figure 5). If the value *u*_{i} does not belong to one of these two windows for some *i*, then our coding procedure fails, and we are not interested in such ‘unencodable’ functions *u*(*x*). Since , not more than one code can correspond to a particular function *u*(*x*). Now the compatibility condition (C1) implies that *any random sequence ω appears as a code for some steady profile*. (Indeed, we can introduce a sequence *ω*^{*} by substituting *n*/*n*_{0} ones (*n*/*n*_{1} zeros) instead of any one (any zero) entry of the sequence *ω*. Then, the function has the required properties.)

Let us now delve a bit deeper. For a given function *u*(*x*) we denote by its (left) *nT*-shift to be defined by *v*(*x*)=*u*(*x*+*nT*). Let us also denote by *σ* the standard (*left*) *shift* on *Ω* given by , where . For instance, if , then .

*Suppose that the equation* (1.4) *is* (*u*_{0}, *u*_{1}, *ϵ*)-*compatible*. *Then to each* *ω*∈*Ω there is a solution v _{ω}*(

*x*)

*of the equation*(1.4),

*such that v*

_{ω}(.)

*has the code ω*,

*and this correspondence*

*is shift invariant*:

*for ω∈Ω*.

Proposition 1.3 can be proved in a standard way using the Zorn lemma. In technical terms, the above proposition means that the restriction of the shift to the set is *semi-conjugate* to the shift *σ* in *Ω*. Such semi-conjugacy is widely recognized as the most important attribute of topologically chaotic behaviour (for example, Katok & Hasselblatt 1995).

It is also important to note that (*u*_{0}, *u*_{1}, *ϵ*)-compatibility implies the existence of waves with a wide range of non-trivial *regular* patterns. For instance, one more basic characteristic of chaotic behaviour is an abundance of periodic solutions and, in particular, an exponential rate of increase of the maximal number of pairwise *ϵ*-separated1 *kT*-periodic solutions when *k* increases (see Katok & Hasselblatt 1995). On the physical side, this characteristic is important in the context of the article by Cox & Mortell (1986).

In this context, *kT*-periodic solutions *u*(*x*) of the equation (1.4) describe *kT*-periodic (in time) oscillations of the liquid level in a tank, where 1/*T* is proportional to the fundamental frequency of the tank. This physical context has played a major role in the genesis of our paper, and we mention two important related problems. The first problem is the stability in a physical sense of the long-period solutions *u*(*x*) of the equation (1.1). We note immediately that this bears no relation to Lyapunov-type stability of those functions as solutions of the ordinary differential equation (1.4) (in that sense all solutions are, of course, exponentially unstable). Instead, it seems to be related to the (Lyapunov) stability of the corresponding solutions *u*(*t*, *x*)≡*u*(*x*) of the initial boundary-value problemfor the equation (1.1). Tricky pilot numerical experiments performed by James Gleeson suggest that some of the long-period solutions *u*(*x*) could be stable in this sense. The second problem is whether each *kT*-periodic wave profile *u*(*x*) can be obtained as the result of a certain strategy of gradual non-monotone changing of the frequency *ω*=*ω*(*t*) of the wave-maker in a tank, where initially the liquid is still. In an idealized set up, it is the question whether there exists a function *ω*(*t*) such that the solution of the boundary-value problemfor the equation (1.1), where *ω*=*ω*(*t*), gets close to *u*(*x*) as time *t* increases. Again, we note that some recent observations of Cox *et al*. (2005) concerning multi-stability indicate that the answer could be positive. See Broer *et al*. (1996) and bibliography therein for discussions on different aspects of stability.

*Suppose that the equation* (1.4) *is* (*u*_{0}, *u*_{1}, *ϵ*)-*compatible and denote by n the least common multiple of the numbers n*_{0}, *n*_{1}. *Then it has at least* 2^{k} *periodic solutions of period kn which are pairwise ϵ-separated*.

This assertion follows immediately from the definitions.

Another example of functions with a regular pattern is given by the so-called recurrent functions, which exhibit fractal type features. The recurrence property of a function *u*(*x*) with −∞<*x*<∞ means that for each pair of positive numbers (*ϵ*, *X*) there exists *A*>0, such that for each real *τ*, any interval which is longer than *A* contains *σ*, such that the function *x*→*u*(*x*+*τ*) is *ϵ*-close to the function *x*→*u*(*x*+*σ*), *x*∈[−*X*,*X*]. Recurrent non-periodic functions are important today in the analysis of desynchronized systems, various new models in biology etc.

The notion of a recurrent sequence *ω* is defined analogously. Recall, in particular, the classical Morse example of a recurrent sequence. We begin from the one-element sequence *s*_{1}=0 and then substitute at each step the two-element sequence 0, 1 instead of 0 and the two-element sequence 1, 0 instead of 1. Thus, *s*_{1}=0, 1; *s*_{2}=0, 1, 1, 0; *s*_{3}=0, 1, 1, 0, 1, 0, 0, 1; *s*_{4}=0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0 etc. After many iterations we obtain the famous forward Morse sequence *M*. Considering backward shifts of this sequence and their weak limits we arrive at the set of bi-infinite Morse sequences.

*Suppose that the equation* (1.4) *is* (*u*_{0}, *u*_{1}, *ϵ*)-*compatible and the sequence ω is recurrent in the Morse sense* (*for instance*, *one of the set* ). *Then there exists a recurrent*, *non-periodic*, *wave profile satisfying equation* (1.14).

The proof follows the standard constructions (Sell 1971) and so is omitted.

In conclusion we discuss the advantages and drawbacks of our results and techniques compared with the Melnikov method.

The Melnikov method provides a tool to prove the existence of horseshoe-like structures for the corresponding Poincaré mappings. It has the following features.

The method is applicable for infinitesimally small perturbations of Hamiltonian systems.

The Melnikov method guarantees the existence of the Smale horseshoe for some (large) iteration of the corresponding Poincaré mapping. However, the number of iterations required becomes infinitely large as the perturbation becomes infinitesimally small.

Owing to (ii), fragments of trajectories that correspond to 0 and 1 in bi-infinite sequences generated by the horseshoe are very long and hardly distinguishable from each other.

In summary, the Melnikov method is a powerful tool for the analysis of abstract dynamical systems. It does not, however, yield information about trajectories of a given system that could be observed in experiments or verified by numerical simulations.

In this sense the Melnikov method does not bridge the gap between system analysis and observable experiments. One of the central tasks of applied mathematics is surely to do so.

In contrast to the Melnikov method, the approach presented in our article allows us to

consider a given dynamical system with specified parameter values (not necessarily approaching zero);

specify a relatively small number of iterations of the Poincaré mapping and the sets for which there is a horseshoe-like dynamics;

give distinguishable wave profiles that correspond to 0 and 1 in bi-infinite sequences.

Our method therefore provides results that are verifiable and (maybe more importantly) falsifiable by straightforward experiments. Furthermore, we can now extend our techniques to analyse ordinary differential equations of up to fifth order.

Another approach for establishing the existence of chaotic behaviour is to locate a transverse homoclinic point for the Poincaré map. A transverse intersection of stable and unstable manifolds of a hyperbolic fixed point guarantees the existence of the Smale horseshoe. Since there is no analytical representation of the Poincaré map, we need to find a hyperbolic fixed point and numerically construct both its stable and unstable manifolds. Unfortunately, guaranteed numerical construction of the reasonably long pieces of the manifolds and a check on their transverse intersections are beyond the computational power of modern computers.

### (c) Paper's structure and remarks

The paper is structured as follows. In the next section we formulate our main results: theorem 2.2 for the case *α*=1 is in §2*a*, and theorem 2.4 for the case *α*=2 is in §2*b*. In §3 we give the proof of theorem 2.2, excluding computer aided estimates that are partially described in §4. The full proof can be found in the preprint Cox *et al*. (2002), which is available online.

In the preprint Cox *et al*. (2002), we explain how the corresponding function *u*_{0}(*x*) is constructed and discuss changes converting the proof of theorem 2.2 to the proof of theorem 2.4.

We make a remark to conclude the introduction. We have chosen and discussed the problem, designed the proofs and written this paper together, and the authors are listed alphabetically. However, the three older authors would like to highlight the role of our younger colleague, Oleg Rasskazov; in particular, the results of §4, as well as difficult computing, are mainly his.

## 2. Principal results

### (a) Results for the case *α*=1

This is the case of the steady state periodically forced KdVB equation. For *α*=1 we will be interested in the following set of parameters(2.1)and the forcing term *r*′(*x*) is given by(2.2)Thus, equation (1.4) is(2.3)The parameters (2.1) and the forcing term in equation (2.2) coincide with those suggested in Grimshaw & Tian (1994) where chaotic-like behaviour was indicated.

Equation (2.3) is equivalent to the ‘union’ of the second order equations(2.4)where *c* is a constant: that is, a function *u*(.) satisfies equation (2.3) if and only if it satisfies equation (2.4) for some value of *c*. A special role will be played by the equation(2.5)Below we will use the valuesWe define the function *u*_{0}(*x*) on the interval [*iT*/2,(*i*+1)*T*/2), *i*=0, …, 25, as the solution of equation (2.5) satisfying(2.6)All *c*_{i}, are given in table 1. The function *u*_{1}(*x*), by definition and *n*_{1}=1, coincides with *u*_{0} on [0,*T*).

These functions are easy to calculate to any reasonable accuracy; for instance, in §4 we will prescribe a method that has a guaranteed accuracy no worse than 0.0005. See also figures 6 to 8 for a graphic representation of functions *u*_{0} and *u*_{1}. Note that, despite appearances, the functions *u*_{0} and *u*_{1} are not continuous; they have very small jumps at points *iT*/2 for integer *i*, where *T*=40*π*/7.

We recall that **U**_{Ω} denotes the totality of all shuffings of functions *u*_{0}(.), *u*_{1}(.) and that the separation threshold is defined by equations (1.10) and (1.11).

Let *ω*^{(1)}, *ω*^{(2)}∈*Ω* and *ω*^{(1)}≠*ω*^{(2)}. We must establish the estimate(2.8)Consider the set of functions *v*_{i}(*x*)=*u*_{0}(*x*+*iT*), *i*=0, …, 12, 0≤*x*<*T*.

Introduce the number(2.9)By construction, for any *ω*^{(1)}≠*ω*^{(2)}. By equation (2.9) it remains to prove(2.10)On a heuristic level this estimate can be seen in figure 8. The inequality (2.10) will be proved rigorously as part of lemma 3.5. ▪

By this lemma the (*u*_{0}, *u*_{1}, *ϵ*)-compatibility is well defined whenever *ϵ*<0.05. Our principal result for the case *α*=1 is as follows.

*The equation* (2.3) *is* (*u*_{0}, *u*_{1}, 0.006)-*compatible*.

We rewrite the equation (2.3) in the form(2.11)It is important to note that the set of *g*(*u*, *u*′, *x*), such that the conditions of the above theorem hold, is open with respect to the uniform norm. Thus if the theorem is applicable to some function *g*, then it is also applicable for any sufficiently small perturbation .

*The equations* *are* (*u*_{0}, *u*_{1}, 0.006)-*compatible for all* *satisfying*

The proof and the estimates follow Pokrovskii *et al*. (2001). ▪

Although 10^{−10} seems to be a very small zone, numerical experiments show that the -compatibility property holds for a much larger perturbation of parameters and guide functions , that differ slightly from *u*_{0}, *u*_{1}.

We fix the frequency of the forcing term in equation (2.3) and consider a four-dimensional space of parameters of *g*(.). We have proved that -compatibility holds for all 16 corners of the hypercube in the parameter space with a side length of 2×10^{−4} centred at (−3, −0.2, −0.194, 0.2). This can be viewed as evidence that -compatibility and associated chaotic behaviour exist for parameter perturbations of size 10^{−4}, and the results are robust with respect to small perturbations of the coefficients. Unfortunately, the computational time needed to prove such a statement is beyond any reasonable bounds for modern computers.

### (b) Results for the case *α*=2

This is the case of the steady state periodically forced eKdVB equation. For *α*=2, we consider the following set of parameters(2.12)and the forcing term *r*′(*x*) given by(2.13)Equation (1.4) then is(2.14)The parameter values in equation (2.14) almost coincide with those suggested in Mackey & Cox (2003) where the chaotic-like behaviour was indicated. The reason for a small correction to these values is explained in the preprint Cox *et al*. (2002). Again, equation (2.14) is equivalent to the union of the second order equations(2.15)where *c* is real parameter. We consider the specific equation(2.16)We choose *n*_{0}=17, *n*_{1}=1 and define the function *u*_{0}(*x*) on each intervalwhere *T*=2*π*, as the solution of equation (2.16) satisfying(2.17)and the numerical values of *c*_{i}, are given in table 2. The function *u*_{1}(*x*) coincides with *u*_{0} at [0,*T*). See figures 9 and 10 for the corresponding graphs.

The estimate *r*(**U**_{Ω})>0.2 can be proved in a manner similar to equation (2.7). Therefore, (*u*_{0}, *u*_{1}, *ϵ*)-compatibility is well defined whenever *ϵ*<0.1.

*The equation* (2.14) *is* (*u*_{0}, *u*_{1}, 0.01)-*compatible*.

*Let the equation* (2.14) *be written in the form u*″+*g*(*u*, *u*′, *x*)=0. *Then the equations* *are* (*u*_{0}, *u*_{1}, 0.01)-*compatible for all* *satisfying*

The proof and the estimates follow Pokrovskii *et al*. (2001). ▪

## 3. Proof of theorem 2.2

### (a) The key proposition

In this subsection we introduce another, less intuitive although technically more convenient, notion of compatibility. We then reduce the proof of theorem 2.2 to verification of this new compatibility property.

Denote by *Ω*_{26} the totality of all bi-infinite sequences with *ω*_{i}∈{0, 1, …, 25}, *i*=0, ±1, ±2, …. Denote by *Ω*^{*} the totality of the sequences *ω*^{*}∈*Ω*_{26}, such that , and for each integer *i* either the congruence (mod 26) holds or , and . Figure 11 illustrates the possible transitions from to .

Let =(*U*_{0}, …, *U*_{25}) be a finite family of compact connected subsets of . The equation (2.4) is (, *Ω*^{*})-*compatible* if the following two conditions hold:

For each

*ω*^{*}∈*Ω*^{*}there exists a solution*u*(*x*) satisfying for all integers*i*.Additionally, if

*ω*^{*}∈*Ω*^{*}is 2*p*^{*}-periodic, then the solution*u*(*x*) can be chosen to be*p*^{*}*T*-periodic.

The geometrical meaning of condition (C1^{*}) is the following. Let *ω*^{*} be any sequence from *Ω*^{*}. Consider the planes and cut the window in the plane *Π*_{i}, see figure 12. Then compatibility means that for any *ω*^{*}∈*Ω*^{*} there exists a solution *u*(*x*) such that the trajectory (*u*(*x*), *u*′(*x*)) passes through all windows.

Conditions (C1^{*}), (C2^{*}) are similar to conditions (C1), (C2) from the definition of (*u*_{0}, *u*_{1}, *ϵ*)-compatibility. At the end of this subsection (after the formulation of proposition 3.1), we show that the proof of theorem 1 can be disaggregated into a proof of (, *Ω*^{*})-*compatibility* for a specific family , plus a straightforward verification of simple explicit estimates. On the other hand, some simple topological methods can be applied to prove (*u*_{0}, *u*_{1}, *ϵ*)-compatibility; this will be done in subsections *b* and *c*. Note also that the definition of (*u*_{0},*u*_{1}, *ϵ*)-compatibility is similar to that used in Pokrovskii (1997) and Zgliczyński (1997), but we use only those features that are pertinent to this paper.

We introduce the parallelograms _{i}, *i*=0, …, 25, in the phase plane (*u*, *u*′) given by(3.1)These parallelograms are centred at the points given in table 1, with the other parameters given in table 3, and are graphed in figure 13.

The following statement will play a key role.

Theorem 2.2 follows directly from proposition 3.1.

Indeed: let *ω*∈*Ω*, and *c* be as close to 0.2 as required in proposition 3.1. To prove the theorem we must construct a solution *u*(*x*) of the equation (2.4) satisfying the uniform estimate (1.14) for *ϵ*=0.006, i.e.(3.3)Additionally, the function *u*(*x*) should be *T*_{ω}-periodic if the sequence *ω* is *p*-periodic. Here *T*_{ω} is given by equation (1.12) with *n*_{0}=13, *n*_{1}=1.

Firstly, we construct an auxiliary sequence *ω*^{*}∈*Ω*^{*} by substituting each 0∈*ω* by the sequence 0, 1 and substituting each 1∈*ω* by the sequence 0, 1, …, 25. Define(3.4)or, what is the same,(3.5)By equation (3.4), *ω*^{*} is 2*p*^{*}-periodic if *ω* is *p*-periodic.

By proposition 3.1*a*, there exist compact sets , such that the equation (2.4) is (, *Ω*^{*})-compatible. Thus, by condition (C1^{*}) there exists *u*(*x*) satisfying for all integers *i*. Moreover, by condition (C2^{*}), the function *u*(*x*) can be chosen to be *p*^{*}*T*-periodic if the sequence *ω*^{*} is 2*p*^{*}-periodic. By equations (1.12) and (3.5) this means that the solution *u*(*x*) can be chosen *T*_{ω}-periodic, if *ω* is *p*-periodic. All that remains is to note that the estimate (3.3) follows immediately from proposition 3.1*b*. Thus, we have reduced the proof of theorem 2.2 to the proof of proposition 3.1. The rest of this section and all of the next are devoted to the proof of this proposition.

### (b) Topological hyperbolicity

In this section we remind the reader of the definitions of some topological tools, which will play a major role below.

If is a continuous mapping, is a bounded open set, does not belong to the image *f*(∂*U*) of the boundary ∂*U* of *U*, then the symbol deg(*f*, *U*, *y*) denotes the *topological degree* of *f* at *y* with respect to *U* (see Deimling 1980). If , then the number is well defined and it is called the *rotation of the vector field f at* ∂*U*. The properties of *γ*(*f*, *U*) are described in detail in Krasnosel'skii & Zabreiko (1984). For an isolated root *a* of the equation *f*(*x*)=0, the Kronecker index, ind(*a*, *f*), is defined as the common value of the numbers *γ*(*f*, *B*_{a}(*ϵ*)). Here *ϵ*>0 is sufficiently small, and *B*_{a}(*ϵ*) denotes the open ball of radius *ϵ* centred at *a*. The Kronecker index counts the generalized multiplicity of a root of the equation *f*(*x*)=0; in this context, due to the Kronecker formula (Krasnosel'skii & Zabreiko 1984), *γ*(*f*, *U*) can be interpreted as the algebraic number of roots of the equation *f*(*x*)=0 located inside *U*.

We fix two positive integers *d*_{u}, *d*_{s} with *d*_{u}+*d*_{s}=*d*. Let *V* and *W* be bounded, open and convex product-setssatisfying the inclusions 0∈*V*, *W* and let be a continuous mapping. It is convenient to treat *g* as the pair (*g*^{(u)}, *g*^{(s)}) where and . The mapping *g* is (*V*, *W*)-*hyperbolic*, if the equations(3.6)hold, and(3.7)Here denotes the closure of a set *S*.

The first relationship in equation (3.6) means geometrically that the image of the ‘u-boundary’ of *V* does not intersect the infinite cylinder ; similarly, the second part of equation (3.6) means that the image of the whole set can intersect the cylinder *C* only by its central fragment . Thus the first equation (3.6) means that the mapping expands in a rather weak sense along the first coordinate in the Cartesian product , whereas the second one confers a type of contraction along the second coordinate (the indices ‘(s)’ and ‘(u)’ refer to the adjectives ‘stable’ and ‘unstable’, respectively). Figure 14 illustrates the geometrical meaning of the relationships (3.6) in the two-dimensional case.

The application of the concept of topological hyperbolicity is simplified significantly if *d*_{u}=1. In this case, the mapping *g*^{(u)}(0, *x*^{(u)}) is one-dimensional, *V*^{(u)} is an interval (*α*, *β*) with *αβ*<0, and verification of the inequality (3.7) is trivial.

*The inequality* (3.7) *holds if and only if* .

### (c) How topological hyperbolicity can be used in the proof of the key proposition

In this subsection we will link the notions of topological hyperbolicity and (, *Ω*^{*})-compatibility. However, first it is convenient to present one further lemma—lemma 3.3.

Let *m* be a positive integer. Denote by *Ω*_{m} the totality of all bi-infinite sequences with *ω*_{i}∈{0, 1, …, *m*−1}, *i*=0, ±1, ±2, …. This is consistent with the definition of *Ω*_{26} in §3*a*.

Letbe square *m*-matrices with binary entries (labelling *a*_{0,0} etc. for matrix entries is usual and convenient in symbolic dynamics, see Katok & Hasselblatt 1995). We will use the notation *A*^{(0,1)} for the pair (*A*^{(0)},*A*^{(1)}) with analogous shorthand below. Denote by the set of sequences *ω* from *Ω*_{m} satisfyingFor instance, if *m*=3 and(3.8)then the sequence with the fragment 0, 0, **1**, 1, 2, 2, 0, 0, 0, 0, 1, 1, 2, 2 can belong to (recall that we type the element *ω*_{0} in bold face), although the sequences with any one of fragments **1**, 2 or 2, 1 cannot: the first fragment is impossible by , and the second by .

If we have two mappings , then a bi-infinite sequence is called an *f*^{(0,1)}-trajectory, if for even *i* and for odd *i*. Finally, if we have two families , of compact sets in , we say that the pair *f*^{(0,1)} is -compatible, if

for any

*ω*∈*Ω*_{m}there exist an*f*^{(0,1)}-trajectory such that for even*i*, and satisfies for odd*i*;additionally, for a 2

*p*-periodic*ω*this sequence can also chosen to be 2*p*-periodic.

We illustrate this definition graphically in figure 15 for the matrices defined by equation (3.8). The transitions that are allowed by the matrix *A*^{(0)} are indicated by the solid arrows, and those allowed by the matrix *A*^{(1)} by the dashed lines.

*Let* *d*=*d*_{u}+*d*_{s}, *and let* , *i*=0, …, *m*−1, *be homeomorphisms*, *and* , *i*=0, …, *m*−1, *be bounded*, *open and convex product sets*. *Suppose that* *is* -*hyperbolic whenever* , *and* *is* -*hyperbolic whenever* . *Then there exist compact sets**such that f*^{(0,1)} *is* -*compatible*.

The proof of this lemma follows closely the proof of theorem 1 in Rasskazov *et al*. (2001). The details of the proof can be found in the preprint Cox *et al*. (2002).

We denote by the shift operator along the trajectories of equation (2.4) on the time-interval [0,*T*/2], i.e. maps a pair on to the pair (*u*(*T*/2), *u*′(*T*/2)), where *u*(*x*) is the solution of equation (2.4) with the initial conditions *u*(0)=*u*_{0}, . Similarly, we denote by the shift operator along the trajectories of the equation (2.4) on the time-interval [*T*/2,*T*]. Then we have the following.

*Let* , *i*=0, …, 25; *be homeomorphisms*, *and* , *i*=0, …, 25 *be bounded*, *open and convex product sets*. *Suppose that the mappings*(3.9)*are* (*V*_{2i},*V*_{2i+1})-*hyperbolic*; *the mappings*(3.10)*are* (*V*_{2i+1},*V*_{2i+2})-*hyperbolic*; *the mapping* is (*V*_{25},*V*_{0})-*hyperbolic*, *and the mapping* is (*V*_{1},*V*_{0})-*hyperbolic*. *Then there exist compact sets* , *such that the equation* (2.4) *is* (, *Ω*^{*})-*compatible*.

Define *d*_{u}=*d*_{s}=1, *m*=13. Let *A*^{(0)} be the identity 13×13-matrix, and the elements of the matrix *A*^{(1)} be given byDenote, further, , , *i*=0, …, 12 and , , *i*=0, …, 12. In this notation the conditions of the corollary mean that is -hyperbolic whenever , and is -hyperbolic whenever . Then, by lemma 3.3, there exist compact setssuch that is -compatible. This implies, however, that equation (2.4) is (, *Ω*^{*})-compatible withand the corollary is proved. ▪

### (d) Completing the proof of proposition 3.1

We introduce the product sets (rectangles in our case)(3.11)where , are defined in table 3. We also introduce the homeomorphisms(3.12)as the affine mappings , where are as in table 1, and , are as in table 3. Note immediately that(3.13)where the parallelograms _{i} are defined by equation (3.1). Finally, we use the notation , for the shift operators along the trajectories of equation (2.5), and the notation , , for the specification of the corresponding functions (3.9), (3.10), etc., in corollary 3.4. For a product set *V*=*V*^{(u)}×*V*^{(s)} we also use the notation .

*The following inequalities hold:*(3.14)(3.15)(3.16)*Further,* *the estimates*(3.17)*hold whenever u*(*x*) *is a solution of the equation* (2.5) *and* . *Finally*,(3.18)

The proof of this lemma is computer aided and is sketched in §4.

*Let c be sufficiently close to* 0.2. *The mappings g*_{i}, *i*=0, …, 25 *and g*_{*} *are as defined in* corollary 3.4. *Then the mappings* *g*_{i}, *i*=0, …, 24 *are* (*V*_{i}, *V*_{i+1})-*hyperbolic*; *the mapping* *g*_{25} *is* (*V*_{25}, *V*_{0})-*hyperbolic*, *and the mapping* *g*_{*} is (*V*_{1}, *V*_{0})-*hyperbolic*.

By continuity, for all *c* sufficiently close to 0.2 the inequalities (3.14) imply(3.19)where *g*_{i}, *g*_{*} are as in corollary 3.4. Similarly, inequalities (3.15) imply(3.20)and inequalities (3.16) imply(3.21)Now the stronger relationships(3.22)can be derived from inequalities (3.19) using the following classical theorem.

(Krasnosel'skii & Zabreiko 1984) *Let a shift operator F along trajectories of an ordinary differential equation be well defined at the boundary of a bounded open set S. Then the operator F is well defined on the whole S*, *and* .

By equation (3.11), the relationships (3.20) can be written as(3.23)(3.24)(3.25)The relationships (3.22) and (3.11) implyand, in particular,(3.26)(3.27)(3.28)Finally, equation (3.21) and lemma 3.2 imply(3.29)(3.30)(3.31)The relationships (3.23), (3.26) and (3.29) imply immediately that the mappings *g*_{i}, *i*=0, …, 24 are (*V*_{i},*V*_{i+1})-hyperbolic. Similarly, equations (3.24), (3.27) and (3.30) imply that *g*_{25}, is (*V*_{25},*V*_{0})-hyperbolic; and equations (3.25), (3.28) and (3.31) imply that *g*_{*}, is (*V*_{1},*V*_{0})-hyperbolic. The corollary is proved. ▪

This corollary together with corollary 3.4 imply the following.

*Let c be sufficiently close to* 0.2. *Then there exist compact sets* , *such that the corresponding equation* (2.4) *is* (,*Ω*^{*})-*compatible*.

Thus the assertion in proposition 3.1*a* holds. The assertion in proposition 3.1*b* follows immediately from the inequality (3.17) and theorem 3.7. Thus proposition 3.1 is proved, and so is theorem 2.2. ▪

## 4. Sketch of the proof of lemma 3.5

The proof is based on rigorous estimates of the numerical integration errors for the shift operator *φ*(*x*_{0}, *y*_{0}; *x*) along trajectories of(4.1)with the initial condition *y*(*x*_{0})=*y*_{0}.

Consider a Runge–Kutta method of 4th order with step *h*. We approximate *φ*(*x*_{0}, *y*_{0}; *nh*) by *η*_{n}(*x*_{0}, *y*_{0}) calculated recursively: *η*_{0}(*x*_{0}, *y*_{0})=*y*_{0},Here *b*_{1}=*b*_{4}=1, *b*_{2}=*b*_{3}=2, *ω* is the error in the IEEE implementation of the method andwhere (*x*, *y*) stands for the right-hand side of equation (4.1).

General estimates of the numerical integration by Runge–Kutta methods can be found, for example, in Hairer *et al*. (1987). Recall that the logarithmic norm (with respect to Euclidean metrics) *μ*(*Q*) of a square matrix *Q* is defined bywhere *I* is the identity matrix and ‖.‖ is the operator norm of the matrix.

The local error *e*(*h*, *Ω*) for the RK4 method in *Ω* is given bywhere *h* is the integration step and *η*_{1}(*y*) is the point calculated by the RK4 method.

*Let y*(*x*) *be an exact solution of equation* (4.1) *belonging to Ω*, *and* [0, *nh*] *be a numerical integration range*. *Assume that h is small enough so that the numerical solution remains in Ω*. *If the local error e*(*h*) *in Ω satisfies e*(*h*)≤*C*_{0}*h*^{5} *and the logarithmic norm satisfies* , *then*

We consider the equation (2.5) in vector form(4.2)in the area *Ω*=(−0.175, 0.55)×(−0.28, 0.39).

*Let* *denote the right-hand side of equation* (4.2) *and let μ*( ) *denote the logarithmic norm. Then*

The proof follows that in Bobylev *et al*. (2000).

*For h*>3×10^{−4} *the local error of the RK4 method applied to equation* (4.2) *satisfies*(4.3)

General estimates of the local error for Runge–Kutta methods can be found, for example, in theorem 3.1 from Hairer *et al*. (1987), p. 157), and we reformulate these for our particular case:(4.4)Here *b*_{i} are taken from the definition of RK4: *b*_{1}=*b*_{4}=1, *b*_{2}=*b*_{3}=2; *ω* is the error in the computer implementation of the RK4 method; all derivatives are taken with respect to the system (4.1). For example,The estimate (4.3) follows immediately from equation (4.4) and the inequalities(4.5)(4.6)The restriction on *h* in the formulation of the lemma results from the estimation of the computational error *ω*.

It remains to justify the estimates (4.5) and (4.6). The inequalities (4.5) were obtained via the following scheme. Using interval arithmetic in Mathematica we obtain the explicit analytical formulae for and evaluate them with respect to intervals of change for *x*, *y*, *Χ*. A similar approach was used in Rasskazov *et al*. (2001) and the actual programme can be downloaded from www.ins.ucc.ie/kdv-chaos.htm.

Here we note that, although the formula for seems to be very simple, the expansion of contains hundreds of thousands of terms and can only be estimated with a computer.

The inequality equation (4.6) follows from the description of IEEE arithmetic, and the manner in which trigonometric and exponential functions are realized in the computer.

The lemma is proved. ▪

## Acknowledgments

This research was partially supported by Enterprise Ireland, grants SC/2000/138, SC/2003/376E and IRCSET, Embark Ireland, grant PD/2004/26.

## Footnotes

↵Two functions

*u*(*x*),*v*(*x*) are*ϵ*-separated if .- Received September 29, 2004.
- Accepted March 23, 2005.

- © 2005 The Royal Society