Shilnikov Chaos, Low Interest Rates, and New Keynesian Macroeconomics

The paper shows that in a New Keynesian (NK) model, an active interest rate feedback monetary policy, when combined with a Ricardian passive fiscal policy, à la Leeper-Woodford, may induce the onset of a Shilnikov chaotic attractor in the region of the parameter space where uniqueness of the equilibrium prevails locally. Implications, ranging from long-term unpredictability to global indeterminacy, are discussed in the paper. We find that throughout the attractor, the economy lingers in particular regions, within which the emerging aperiodic dynamics tend to evolve for a long time around lower-than-targeted inflation and nominal interest rates. This can be interpreted as a liquidity trap phenomenon, produced by the existence of a chaotic attractor, and not by the influence of an unintended steady state or the Central Bank's intentional choice of a steady state nominal interest rate at its lower bound. In addition, our finding of Shilnikov chaos can provide an alternative explanation for the controversial “loanable funds” over-saving theory, which seeks to explain why interest rates and, to a lesser extent inflation rates, have declined to current low levels, such that the real rate of interest is below the marginal product of capital. Paradoxically, an active interest rate feedback policy can cause nominal interest rates, inflation rates, and real interest rates unintentionally to drift downwards within a Shilnikov attractor set. Policy options to eliminate or control the chaotic dynamics are developed.

and Shilnikov chaos for a particular scenario/criterion related to the Shilnikov homoclinic orbit. We chose to investigate Shilnikov chaos for multiple reasons. One reason is that it can be detected directly from the Shilnikov criterion. But also, as explained by Alan Champneys (2011), "Over the years, Shilnikov's mechanism of chaos has proven to be one of the most robust and frequently occurring mechanisms chosen by nature." As pointed out by Afraimovich et al (2014, p. 19), "Only starting from mid 70s-80s, when researchers became interested in computer studies of chaotic behavior in nonlinear models, it became clear that the Shilnikov saddle-focus is a pivotal element of chaotic dynamics in a broad range of real-world applications. In general, the number of various models from hydrodynamics, optics, chemical kinetics, biology etc., which demonstrated the numerically or experimentally strange attractors with the characteristic spiral structure suggesting the occurrence of a saddle focus homoclinic loop, was overwhelming. Indeed, this scenario has turned out to be typical for a variety of systems and models of very diverse origins." The relevancy of this general observation to economics has been confirmed by the finding of Shilnikov chaos in an economic growth model by Bella, Mattana, Venturi (2017). We find potentially high relevance of Shilnikov chaos to current problems in the world's macroeconomies, when active Taylor rule monetary feedback policy is adjoined to a NK dynamic macroeconomic model.

Our approach
As stated by Christiano and Takahashi (2018), "Monetary models are notorious for having multiple equilibria. The standard NK model, which assumes that fiscal policy is passive and monetary policy is set by a Taylor rule is no exception." In fact, a large literature exists on complicated dynamics problems produced by NK models with Taylor rule interest rate feedback policies. Our research introduces new problems, which we believe are potentially highly relevant to policy challenges in recent years. We also propose potential solutions to the problems.
Many papers have shown that following an aggressive interest rate policy, in accordance with the Taylor Principle, is not a sufficient criterion for stability in the NK model. 8 One major obstacle to uniqueness is that the stance of fiscal policy may collide with the central bank's inflation objective, when fiscal policy is unable or unwilling to adjust primary surpluses to stabilize government debt (Kumhof et al., 2010). 9 Further limits may result from the way preferences and technologies are introduced into the model.
In this paper, using the path-breaking work of Shilnikov (1965), we find that there may be further reasons to distrust the ability of Taylor rules to be conducive to stability. 10 We show that this policy may induce a class of policy difficulties, emerging from the onset of a chaotic attractor. If the economy becomes enmeshed in a chaotic attractor, the policy maker faces unwanted challenges. Within a chaotic attractor, there is sensitivity to initial conditions, even to infinitesimal changes in initial conditions. Long term predictions become nearly impossible, since an initial condition is known only to a finite degree of precision. It becomes impossible to predict dynamics far into the future. Small changes in initial conditions have major effects on future temporal evolution.
Moreover, given the initial value of the predetermined variable, there would exist a continuum of initial values of the jump variables giving rise to admissible equilibria. Policy options required for recovering uniqueness suggested by the local analysis are exactly those which would cause global indeterminacy of the equilibrium.
Additionally, the qualitative "dimensions" of the chaotic attractor are of great interest in the present context. 11 The relative frequency with which an orbit visits different regions of the attractor is heterogeneous. Then, throughout the attractor, the economy lingers on regions with higher "densities." This is exactly what happens in the numerical simulations developed in this paper. If the initial conditions of the jump variables are chosen far enough from the target steady state, then the emerging aperiodic dynamics continue to evolve over a long period of time around lower-than-targeted inflation and nominal interest rates. This can be interpreted as a liquidity trap phenomenon that, in our case, will depend on the presence of a chaotic attractor and not on the influence of an unintended steady state. 12 The mathematical underpinnings behind these results exploit the presence of a family of homoclinic orbits, double asymptotic to a saddle-focus, in a three-dimensional ambience. The striking complexity of the dynamics near these homoclinic orbits has been discovered and investigated by Shilnikov (1965), who has shown that, if the associated saddle quantity is positive, infinitely many saddle limit cycles coexist at the bifurcation point. Each of these saddle limit cycles has both stable and unstable manifolds, which determine high sensitivity to initial conditions and irregular transitional dynamics. To the best of our knowledge, the Shilnikov homoclinic bifurcation theorem, largely used in physics, biology, electronic circuits, chemistry and mechanical engineering, has recently found application in economics only in a growth theory paper (Bella, Mattana and Venturi, 2017).
The fourth section of our paper discusses an innovative solution to these unfamiliar problems, if the 10 Consider, for example, the case in which the policy maker runs an active fiscal-monetary regime. Assume further that a change in the conduct of fiscal policy induces uniqueness of the equilibrium around the intended steady state. Then, the policy maker may be pressured to renounce discretion in fiscal policy by committing to a marginal tax rate above the real interest rate. As we show, a consequence could be Shilnikov chaos. 11 Cf. Farmer et al. (1983) for a classical discussion on the relevant dimensions of a chaotic attractor. 12 In contrast, as discussed below in Section 2, Benhabib et al. (2001 a,b) found that when the zero bound on nominal interest rate is explicitly taken into account, aggressive interest rate policies may lead the economy to an unintended equilibrium at a liquidity trap or to a limit cycle characterized by Hopf bifurcation. The low inflation rate and low interest rate phenomenon arising in our research, as a consequence of density heterogeneity in the Shilnikov chaotic attractor, is disconnected from the liquidity trap that can emerge because of the influence of an unintended steady state, as in Benhabib et al. (2001a,b). In fact, the two types of liquidity trap may even co-exist for a while, depending on the initial conditions of the economy.
Central Bank chooses to retain the Taylor rule and its consequent Shilnikov chaos. Specifically, we show that the chaotic dynamics can be controlled, in the sense of Ott, Grebogi and Yorke (1986), henceforth OGY. Under specific conditions, the announcement of a higher nominal interest rate at the steady state anchors expectations to the long-run target. More generally, the long run nominal interest rate can be treated as an intermediate target of policy, with the instrument being one of the new policy instruments available, such as forward guidance or quantitative easing. Undesired irregular and cyclical behavior can be superseded, and the intended fixed point can be targeted and attained in a relatively short time.
We now present the plan of the paper. The second section presents the model and the implied threedimensional system of first-order differential equations. We also obtain stability results for the intended steady state, when monetary policy is active. In the third section, we show that the three-dimensional dynamics, characterizing the solution of the model, can satisfy the requirements of the Shilnikov (1965) theorem under plausible calibration settings of the NK model. An example of chaotic dynamics is also discussed, along with its sensitivity to perturbations of the bifurcation parameter and the initial conditions. In section 4 we consider policy approaches to solving the problems produced by the dynamics of the economy within the Shilnikov attractor set. We consider approaches to eliminating the chaos by replacing the Taylor rule by an alternative policy design without interest rate feedback. We also consider approaches that retain the Taylor rule and the associated Shilnikov chaos, while controlling the chaos through the OGY algorithm using a second policy instrument. The conclusion reassesses the main findings of the paper.

The model
Consider the optimization problem faced by household-firm i in the sticky-price, money-in-the utilityfunction, NK model in continuous time (cf., inter al. Benhabib et al., 2001a,b;and more recently Tsuzuki, 2016). 13 We shall call this problem Decision P.

Decision P:
The money in the utility function approach implicitly uses the derived utility function shown to exist by Arrow and Hahn (1971), if money has positive value in equilibrium. A long literature has repeatedly confirmed this existence from models having various explicit motives for holding money, such as transactions or liquidity constraints (e.g., Feenstra (1986), Poterba and Rotemberg (1987), and Wang and Yip (1992). Recently, in a dynamical framework, Benhabib, Schmitt-Grohè, and Uribe (2001a,b; have shown equivalence to a money in the production function model. The mapping from explicit motives for holding money to the derived utility function does not have a unique inverse. Hence, money in the utility function models cannot reveal the explicit motive for holding money. But the ability to infer the explicit motive is not relevant to our research. Hence, for our purposes, we can assume that money has positive value in equilibrium, without conditioning upon an explicit motive.

( ) ( )
The objective of the household-firm optimizer is to maximize the discounted sum of a net utility stream, where ( ) measures utility derived by household-firm i from consumption of the composite good, , and from real money balances, , under the time discount rate, . It is assumed that ( ) is twice continuously differentiable in all its arguments and that where the function subscripts denote partial derivatives.
The function ( ) measures the disutility of labor, where ( ) is twice continuously differentiable, with and .
The term ( ) is standard to account for deviations of the price change, ̇ , with regard to the intended rate , where is the price charged by individual on the good it produces, and where the parameter measures the degree to which household-firms dislike to deviate in their price-setting behavior from the intended rate of inflation, .
In the household-firm budget constraint, denotes real financial wealth, consisting of interest-bearing government bonds, where is the nominal interest rate and ( ) is an endowment of perishable goods, produced according to a production function using labor, , as the only input. Real lump-sum taxes are denoted by . Therefore, the instantaneous budget constraint says that the change in the firm-household real wealth equals real interest earnings on wealth, plus disposable income net of the opportunity cost of holding money minus consumption expenditure.
Before applying the Maximum Principle, it is important to recall that in the NK model, sales of good are demand determined, where is the elasticity of substitution across varieties, and is the aggregate price level.
Taking into account (2), the discounted Hamiltonian can be set as where and are the costate variables; and are the control variables; and and are the state variables.
The necessary first order conditions are Second order conditions also require Consider now a symmetric equilibrium in which all household-firm units' behaviors are based on the same equations. Then, recalling that in equilibrium ( ) the equations from (3.a) to (3.e) allow us to derive the following three-dimensional system of differential equations, which we shall call System M.
System M: where the subscripts are dropped to simplify notation (cf. Tsuzuki, 2016, andBenhabib et al., 2001a,b for details on the derivation). The first equation denotes the time evolution of the Lagrange multiplier associated with the continuous time budget constraint (or shadow price of the real value of aggregate per capita government liabilities, real balances and bonds) at instant of time t. 14 The second equation is the well-known New Keynesian Phillips Curve. The third equation is the budget constraint at time t.
14 Notice that in the Tsuzuki (2016) formulation of the model, there is also a term representing real government spending, which however is held constant. Since the term has no qualitative relevance for the results in this paper, we neglect it, in line with the Benhabib et al. (2001a,b) formulation.
Solutions of system M are admissible equilibrium paths, if the Transversality Condition (TVC) is satisfied. 15 We now turn our attention to the behavior of the public authorities. Following Benhabib et al. (2001a,b), we assume that the monetary authority adopts an interest rate policy described by the feedback rule, The function ( ) is continuous, strictly convex, and satisfies the following properties.

Assumption 1. (Zero lower bound on nominal rates and Taylor principle). Monetary authorities set the nominal interest rate as an increasing function of the inflation rate, implying that
It is further assumed that there exists an inflation rate, , at which the following steady-state Fisher equation is satisfied: Consider, moreover, the following definition (cf. Benhabib et al., 2001a,b).

Definition 1. Let ( )
. Then the policy maker reacts more than proportionally to an increase in the inflation rate (active monetary policy). If, conversely, ( ) , the policy maker reacts less than proportionally to an increase in the inflation rate (monetary policy is passive).
Let us now turn our attention to fiscal policy. We assume that taxes are tuned according to fluctuations in total real government liabilities, a, so that Similarly, for monetary policy, it is further assumed that there exists a tax rate corresponding to the steady-state state level of real government liabilities ( ) .
As in Leeper (1991), Woodford (2003), and Kumhof et al. (2010), we provide a definition of the fiscal policy stance. Let us consider the responses of to its own variations. We have The dynamic path of total government liabilities is locally stable or unstable, according to the magnitude of the marginal tax rate, ( ). Therefore, we have the following useful definition.
Definition 2. Let ( ) ( ) . Then, since the dynamic path of total government liabilities is stable, fiscal policy is passive. Let ( ) ( ) . Then the dynamic path of total government liabilities is unstable, and the fiscal policy is of active type.
Notice that adopting a passive fiscal policy is tantamount to committing to fiscal solvency under all circumstances.

Steady states and local stability properties
The long-run properties of system M are well understood. Benhabib et al. (2001a,b) show that if Assumption 1 holds, then, in general, two steady states exist: one where inflation is at the intended rate and one where ̄ . 16 The unintended steady-state is labelled as a liquidity trap, in which the interest rate is zero or near-zero, and inflation is below the target level and possibly negative. Moreover, at the steady-state where inflation is at the intended rate, , it follows that exists and is unique.
The local stability properties around the intended steady-state are also well described in the literature. A complete picture is provided in Tsuzuki (2016), where the following are clear.
1. When monetary policy is passive, an active fiscal policy induces uniqueness of the equilibrium.
Conversely, a passive fiscal policy commitment to preserve fiscal solvency under all circumstances leads to an indeterminate equilibrium.
2. When monetary policy is active, the stability properties are more mixed. Using the steady state degree of complementarity/substitutability between money and consumption in the utility function, , to characterize the results, we have the following.
2a. When money and consumption are Edgeworth complements in the utility function ( ), the combination of an active monetary policy regime with a passive fiscal rule still induces uniqueness of the equilibrium. Conversely, no equilibria exist in the neighborhood of the steady state in the case of an active fiscal policy.
2b. When money and consumption are Edgeworth substitutes in the utility function ( ), there exists a critical threshold, ̂ such that if | | | ̂ |, then the same consequence as in (2a) occurs. Conversely, when | | | ̂ |, full stability of the intended steady state is established, when fiscal rule is passive, while indeterminacy of the equilibrium prevails, when fiscal policy is active.
For the sake of a clear discussion of the main point of this paper, we shall assume the following.
Assumption 2. Money and consumption are Edgeworth substitutes in the utility function, i.e.
. 17 defined as the intended steady state. Then, we prove the following result.

Proposition 1. (Local stability properties of the intended steady state under Assumption 2). Recall
Assumption 2. Assume monetary policy is active. Then two stability cases can occur according to the magnitude of | |. Consider, first, the case | | | ̂ |. If fiscal policy is also active, is a repellor and there are no equilibrium paths besides the steady-state itself. If fiscal policy is passive, is a saddle of index 2, and the equilibrium is locally unique. Consider now the case, | | | ̂ | If fiscal policy is passive, is an attractor, whereas when fiscal policy is active, there is a continuum of equilibria that converge to the steady-state (local indeterminacy).
Proof. These results are obtained by applying the Routh-Hurwitz stability criterion to system M, evaluated at the steady state. Cf. Appendix 1. ∎

Shilnikov chaos
Let us now focus on the case, | | | ̂ |. Consider a scenario where the policy-maker runs an active fiscal-monetary regime. Then, by Proposition 1, the policy maker may be pressured to increase the marginal tax rate above the real interest rate. In this Section, we show that following this policy prescription may induce another class of difficulties.

An explicit variant of the model
Before proceeding with our analysis, we need to provide specific forms for the implicit functions presented in system M. Following the standard literature, we first assume that the utility function has constant relative risk aversion in a composite good, which in turn is produced with consumption goods and real balances via a CES aggregator as follows: where is a share parameter, measures the intra-temporal elasticity of substitution between the two arguments, and , and is the inverse of the intertemporal elasticity of substitution. Since we have, for now, assumed that consumption and real money balances are Edgeworth substitutes, the following parametric restriction is implied.

Remark 1.
( ) ( ) . Therefore, Assumption requires . 18 Moreover, it is standard to assume that the disutility of labor is captured by the following functional form where measures the preference weight of leisure in utility.
Furthermore, following Carlstrom and Fuerst (2003) and more recently Tsuzuki (2016), we also assume that production is linear in labor, where A denotes the productivity level in the composite goods production. Without loss of generality, we will also set . 19 Additionally, we use the specification of the Taylor principle in Benhabib et al. (2001a,b), and assume that monetary authorities observe the inflation rate and conduct market operations to ensure that where C is a positive constant. Notice that, from (6), our chosen functional form implies Finally, in order to avoid violating the Transversality Condition, we assume that the economy satisfies a Ricardian monetary-fiscal regime. More specifically, equation (16) is complemented by the fiscal rule where the marginal tax rate ( ) is a positive constant.

Conditions for the existence of Shilnikov chaos
In this section, we provide the mathematical underpinnings that guarantee the existence of a chaotic regime in system M. Consider the following Theorem (Chen and Zhou, 2011), which is a generalized version of the original result of Shilnikov (1965 . Assume that the following conditions also hold: ( ) The saddle quantity, | | | | ( ) There exists a homoclinic orbit, , based at .
Then the following results hold: ( ) The Shilnikov map, defined in the neighborhood of the homoclinic orbit of the system, possesses an infinite number of Smale horseshoes in its discrete dynamics; ( ) For any sufficiently small -perturbation, , of , the perturbed system has at least a finite number of Smale horseshoes in the discrete dynamics of the Shilnikov map, defined in the neighborhood of the homoclinic orbit; ( ) Both the original and the perturbed system exhibit horseshoes chaos.
The application of Theorem 1 to system M requires that several conditions be fulfilled, gradually restricting the relevant parameter space. Specifically, we need the parameters to be such that: (i) the system possesses a hyperbolic saddle-focus equilibrium point; (ii) in the case of the saddle-focus equilibrium, the saddle quantity is positive; and (iii) in the case of the saddle-focus equilibrium, with positive saddle quantity, there exists a homoclinic orbit connecting the saddle-focus to itself. System M is highly non-linear and heavily parametrized.
Our attempts to obtain a general result on the critical parametric bifurcation surfaces have been frustrated by frequent numerical anomalies. In order to show that there are regions in the parameter space such that system M may satisfy the conditions of Theorem 1, we therefore propose a numerical strategy based on the parametrization of the US economy for the period 1960(Q1) to 1998(Q3), proposed by Benhabib et al. (2001a,b) Therefore, if we set , the saddle quantity is positive.
We are now ready to propose the following result. Proof. Set and as in Example 1. Then, the eigenvalues associated with system M are of the form required for to be a saddle-focus equilibrium with . ∎ In order to verify the robustness of the results in Lemma 1 to changes in the parameters, we conducted some further numerical simulations. First, we obtained a more general form of the eigenvalues by relaxing the parameters one-by-one from the set ( ). Results, not reported but available upon request, indicate that there is always a small range of the parameter C above unity, for which eigenvalues are all real and there exist values of the marginal tax rate, such that .
Furthermore, we kept ̄, took and studied the surface in the remaining ( ̄ ) parameter space. As shown in Bella, Mattana and Venturi (2017), the vanishing of corresponds to the critical parametric surface at which a generic steady state is a saddlefocus equilibrium with null saddle quantity. Figure 1 depicts the parametric surface in the ( ̄ ) space such that is a saddle-focus equilibrium at the bifurcation point = 0. Above the surface, the saddle quantity is positive. Below the surface, the saddle quantity is negative. Interestingly, the figure shows that a positive saddle quantity can be determined exactly, when the pair ( ̄ ) is plausibly low and We end this section by noticing some further details regarding the form of the eigenvalues in Example 1. It is clear that, for , and irrespective of the stance of fiscal policy, one eigenvalue is real, and the remaining two eigenvalues are complex conjugate. This means that locally, when monetary policy is active, convergence towards occurs typically through (damped) oscillating paths. 20 It is also useful to observe the following.
Remark 2. In our simulations, the structure of the eigenvalues derived in Example 1 survives wide variations of the parameters. More specifically, when C > 1 (active monetary policy), there is always a small right neighborhood of C = 1 such that eigenvalues are all real. Things are different when C < 1 (passive monetary policy). In this case, eigenvalues are always real, and the convergence towards generally takes place along the monotonic perfect-foresight path.
Once it has been established that there are regions in the parameter space such that is a saddle-focus equilibrium with , we need to show that system M admits homoclinic solutions (pre-condition H.2 in Theorem 1). Bella, Mattana and Venturi (2017) describe in detail the necessary steps required to establish whether a given dynamical system supports the existence of a family of homoclinic orbits doubly asymptotic to a saddle-focus in . The application of the method is very lengthy. Because of the space constraint, we do not report those computations, which remain available upon request. For details, please refer to Bella, Mattana and Venturi (2017).
A preliminary step requires translation of the system, M, to the origin and putting the system into normal form by using the associated eigenbasis. We thereby obtain the following (truncated) normal form of system, M, where ( ) is the vector of transformed coordinates, and where the coefficients, with and , are combinations of the original parameters of the model, also depending on the values of three free constants, arising in the computation of the eigenbasis. Following Freire et al. (2002), system (20) can be put into the hypernormal (truncated) form where ( ) ( ) ( ) , and where d and k are combinations of various coefficients.
Once the hypernormal form has been obtained, the method of undetermined coefficients (Shang and Han, 2005) is applied to obtain a polynomial approximation of the analytical expressions of both the two-dimensional unstable manifolds associated with and , and of the one-dimensional stable manifold associated with . The procedure leads to the following split function 21   The reason why the three constants ( ) are bound to belong to the cube (0,1) 3 is strictly related to the geometry of the stable and unstable manifolds which intersect near the origin (in the transformed eigenspace) and forms the homoclinic loop. The issue is well identified in Kuznetsov (1998, p. 259). 23 In order to identify the monetary policy and fiscal policy regimes that prevailed in the US, Bhattarai et al. (2012) considered 90 percent prior probability interval for the parameter C to be (1.189, 1.811) under active monetary policy regimes and for marginal tax rates to be (0.003, 0.107) under passive fiscal policy regimes in their calibrations. The intervals cover the range of values found in the literature (e.g., Davig and Leeper (2011), Xu and Serletis (2016), Ascari et. al. (2017) etc.).Using more recent US data with a superior policy rule that incorporates time varying disturbance variances in interest rate rules, Xu and Serletis (2015) found parameter C = 1.655 and marginal tax rate = 0.017 under the activepassive monetary-fiscal regime. However, note that the extent to which the marginal tax rate can be revised upwards depends on where the economy is located on its Laffer curve and the political resistance to higher taxes on the economy.

Remark 3. For this calibration of the economy, since
for all values of ( ) ( ) , there exists a unique critical value of solving the split function (22).
The following statement is therefore implied. Without showing the computations, we also point out the following.

Remark 4.
Alternative calibrations of the economy show that the result is qualitatively robust.

Existence and properties of the chaotic attractor
We can now go to the main result in this section. Let ̄ , where ̄ is the critical value of the marginal tax rate, such that an admissible solution of the split function exists for given coordinates, The attractor generated by this specific example is represented in Figure 3.

Economic implications
Economic implications of Proposition 2 are very important to the dynamics implied by NK models. The existence of a chaotic attractor implies that small changes in initial conditions can produce large changes in dynamics over time. Two economies, starting contiguously in the space of initial conditions, 24 Is the chaotic attractor in Figure 3 a global absorbing set in which trajectories fall over time for any initial data, or does the considered attractor have only a bounded basin of attraction? A way to answer these questions is to follow Bella, Mattana, and Venturi (2017) and perform a numerical scanning of the initial conditions space in which the attractor is observed. However, since initial conditions are given in the transformed eigenspace ( ( ) ( ) ( )) and since it is interesting to understand the boundaries of the basin of attraction in the original ( ) coordinates, we retrace the transformation matrix for the case of parameters as in Example 3. If we consider inflation, we have where the weights in the formula depend on the structure of the chosen eigenvectors. Applying the iterative procedure, and starting from the vector ( ( ) ( ) ( )) ( ) in Example 3, we find that the attractor survives any variation of ( ) ( ) This means that any perfect foresight path originating in this interval for inflation, is captured by the chaotic attractor. Notice that, these findings imply that the region of the phase space around the homoclinic orbit, which also belongs to the basin of attraction of the chaotic set, is very narrow. Therefore, as is customary in the literature discussing the characteristics of Shilnikov chaos, it suffices for a small perturbation of the system to make the attractor disappear.
can follow completely different patterns over time. Since an initial condition is known only to a finite degree of precision, it is impossible to predict dynamics deterministically over extended periods of time.
Moreover, within a chaotic attractor, given the initial value of the predetermined variable, there exists a continuum of initial values of the jump variables giving rise to admissible equilibria. Therefore, the policy options required to recover the uniqueness suggested by the local analysis are exactly those which may cause global indeterminacy of the equilibrium. In this regard, showing that the equilibrium is globally indeterminate requires the proof that, given an initial condition in terms of the predetermined state variable, ( ), there exist multiple choices of the jump variables, ( ) and ( ), lying outside the small neighborhood relevant to the local analysis. Our analysis is able to give rise to recurrent equilibria, namely solution trajectories of system M that stay in a fixed tubular neighborhood of a given homoclinic orbit for all times.  Barnett and Duzhak (2008, 2019 found Hopf bifurcation boundaries and Period Doubling (Flip) bifurcation boundaries in discrete time NK models. Benhabib et al. (2001a,b) and Tsuzuki (2016) located the Hopf bifurcation boundary in the continuous time version of the NK model. These bifurcation boundaries in the NK model parameter space represent different qualitative dynamics within the class of such models. Bifurcation boundaries are in fact commonly found in the parameter spaces of all credible macroeconomic models, such as optimal growth models and overlapping generations models. See, e.g., Grandmont (1985), Geweke, Barnett and Shell (1989), Barnett and Chen (2015); Barnett and Ghosh (2014); and Bella, Mattana and Venturi, (2017).
context. 26 Assume that the relative frequency at which an orbit visits different regions of the attractor is largely heterogeneous. Then, across the volume of all possible coordinates contained in the attractor, the economy lingers on particular regions with higher "density." In the numerical simulations developed in the paper, it is evident that if the initial conditions of the jump variables are chosen far enough from the target steady-state, then the emerging aperiodic dynamics tend to evolve for a long time around lower-than-targeted inflation and nominal interest rates. This can be interpreted as a liquidity trap phenomenon, which now depends on the existence of a chaotic attractor and not on the influence of an unintended steady state.
Consider the reconstructed time profile of the inflation rate (Figure 4a). The time span is 10 years. First, observe the distinct shape of Shilnikov chaos. The wave train generated by the spiral attractor has long quiescence periods, when the phase point approaches the saddle-focus, followed by bursts of oscillatory activity. The average inflation rate can be persistently higher/lower than the steady-state value. This implies the possibility of long periods, during which inflation is stubbornly high, or long periods during which inflation is stubbornly low (akin to a deflationary equilibrium), and periods during which inflation is volatile.  The relevance of periods of persistently low inflation rates is strengthened by a further phenomenon.
The time profile for inflation shows a slight negative drift. 27 Consider in particular Figure 4b, depicting the moving average of the inflation rate (window = 100 iterations), which we detrend using the steady state value of 0.042. The figure reveals a robust and persistent downward deviation of the moving average from the (de-trended) target inflation rate.
These empirical characteristics of the dynamic pattern imply the following statement.

Corollary 2. (The existence of persistent inflationary/deflationary perfect-foresight equilibrium paths).
Assume that the dynamics of the system evolve along the attractor set. Then, persistently high/low inflation rates with regard to the (unique) steady-state value can emerge. 28 Corollary 2 has important implications for the debate regarding liquidity traps. As discussed in our introduction, this phenomenon has previously been linked mainly to the existence of a low-inflation steady state (cf., in particular, Benhabib et al., 2001a,b) and to its basin of attraction. We offer an alternative explanation, based on the long-run peculiarities of a chaotic attractor and the evolution of the dynamics within that attractor set, such that the economy drifts into the liquidity trap without any policy intent. 29 The differences in the qualitative dynamics arising because of an unintended steady state or because of the existence of a chaotic attractor are remarkable. The time profile for inflation featured in Benhabib et al., (2001a,b) for the case of , so that consumption and real balances are substitutes, presents higher and higher amplitude oscillations around the active steady. Then inflation suddenly arrives to the passive (lower) steady state value, when the saddle connection is established. This kind of predictable/regular behavior of the economy could be traced out by an econometric exercise. In our case, inflation, along the spiral attractor, has long quiescence periods, possibly characterized by a persistent and steep monotonic behavior, followed by bursts of irregular oscillatory activity.
This kind of pattern is largely unpredictable and cannot be inferred by conventional econometric tools, since such behavior violates the regularity conditions for available statistical inference methodologies, such as the usually assumed properties of the likelihood function and polyspectra (see, e.g., Barnett, Gallant, Hinich, Jungeilges, Kaplan, and Jensen (1997) and Geweke (1992)).
As a check on the robustness of our conclusions to our assumption of money in the utility function, we repeat our analysis of appearance of Shilnikov chaos to the alternative specification of money in the production function in Appendix 6.

Ending the chaos
Potential policy solutions to the problems produced by Shilnikov chaos can be divided into two groups. One group of approaches ends the Shilnikov chaos by removing the Taylor rule and its closed-loop interest rate feedback dynamics, while introducing a fundamentally different monetary policy design. The other approach is to retain the imposed Taylor principle and thereby the Shilnikov chaos, while imposing an algorithm to control the chaos. This latter approach requires introduction of a second policy instrument in addition to the interest rate that appears in the Taylor rule.
To end the chaos, the Central Bank could adopt any of the policy approaches that do not use a Taylor rule. There are many such policy designs in the literature that best could be selected by the Central Bank in accordance with the Central Bank's mechanism design. t is not the purpose of this paper to advocate any one of those alternatives.
Examples could include using an active fiscal policy and a passive monetary policy. That approach produces its own dynamical problems in a NK Model, but not Shilnikov chaos. Another example could include monetary policy without interest rate feedback. An open loop fixed monetary quantity growth rate would be the simplest approach. More sophisticated modern approaches could include those using Divisia monetary quantity aggregates, such as those proposed by Belongia (1996), Serletis (2013), or Belongia and Ireland (2014 and as advocated by Peter Ireland in his role on the Shadow Open Market Committee. A long literature exists on alternatives to Taylor rule policies, such as Cochrane (2011). high. This tendency to switch between high and low values is also expected to last a long time into the future. Our computation (using 4000 iterations) leads to Hq ≃ 0.880417, which suggests high persistence in the time series.

Controlling the chaos
If the Central Bank were to decide to retain imposition of the Taylor Principle and thereby the resulting Shilnikov chaos, a second instrument of policy would need to be introduced to deal with the consequent liquidity trap. The need for such a second instrument, in addition to the interest rate in the Taylor rule, is widely accepted and has been applied by most central banks in recent years. A survey of such new tools of monetary policy, such as forward guidance and quantitative easing, has been provided by Bernanke (2020) in his Presidential Address to the American Economic Association. But what is less well established is how to design a rule for use of such an alternative instrument of policy. Under those circumstances, we would propose one of the available algorithms for controlling chaos. In addition, we find that when the economy is on a Shilnikov chaos attractor set associated with imposition of a Taylor rule, the need for a second instrument of policy to control the undesirable properties of chaos should be retained, even if the economy is not in a liquidity trap.
We now consider policy to control chaos. Assume the economy is enmeshed in a chaotic attractor. What should a policy maker do in order to alleviate the implied economic uncertainty, and bring agents' inflationary expectations back in line with those coherent with the intended steady state? In each such approach, one of the new tools of policy would be adopted as a second instrument of policy to target, as an intermediate target, a long run anchor consistent with an available algorithm for controlling chaos.
The methods of controlling chaotic dynamics in the engineering literature provide useful tools in this regard. Under certain conditions, undesired irregular or even cyclical behavior can be switched off. 30 An ingenious and well-known method for doing so is proposed by Ott, Grebogi and Yorke (1990), also known as the OGY algorithm. It enables one to force a chaotic trajectory onto a desired target (a periodic orbit or a steady state of the system) by a correction mechanism. This mechanism has the form of a small, time-dependent perturbation of a certain control parameter. Suppose also that a neighborhood of the desired fixed point can be found, such that the system is guaranteed to be driven to the fixed point. If this neighborhood has points in common with a chaotic attractor, it may be used as a controllable target for the fixed point.
To achieve control over a chaotic solution trajectory, the control parameter must be accessible to the Central Bank. This is a relevant point in our NK sticky-price model with Taylor rule interest rate feedback and a Ricardian fiscal policy. If we go back to the form of the eigenvalues in Example 1, it is clear that fiscal policy parameters can only govern the sign of the real eigenvalue. 31 As a consequence, fiscal policy is ineffective in controlling chaos in the present setting. 32 30 There are examples of chaos control in the literature on optimal monetary models (cf., inter al., Mendes and Mendes, 2006). However, those examples are developed in a discrete-time environment. To the best of our knowledge, this is the first attempt in economic theory with continuous-time dynamics. Experiments of chaos control are widespread in other fields of the economics literature. For example, there are the very recent contributions in tâtonnement processes (cf. inter al. Naimzada and Sordi, 2017) and in disequilibrium macroeconomic models (Kaas, 1998). 31 Recall that τ cancels out in the computation of the characteristic equation. See also (A.2), (A.3), and (A.4) in Appendix 1. 32 A similar problem is discussed in Benhabib, Schmitt-Grohè, and Uribe (2002). The authors describe the characteristics of fiscal policy schemes capable of eliminating the liquidity trap, while maintaining the assumed monetary policy stance. More In this regard, we choose to operate through the manipulation of the nominal interest rate, ̄, at which the steady-state Fisher equation in (8) is satisfied. Since it is not under the direct control of the Central Bank, our algorithm treats manipulation of that interest rate as an intermediate target, rather than as an instrument of policy. 33 Before proceeding with the implementation of the OGY algorithm, some preliminary steps need to be taken. First, we need to show that system M is controllable. Then, we will need to discuss the region of the parameter space supporting application of the OGY algorithm. Consider the following initial result.

Lemma 3. System M satisfies the conditions for controllability.
Proof. Cf. Appendix 4. ∎ Once controllability of system M can be established, the OGY algorithm requires that the eigenvalues of the controlled system be chosen such that stability is implied. Stabilizing a system is thus translated into searching for values of the nominal, steady state interest rate, ̄, such that all eigenvalues exhibit a negative real part (see Appendix 4).
From Proposition 1, we know that there is a critical value, | ̂ ( ̄) |, such that if | ( ̄) | | ̂ ( ̄) |, then an active-passive monetary-fiscal regime implies stability of the intended steady state. For notational convenience, let us define and denote We are now ready to prove the following.
specifically, they find that by an inflation-sensitive revenue schedule of the generic form τ(a) α(π)a + Rm the Government may manage to avert the unintended low-inflation equilibrium. However, this exercise implies a modification of the standard model presented in Section 1. We therefore leave this type of analysis to future research. 33 The choice of policy instrument or of market intervention operating procedure, to be used in that intermediate targeting, could depend upon the mechanism design of the central bank, which is not a topic of this research. An alternative OGY procedure could use the long run inflation rate, , instead of the nominal interest rate, ̄. Although we believe that the two procedures would likely prove to be mathematically equivalent, we anticipate that an intermediate targeting OGY procedure using ̄ would be more easily implemented by a Central Bank than an OGY procedure using the long run inflation rate, which is a final target of policy. From a more technical point of view, and in order to operate an informed choice between and ̄, we have also computed and evaluated at the intended steady-state, the partial derivatives of ( ) in (A.5) with regard to and This computation is helpful since in correspondence of a saddle-focus, the Jacobian of system M presents negative ( ) ( ) at the intended steady-state. Therefore, chaos control according to the OGY mechanism translates into varying ̄ or in such a way that ( ) becomes negative. We found that, for parameters as in Example 3, ⌈ ( ) ⌉ ⌈ ( ) ⌉ , implying that using ̄ has proportionally much higher stabilization power than varying  4) to obtain the controlled system. In Figure 5, we superimpose the time profile of inflation (red curve), where the control has been initiated at iteration 800. The result in Example 4 might appear puzzling when compared to the policy prescriptions suggested by Benhabib, Schmitt-Grohè, and Uribe (2002). However, a closer look at the different comparative statics 34 Since the two elasticities, and are very close, there is a very large divide between | ( ̄) | and | ̂ ( ̄) |; and ̄ has to undergo a too large jump to make | ( ̄) | | ̂ ( ̄) |. In this example, we have therefore slightly decreased . implied by our model provides a full explanation of this apparent contradiction. 35 Recall that Benhabib, Schmitt-Grohè, and Uribe (2002) maintain throughout the paper the complementarity condition in the utility function between real balances and consumption, ; this implies that a drop of the interest rate increases consumption and real economic activity in the model, via the implied increase in money holdings. In our case, the same stimulus to the real activity of the economy is obtained by an increase of the interest rate, provided that we have assumed (cf. Assumpion 2 above). 36 In Benhabib, Schmitt-Grohè, and Uribe (2002), stability is achieved through the simultaneous modulation of nominal interest rates. 37 In our case, instead, it is the commitment to a long-run easing or tightening of the monetary policy stance, which is able to re-anchor expectations to the long-run target of inflation.
As a check on robustness of our conclusions to our assumption of money in the utility function, we repeat our analysis of OGY chaos control to the alternative specification of money in the production function in Appendix 6.

Conclusions
Using the Shilnikov criterion, we find bifurcation to chaos in a NK model at plausible settings of parameters with common NK policy design. The existence of chaos is consistent with the fact that economists who provide short term forecasts are rarely willing to provide long term forecasts. Within the Shilnikov chaos attractor set, we find a downward bias in the interest rate and inflation orbits producing a phenomenon similar to a liquidity trap. The problems associated with the zero-lower bound on nominal interest rates would thereby not be an intentional objective of central bank policy but of the dynamics of the system within the attractor set. The existence of this downward bias, evident for four decades of declining interest rates and inflation rates, has produced the puzzle of very low real rates of interest substantially below the marginal product of capital.
That puzzle has often been observed and frequently unconvincingly imputed to oversaving. 38 Our explanation is different and has different policy implications. Paradoxically, an active interest rate feedback policy can cause nominal interest rates, inflation rate, and real interest rates unintentionally to drift downward, as a result of the dynamical response of the economy within a Shilnikov attractor set. Unlike an open loop interest rate rule, which would directly control an interest rate, active interest rate feedback rules are closed loop rules that link an interest rate to the dynamics of the rest of the economy.
Our results strikingly resemble experience in developed economies in recent decades. We also find and investigate other complications for policy associated with Shilnikov chaos of a NK model.
We propose two potential classes of solutions to the problem: (a) The policy design could be altered from an active NK interest rate policy to a fundamentally different design, such as active fiscal policy with passive monetary policy. That alternative is known to produce its own problems in economic dynamics with NK models, but not Shilnikov chaos. Among the many other alternatives in the literature is the targeting of a Divisia monetary aggregate, in accordance with such proposals as those in Belongia (1996), Serletis (2013), or Belongia and Ireland (2014 and as advocated by Lucas (2000, p. 270) in measuring welfare loss from inflation. 39 By removing the Taylor rule from the NK model, such alternative approaches could prevent chaotic dynamics from occurring. A long literature exists on alternatives to Taylor Rule policies. See, e.g., Cochrane (2011).
(b) A Taylor rule with interest rate feedback could continue to be used, but with the resulting Shilnikov chaos controlled through the use of a second policy instrument applied in accordance with the policy procedures advocated by engineers in the literature on controlling chaos. We find that the Ott, Grebogi and Yorke (1990) algorithm could be particularly well suited to that objective.
In subsequent research, we plan to explore robustness of our conclusions to different Taylor rules and to alternatives to the Ricardian fiscal policy laws. 40 But we do not expect fundamental changes in our conclusions, which are systems theory properties of New Keynesian macroeconomic dynamics, when augmented by the closed loop interest rate feedback rules, characterizing all Taylor rules.
Let J denote the Jacobian matrix of system M, evaluated at the long-run equilibrium, and let starred values denote steady-state state levels. Simple algebra leads to the following ( ) matrix, By relations in (1) and (4.b), and since , we know that . The eigenvalues of J are the solutions of the characteristic equation where I is the identity matrix and where Under the active monetary policy ( ( ) ) , we now determine the sign of the real parts of the eigenvalues with active or passive fiscal policy. Consider first the case in which the fiscal policy is passive, implying ̄ ( ) . In this case, ( ) . Consider instead the case of active fiscal policy, implying ̄ ( ) . Then, both ( ) and ( ) are positive. In this case, irrespective of the sign of ( ), we have one eigenvalue with negative real part and two eigenvalues with positive real parts: is a saddle of index 2, and the equilibrium is locally unique.
We now consider the local stability properties of system M in the neighborhood of , when monetary policy is active. Assume that the monetary policy is active. Then, if the fiscal policy is also active, is a repellor, and there are no equilibrium paths besides the steady-state itself. Conversely, if the fiscal policy is passive, is a saddle of index 2, and the equilibrium is locally unique. ∎

Appendix 2: Proof of Proposition 2.
Assume that the conditions in Lemmas 1 and 2 are satisfied. If we start in the neighborhood of the origin, we know from Theorem 1 that, in the phase space of system (22), the solution trajectories are bounded to evolve forever in the neighborhood of the origin and are therefore valid equilibria. By construction, the results obtained for system (21) also apply to the original system of differential equations, M. ∎ Appendix 3: Proof of Corollary 1.
Recall that ( ) is the predetermined variable of the system, and that ( ) and ( ) are jump variables. If we choose ( ) to belong to the tubular neighborhood of the homoclinic orbit, as defined above, we know that there must be a continuum of possible choices of ( ) and ( ), capable of giving rise to recurrent paths, which are bound to stay forever within . Since all these recurrent paths are bound to stay in a neighborhood of , such that the neighborhood can well exceed the small neighborhood valid for the local analysis, we have in fact global indeterminacy of the equilibrium. ∎

Appendix 4: Proof of Lemma 3.
The algorithm for proving controllability of a given system requires that the nonlinear system be written in state-space notation. We first put the linear part of system (20)  Where, as in system M, the parameter A has been set to one. The calibration of this economy presents several complications. Since we have labor in the production function, we are faced with the calibration of the parameter θ in a constant-returns-to-scale setting. This is problematic since, to the best of our knowledge, that has not previously been done with Cobb-Douglas technology. Therefore, we proceed according to the following steps. We first use θ as a free bifurcation parameter and numerically identify the values at which the chaotic attractor is found. Then, we discuss plausibility of these values with regard to the existing econometric evidence (cf., footnote 28).
Consider the following Example. ). We therefore look for values of the free parameter θ such that the steady-state is a saddle-focus equilibrium. We find that this requires θ (0.9325, 1). Re-running the algorithm in search of a chaotic attractor, we find that the intended steady state in system S satisfies conditions of Theorem 1 for all ( ). Set therefore as in Example 3. Then, given a triplet of the initial conditions ( ( ) ( ) ( )) sufficiently close to the origin, system S admits perfect-foresight chaotic equilibrium solutions. Therefore, by Example 5, when money is productive, the onset of a chaotic attractor in a sticky-price NK model with a Ricardian fiscal policy cannot be excluded. Comparing the regions of the parameter space, where the intended steady-state is a saddle-focus equilibrium with a positive saddle quantity in systems M and S, we find the following.
Remark 5. Either when money enters the utility function or the production function, the accompanying coefficient must be sufficiently low for the intended long-run equilibrium to be a saddle-focus with a positive saddle quantity. Specifically, given the parameters in the examples above, we have 0.0901 and 0.0675, for money in utility or in production, respectively. 42 Moreover, to verify the robustness of the results to changes in the parameters, we parallel the analysis of Sub-Section 3.2 and study the surface ( ) ( ) for the money in the production function case. Setting ̄, set θ = 0.95 and using we obtain Figure 6. The combinations of the remaining ( ̄ ) policy parameters, such that the intended steady-state of system S is a saddle-focus equilibrium with no saddle-quantity, are represented. Above the surface, the saddle quantity is positive. Below the surface, the saddle quantity is negative. Figure 6 matches up well with the analogous Figure 1: to have a saddle-focus equilibrium with a positive saddle-quantity, in presence of an active monetary policy ( ), the pair ( ̄ ) can assume plausibly low values.
We now investigate whether the chaotic motion arising from system S can be OGY-controlled.
Recalling that the control parameter must be accessible to the Central Bank, stabilizing chaotic solutions implies the search for long-run values of the policy parameters ̄ and π, such that all the eigenvalues of the Jacobian associated with system S have a negative real part.
Consider, therefore, the following numerical example, where we first take ̄ as the control parameter.
Example 6. Set ̄ and θ = 0.95 as in the preceding Example 5. Consider the case of ( ) ( ) and . Then, as shown in Example 5, if ̄ the intended steady state of system S is a saddle-focus equilibrium with a positive saddle quantity. First, we check that system S is controllable. Then, using ̄, as the bifurcation parameter, we see that the Jacobian of S, evaluated at the intended steady-state, has three eigenvalues with negative real parts for any ̄ belonging to the interval ̄ ( ). Suppose now the policy-maker announces a commitment to ̄ ̄.
Then, by the OGY algorithm, the economy supersedes irregular and cyclical behavior and approaches the intended steady state.
Notice that, contrary to the money in utility function model, the policy-maker's commitment must be to lower the steady state nominal interest rate. 43 Again, as discussed at the end of Section 4, what is ultimately required for the economy to be stabilized is that the monetary impulse be positive for real economic activity. In the present case of productive money, the comparative statics invariably implies a positive partial derivative between c (and therefore y) and money holdings. Therefore, a commitment to a reduction of the long-run interest rate immediately raises money balances.