In crisis-induced intermittency, the systems switch between weakly chaotic and strongly chaotic behaviors Grebogi, Ott and Romeiras There are many examples of experimental observations of chaos-driven intermittency. For example, Hayashi, Ishizuka and Hirakawa observed a transition from order to chaos via type-I Pomeau-Manneville intermittency in the onchidium pacemaker neuron. Ditto et al. Stable and unstable periodic orbits are the basic elements of complex dynamical systems, and are the key to explain the origin and nature of chaos-driven intermittency.
A complex system consists of order and chaos; order is governed by stable periodic orbits, whereas chaos is governed by unstable periodic orbits. In particular, unstable periodic orbits are the skeleton of chaotic attractors and chaotic saddles Auerbach et al. There is experimental evidence of unstable periodic orbits, chaotic transients and chaotic saddles. For example, Schief et al. Chaotic transients and chaotic saddles are fundamental to the understanding of complex economic dynamics. Lorenz observed chaotic transient motion in a Kaldorian model of business cycles.
Apart from the works by Lorenz and Lorenz and Nusse , most economic literature and books on complex economic dynamics Puu , Chiarella , Zhang , Benhabib , Brock, Hsieh and LeBaron , Rosser , Medio , Day , , Thomas, Reitz and Samanidou have only dealt with chaotic attractors, paying no attention to chaotic transients and chaotic saddles. In Chapter 2, a forced van der Pol oscillator model of economic cycles is formulated as the prototype model to describe the complex economic dynamics. This type-I economic intermittency arises from a local bifurcation known as the saddle-node bifurcation. An economic path evolves from a periodic to an aperiodic pattern when the exogenous forcing amplitude passes a critical value whereby the system loses its stability due to a saddle-node bifurcation.
The power spectrum of the type-I intermittent time series is broadband and displays power-law behavior at high frequencies, similar to the real data of foreign exchange and stock markets. The characteristic intermittency time, measuring the average duration of quiescent periods in the intermittent economic time series, is a function of the exogenous forcing 6 1 Introduction amplitude. The scaling law of the characteristic intermittency time is useful for forecasting the turning points of nonlinear economic cycles.
In Chapter 4, a new type of crisis-induced intermittency in nonlinear economic cycles is discussed. It is shown that after an economic system undergoes a global bifurcation known as attractor merging crisis, the system has the ability to keep the memory of its weakly chaotic state before crisis.
As the system moves away from the crisis point, it becomes more chaotic, consequently the discrete spikes of the power spectrum become less evident due to increasing multiscale information transfer in the complex economic systems.
The exponent of the scaling law of the characteristic intermittency time of the crisis-induced economic intermittency is much larger than that of the type-I economic intermittency. In Chapter 5, an attractor merging crisis in chaotic economic cycles is characterized. Symmetry is a common property of complex systems that exhibit attractor merging crisis. The analysis is performed in a complex region within a periodic window of the bifurcation diagram determined from the numerical solutions of a forced oscillator, where a saddle-node bifurcation marks the beginning of the periodic window.
As the exogenous forcing amplitude increases after the saddle-node bifurcation, two coexisting periodic attractors of period-1 undergo a cascade of period-doubling bifurcations leading to two weakly chaotic attractors. An attractor merging crisis occurs when two coexisting weakly chaotic attractors merge to form a single strongly chaotic attractor, which marks the end of the periodic window.
The onset of attractor merging crisis is due to the head-on collision of the pair of coexisting weakly chaotic attractors, respectively, with a pair of mediating unstable periodic orbits of period-3 and their associated stable manifolds. In addition, it is demonstrated that the two coexisting weakly chaotic attractors also collide with the boundary of the basins of attraction that separates the two weakly chaotic attractors. The aim of Chapter 6 is to perform an in-depth study of unstable periodic orbits and chaotic saddles in complex economic dynamics.
In 1 Introduction 7 particular, the roles of unstable periodic orbits and chaotic saddles in crisis and intermittency in complex economic systems are investigated. The technique of numerical modeling is applied to characterize the dynamics and structure of unstable periodic orbits and chaotic saddles within a periodic window of the bifurcation diagram, at the onset of a saddle-node bifurcation and of an attractor merging crisis, as well as in type-I intermittency and crisis-induced intermittency, of a forced oscillator model of economic cycles.
The links between chaotic saddles, crisis and intermittency in complex economic dynamics are analyzed. The conclusion is given in Chapter 7. By introducing an exogenous driver, the forced van der Pol equation can be adopted as a prototype model for complex economic dynamics. Numerical solutions of this model can elucidate the fundamental properties of complex economic systems which exhibit a wealth of nonlinear behaviors such as multistability as well as coexistence of order and chaos. Unstable periodic orbits are the skeleton of chaotic attractors in complex economic systems.
Actual economic time series are rarely characterized by regular periodic, sinusoidal dynamics typical of linear systems. Empirical evidence of complex behaviors of nonlinear deterministic systems can be obtained by calculating statistical quantities such as Lyapunov exponents, entropies, fractal dimensions, and correlation dimensions. In practice, large amount of data points are often unavailable in macroeconomic time series. This imposes severe limitation on the accuracy of nonlinear analysis of economic data.
In view of this limitation, additional tests are desirable. Brock performed a test for chaos in detrended quarterly US real GNP data from to by calculating the correlation dimension and the largest Lyapunov exponent and applying an additional residual test, and concluded that chaos should be excluded in the GNP data. Barnett and Chen examined several monetary aggregates and found positive values for the largest Lyapunov exponents in some of their data, which provides evidence of chaos. Sayers calculated the correlation dimension and the Lyapunov exponents and applied the additional residual diagnostics to U.
Further literature survey on empirical evidence of nonlinearity and chaos in economical data will be given in the remaining chapters of this monograph. The complex behaviors of nonlinear economic systems restrict the use of purely analytical methods to investigate nonlinear economic models. In contrast to nonlinear analysis of economic data which are restricted by the small sample size as well as noise, numerical modeling of economic systems can provide large sample size required to characterize chaotic behaviors, and determine the dynamical behaviors of economic systems in the absence and in the presence of noise.
Economic models can be formulated by either discrete-time or continuous-time approaches Puu ; Lorenz The main reason for taking the discrete-time approach is the relative facility to handle these models, without the need of heaving computation. For example, Stutzer characterized the qualitative dynamics of a discrete-time version of a nonlinear macroeconomic model, which shows complex periodic and random aperiodic orbit structures.
Xu et al. New econometric techniques emerged recently permit direct empirical testing of continuous-time economic models. In this monograph, the continuous-time approach will be adopted. Additional literature survey on nonlinear economic models will be discussed in the remaining chapters of this monograph. By noting that the linear forms of I Y and S Y fail to produce cyclical motions, Kaldor proposed a S-shaped sigmoid nonlinear form for I Y and a mirror-imaged S-shaped nonlinear form for S Y Gabisch and Lorenz , which yields oscillatory motion of business cycles.
Lorentz b and Lorenz and Nusse considered the following generalization of equation 2. The modern economy consists of a great variety of separate sectors and activities closely coupled to each other. For example, Puu showed that the forced Van der Pol equation similar to equation 2. In this monograph, we will investigate the numerical solutions of the forced van der Pol model of business cycles, equation 2. In the presence of an exogenous forcing, equation 2. Parlitz and Lauterborn gave examples of the bifurcation diagrams of equation 2. For large driving amplitudes, they found that many periodic, quasiperiodic and chaotic attractors coexist.
A systematic analysis of equation 2. Xu and Jiang performed a global bifurcation analysis of equation 2. They studied the evolution of the global structures in simple and complex transitional zones, and the number of coexisting attractors in overlaps of mode-locking subzones. In this chapter, we use the numerical solutions of equation 2. In an ordered dynamical system, for arbitrary initial conditions, after going through a transient period the system approaches a periodic behavior with a predictable periodicity.
In contrast, a chaotic dynamical system exhibits behavior that depends sensitively on the initial conditions, thereby rendering long-term prediction impossible Strogatz Figure 2. Since a complex system 2. Periodic and chaotic time series. Periodic attractor and chaotic attractor. When the attractor is an isolated closed trajectory, it is called a periodic attractor or limit cycle ; when an attractor is a fractal set of points, it is called a strange attractor or chaotic attractor Ott Figures 2.
The average rate of divergence can be measured by the Lyapunov exponents Ott Bifurcation diagram and maximum Lyapunov exponent: global view. As a control parameter varies, the stability of a dynamical system changes due to a local or a global bifurcation. The bifurcation diagram provides a general view of the system dynamics by plotting a system variable as a function of a control parameter Alligood, Sauer and Yorke Bifurcation diagram and maximum Lyapunov exponent: periodic window.
SNB denotes saddle-node bifurcation, MC denotes attractor merging crisis. The phase space of equations 2. For the remaining two exponents, for a stable periodic orbit the maximum Lyapunov exponent is less than zero, for a quasiperiodic orbit the maximum Lyapunov exponent is zero, whereas for a chaotic orbit the maximum Lyapunov exponent is greater than zero. Due to the symmetry of equation 2. The same is true for attractors A3 and A4. The rich dynamics found in this periodic window demonstrates the basic features of multistability and coexistence of order and chaos in complex economic systems.
The basin of attraction for a given attractor is the set of initial conditions each of which gives rise to a trajectory that converges asymptotically to the attractor Hilborn For the initial conditions starting from the light gray 22 2 Nonlinear Dynamics of Economic Cycles Fig. Basins of attraction: multistability. Attractor A0 light gray , attractor A1 dark gray , attractor A2 white. This dramatic change is due to the destruction of the chaotic attractor A0 and its basin of attraction by a boundary crisis.
In contrast to a periodic attractor whereby all trajectories initiated from any point in the state space are attracted to a stable periodic orbit e. Hence, a chaotic trajectory is chaotic because it must weave in and around all of these unstable periodic orbits yet remain in a bounded region of state space Hilborn Unstable periodic orbits can be numerically found by the Newton algorithm Curry Note that the stable and unstable periodic orbits are identical at the onset of a saddle-node bifurcation. The unstable periodic orbits are robust.
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Unstable periodic orbits: skeleton of chaotic attractor. Unstable periodic orbit: stable and unstable manifolds. Trajectories on the in-set converge to the saddle point as time goes on, while trajectories on the out-set diverge from the saddle point as time goes on Hilborn We showed that a complex economic system exhibits multistability behavior with coexistence of attractors, including the possibility of coexistence of order and chaos periodic attractors and chaotic attractors.
In addition, we showed that unstable periodic orbits are the skeleton of a chaotic attractor. Numerical simulations show that after an economic system evolves from order to chaos, the system keeps its memory before the transition and its time series alternates episodically between periods of low-level apparently periodic quiescent and high-level turbulent bursting activities.
Thus, two fundamental attributes of business cycles are: comovement i. Periodic ordered solutions appear when coupled oscillators are phaselocked due to phase synchronization; moreover, phase synchronization can occur in coupled chaotic oscillators Strogatz Selover et al. Financial markets also exhibit intermittent behavior wherein periods of trading frenzy are followed by periods of quiescence; on closer examination the periods of high volatility are themselves consisted of other sub-periods of relative quiet and other sub-periods of relative bursty activities, which is a manifestation of self-similar and scale-invariant properties of nonlinear systems.
Recent statistical analysis of high-frequency data of stock markets and foreign exchange markets have demonstrated the intermittent nature of nonlinear economic time series, which present non-Gaussian behavior in the probability distribution function of price changes and power-law behavior in the spectral density Mantegna and Stanley , ; Ghashghaie et al. There is an increasing interest in applying chaos concept to study nonlinear economic dynamics. Sengupta and Sfeir performed empirical tests of volatility for monthly data of exchange rates from February to August , and concluded that chaotic instability cannot be ruled out in general.
Fernandez-Rodriguez et al. Muckley presented evidence of strange attractor, a long-term memory 3. Intermittency is readily found in nonlinear models of economic dynamics Mosekilde et al. In this chapter, we study an example of economic type-I intermittency based on a model of nonlinear business cycles Chian et al.
We will show by numerical simulations that after a transition from order to chaos due to a saddle-node bifurcation, the time series of business cycles becomes intermittent involving episodic regime switching between quiescent and bursting phases. The characteristic intermittency time will be calculated and its application for economic forecasting will be discussed. The gray white regions denote the basins of attraction of A1 A2. Equation 3. In a nonlinear system, the natural frequency of oscillations changes with the variation of the control parameters.
Hence, in this economic model the dynamical behavior of nonlinear business cycles depends on the competition between 3. The system is phase-locked synchronized if the ratio of these two frequencies is a rational number; its associated solution is then periodic. After the phase-locked solution is destroyed in a saddle-node bifurcation, the solution becomes chaotic. Type-I intermittency results from the transition from order to chaos via a saddle-node bifurcation Strogatz Within the periodic window, two or more coexisting attractors A1 and A2 are found.
To the left of aSN B in the bifurcation diagram, the initial conditions converge to a chaotic attractor A0. Due to the symmetry of equation 3. Figure 3. Note that for values of a between 0. For a 3. This implies that after the transition from order to chaos, the regime switching of intermittent business cycles becomes more frequent as the system moves farther away from the transition point.
Figures 3. The characteristic intermittency time, namely, the average duration of laminar phases in the intermittent time series, depends on the value 36 3 Type-I Intermittency in Nonlinear Economic Cycles of the control parameter a. Close to the transition point aSN B the average duration of laminar phases is relatively longer, and decreases as a moves away from aSN B. The squares circles denote the computed average duration of the laminar phases related to A1 A2.
Note that the circles and the squares coincide most of the time, due to the symmetry of A1 and A2. As an economic system moves farther away from the transition point, the average duration of quiescent periods decreases. In order to understand the nature of economic intermittent behaviors, we performed a study of type-I intermittency in a nonlinear model of business cycles.
In this example of intermittency, an economic path evolves from a regular periodic to an irregular chaotic pattern as the exogenous forcing amplitude a passes a critical value aSN B , where the system loses its stability due to a saddle-node bifurcation. The accuracy of business cycle forecasting relies on a precise estimate of the durations of economic expansions and contractions and of the turning points in business cycles Vilasuso ; Schnader and Stekler ; Diebold and Rudebusch Nonlinear modeling of economic 3.
In particular, the average duration of quiescent phases in business cycles can be determined from the characteristic intermittency time of the simulated time series. Some interesting connections can be made between our results and other papers discussed in the present work. For example, Vilasuso employed nonparametric turning-point tests to investigate the duration of economic expansions and contractions in the United States, which indicated evidence of a turning point to longer expansions in Our work adopted a nonlinear model of business cycles to simulate the duration of expansions and contractions of an open economy driven by a global market, which can be used to predict the turning point to a long period of economic expansions of a nation, such as detected by Vilasuso Moreover, type-I intermittency studied in this chapter demonstrates the ability of a chaotic enonomic system to retain the memory of its system dynamics in the ordered regime.
Following a merging crisis, a complex economic system has the ability to retain memory of its weakly chaotic dynamics prior to crisis. The characteristic intermittency time, useful for forecasting the average duration of contractionary phases and the turning point to the expansionary phase of business cycles, is computed from the simulated time series.
The spectral density of intermittent economic time series indicates power-law behavior typical of mutiscale systems. Kirikos compared a random walk with Markov switching-regime processes in forecasting foreign exchange rates; the results suggested that the availability of more past information may be useful in forecasting future exchange rates. Resende and Teixeira assessed long-memory patterns in the Brazilian stock market index Ibovespa for periods before and after the Real Stabilization Plan, and obtained evidence of short memory for both periods.
Gil-Alana presented evidence of memory in the dynamics of the real exchange rates in Europe using the fractional integration techniques. Intermittency is ubiquitous in chaotic economic systems. In a nonlinear macroeconomic model Mosekilde et al. In a disaggregated economic long wave model describing two coupled industries Haxholdt et al. In a model of an economic duopoly game Bischi et al. An example of type-I intermittency in nonlinear business cycles was studied recently Chian et al. In the economic type-I intermittency, the recurrence of regime switching between bursty and laminar phases indicates that a nonlinear economic system is capable of keeping the memory of its ordered dynamics after the system evolves from order to chaos due to a local saddle-node bifurcation.
Most econometric studies of long memory treat economic data as stochastic processes Granger and Ding ; Resende and Teixeira ; Gil-Alana , however real economic systems are a mixture of stochastic and deterministic processes. In this chapter, we adopt the deterministic approach to study a new type of economic intermittency induced by an attractor merging crisis due to a global bifurcation Chian et al.
A forced model of nonlinear business cycles is formulated in Section 4. Economic crisis-induced intermittency is analyzed in Section 4. Concluding comments are given in Section 4. Equation 4. This symmetry is a typical property of dynamical systems that exhibit attractor merging crises Chian et al. MC denotes attractor merging crisis and SNB denotes saddlenode bifurcation. These changes in the system dynamics can be represented by the bifurcation diagram. A periodic window of the bifurcation diagram determined from the numerical solutions of equation 4.
Within the periodic window, two or more attractors A1 and A2 coexist, each with its own basin of attraction Chian et al.
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Note that CA1 and CA2 are symmetric with respect to each other. In fact, the 4. At the crisis point, each of the two small chaotic attractors simultaneously collide head-on with a period-3 mediating unstable periodic orbit on the boundary which separates their basins of attraction, leading to an attractor merging crisis due to a global bifurcation Chian et al.
After the attractor merging crisis, there is only one large chaotic attractor MCA in the system. The characteristic intermittency time, namely, the average duration of the laminar phases in the intermittent time series, depends on the value of the control parameter a. In the vicinity of the crisis point aM C the average time spent by a path in the neighborhood of pre-crisis CA1 4.
Figure 4. The squares circles denote the computed average time of the laminar phases related to CA1 CA2. Note that circles and squares coincide most of the time, as expected from the symmetry of CA1 and CA2. The squares circles denote the computed average switching time from the laminar phases related to CA1 CA2 to the bursty phases.
Comparing with equation 4. Chaos and nonlinear methods provide powerful tools to achieve this goal. For example, Bajo-Rubio et al. S dollar spot exchange rates using a nonlinear deterministic technique of local linear predictor. This technique may be applied to economic forecasting. In particular, the determination of intermittent features in the modeled economic chaotic attractors, aided by the recognition of regions of high predictability in the chaotic attractors Ziehmann et al. Recurrence of unstable periodic structures is a manifestation of the memory dynamics of complex economic systems.
The characteristic intermittency time given by the scaling relation, equation 4. Modeling of nonlinear economic dynamics enables us to obtain an in-depth knowledge of the nature of regime switiching and memory, in particular, their relation with each other. Econometric literatures on regime switching Kirikos ; Bautista ; Kholodilin and long memory Granger and Ding ; Resende and Teixeira ; Gil-Alana ; Muckley have evolved largely independently, as the two phenomena appear distint. As an economic system evolves, 50 4 Crisis-Induced Intermittency in Nonlinear Economic Cycles microeconomic and macroeconomic instabilities lead to a variety of global and local bifurcations which in turn give rise to chaotic behaviors such as crisis-induced and type-I intermittencies.
In particular, an attractor merging crisis due to a global bifurcation is analyzed using the unstable periodic orbits and their associated stable and unstable manifolds. Characterization of crisis can improve our ability to forecast sudden major changes in economic systems. Complex systems approach provides a powerful tool to monitor and forecast the nonlinear dynamics of business cycles.
For example, Mosekilde et al. Bifurcation diagram of x as a function of a for: a attractors A1 and A3 , b attractors A2 and A4. Puu and Sushko employed a multiplieraccelerator model of business cycles, including a cubic nonlinearity, to study a number of bifurcation sequences for attractors and their basins of attraction. Crisis is a global bifurcation resulting from the collision of a chaotic attractor with a mediating unstable periodic orbit or its associated stable manifold Grebogi, Ott and York ; Grebogi et al.
In this chapter, we show that an attractor merging crisis appears in a forced van der Pol oscillator model of nonlinear business cycles Chian et al. The onset of an attractor merging crisis is characterized using the tools of unstable periodic orbits and their associated stable and unstable manifolds.
In the presence of exogenous forcing, either periodic orderly or aperiodic chaotic solutions appear when we vary any of three control parameters: 5. The VDP equation 5. This symmetry is a typical property of dynamical systems that exhibit attractor merging crises Grebogi et al. The rich dynamical states displayed by the bifurcation diagram indicate that a dynamical system is sensitively dependent on a small variation of its control parameters. Figure 5. Attractor A3 A4 is created by a saddlenode bifurcation, where a pair of p-3 stable solid lines and unstable dashed lines periodic orbits is generated.
Crisis diagram depicting the system dynamics as the control parameter a varies. Saddle-node bifurcations SNB occur at a certain value of a, creating two coexisting attractors A1 and A2 , which via a cascade of period-doubling bifurcations evolve into two chaotic attractors CA1 and CA2. Multistability is a basic feature of complex dynamical systems whereby two or more attractors can coexist for a given value of the control parameter.
The merged attractor after crisis is larger than the union of the two attractors before crisis. The 5. Unstable periodic orbit UPO plays a key role at the onset of attractor merging crisis. We numerically determine UPO from the numerical solution of equation 5. Analysis shows that the mediating p-3 unstable periodic orbits M , evolved from the saddle-node bifurcations at the birth of A3 A4 , are responsible for the attractor merging crisis. This is a manifestation of the symmetry property of the VDP equation 5. Figures 5. Our numerical calculations render support to the conjecture of Parker and Chua Parker and Ott Ott that a chaotic attractor contains the unstable manifolds of every UPO of the chaotic attractor.
Mathematical modelling of crisis can deepen our understanding of sudden major changes of economic variables often encountered in business cycles. The techniques developed in this chapter for crisis characterization e.
Attractor merging crisis appears in systems with symmetry such as equation 5. This type of crisis is absent when the system symmetry is broken. However, other types of crisis phenemena such as boundary crisis Chian, Borotto and Rempel and interior crisis Borotto, Chian and Rempel can be found in asymmetric systems such as the asymmetric van der Pol equation Engelbrecht and Kongas , and are in fact present in the solutions of equation 5.
The techniques developed in this chapter can be readily applied to characterize boundary and interior crises. Hence, crises and global bifurcations are ubiquitous in either symmetric or asymmetric nonlinear economic systems. Chapter 3 showed that saddle-node bifurcation is a route from order to chaos, leading to a chaotic dynamical behavior known as type-I intermittency.
Chapter 5 analyzed an attractor merging crisis in chaotic business cycles which leads to a transition from weak chaos to strong chaos; the strong chaos exhibits a dynamical behavior known as crisis-induced intermittency, as seen in chapter 4. A set of unstable periodic orbits can be chaotic and nonattracting so that the orbits in the neighborhood of this set are eventually repelled from it; nonetheless, this set can contain a chaotic orbit with at least one positive Lyapunov exponent Nusse and York If the chaotic orbit has also one negative Lyapunov exponent the nonattracting set is known as a chaotic 62 6 Chaotic Transients in Nonlinear Economic Cycles Fig.
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Chaotic saddle in bifurcation diagram. Both chaotic saddles and chaotic attractors are composed of unstable periodic orbits. Figure 6. As seen in chapter 2, within this periodic window, two or more attractors can coexist. To plot the chaotic saddle, for each value of the control parameter a, we plot a straddle trajectory close to the chaotic saddle using the PIM triple algorithm Nusse and Yorke , Rempel et al.
As we increase a, the pair of period-1 stable periodic orbits undergoes a cascade of periodic-doubling bifurcations which leads to the formation of a pair of weakly chaotic attractors localized in two separate bands in the bifurcation diagram. Trajectories started 6. The transient time is related to the structure of SCS and its manifolds. Like a saddle point, chaotic saddles possess a stable and an unstable manifold.
The stable manifold of a chaotic saddle is the sets of points that converge to the chaotic saddle in forward time, and the unstable manifold is the sets of points that converge to the chaotic saddle in the time reverse dynamics Nusse and Yorke The closer an initial condition is to the stable manifold, the longer its transient time. Figures 6. Time series of chaotic transient leading to periodic attractor. The empty space between the intersection points along the unstable direction is the origin of the gaps in the chaotic saddle.
Inside the periodic window the gaps of the chaotic saddle are empty in the sense that they do not contain unstable periodic orbits, only nonrecurrent points whose orbits converge very quickly to the small neighborhood of the period-1 attractors Robert et al. Thus, inside the periodic window the surrounding chaotic saddle plays the role of chaotic transient motion before converging to the attractor.
Time series of chaotic transient leading to chaotic attractor. Time series and power spectrum of type-I intermittency. Note that there are 6. Chaotic attractor and chaotic saddle in type-I intermittency. Although the pair of period-1 saddle points M appear only after the saddle-node bifurcation, the system keeps the memory of these saddle points even prior to the occurrence of the saddle-node bifurcation.
In other words, all orbits of the chaotic attractor mimic synchronize with these period-1 unstable periodic orbits M when they come to their neighborhood Kaplan Chapter 5 showed that at aM C an attractor merging crisis occurs due to the collision of two coexisting weakly chaotic attractors CA1 and CA2 with a pair of mediating unstable periodic orbits of period-3 and their associated manifold, which coincides with the boundary of the basins of attraction dividing the two weakly chaotic attractors. As the result of this crisis, two small chaotic attractors combine to form a single large chaotic attractor to the right of aM C.
The stable manifold of the 6. Attractor merging crisis. The stable manifold SM of the mediating saddle divides the merged chaotic attractor into the band region and the surrounding region. Similarly, the surrounding chaotic saddle SCS gray is found by a straddle orbit that never enters the band regions. Actually, the merged chaotic attractor is larger than the union of the surrounding and banded chaotic saddles, since the gaps in the post-crisis chaotic 6.
Chaotic attractor and chaotic saddle in crisis-induced intermittency. Coupling unstable periodic orbits. This set of unstable periodic orbits within gaps, called coupling orbits with components in both band and surrounding regions, are responsible for the coupling between these two regions. Before crisis, for a less than aM C , trajectories on the banded chaotic attractor never abandon the band region. Time series and power spectrum of crisis-induced intermittency. Once inside the surrounding region, the trajectory moves to the neighborhood of the surrounding chaotic saddle SCS. After some time, the trajectory is injected back to the band region.
This process of switching between the band and surrounding regions repeats intermittently. Right after crisis, the coupling orbits created by the explosion have very long period with o et al. However, as the control parameter a is increased further away from the crisis point aM C , shorter coupling orbits are created. This time series alternates episodically between the laminar periods 76 6 Chaotic Transients in Nonlinear Economic Cycles associated with the two banded chaotic saddles and the bursting periods associated with the surrounding chaotic saddle.
The transition between the laminar and bursting periods is linked by the coupling unstable periodic orbits. These unstable structures are the origin of intermittency in nonlinear economic models. We showed that the attractor merging crisis in complex economic systems is due to a chaotic attractor-chaotic saddle collision, whereby two weakly chaotic attractors combine to form a large chaotic attractor.
After the crisis, the pair of pre-crisis weakly chaotic attractors are converted into a pair of banded chaotic saddles. The post-crisis chaotic attractor is composed of the surrounding chaotic saddle, two banded chaotic saddles and coupling unstable periodic orbits in the gap regions which act as the link between the surrounding chaotic saddle and the banded chaotic saddles.
Complex Systems Approach to Economic Dynamics
Characteristic intermittency time, which measures the average duration of the laminar phases of either type-I or crisis-induced economic intermittency, can be calculated from the numerically simulated time series. This result can be useful for forecasting the turning point from bust to boom phases in business cycles. It is important to point out that although we have selected the van der Pol model for its mathematical simplicity and its wide interest in economics, in view of the universal mathematical properties of nonlinear dynamical systems, the dynamical characteristics investigated in this simple model is actually applicable to other more sophisticated economic scenarios.
We succeeded in characterizing the anatomy of a complex economic system by classifying its structure and dynamics. In particular, we showed that unstable periodic orbits are the building blocks of chaotic saddles and chaotic attractors; moreover, chaotic saddles are embedded in a chaotic attractor and are responsible for the transient motion preceding the convergence to an attractor periodic or chaotic.
In terms of the system dynamics our results show that, as the control parameters are varied, a complex economic system undergoes a variety of dynamic transitions which change its stability properties, namely, local bifurcations such as period-doubling bifurcation, saddle-node bifurcation and 78 7 Conclusion Hopf-bifurcation, and global bifurcations such as boundary crisis, interior crisis and attractor merging crisis.
Economic systems are unstable by nature, dominated by instabilities driven by both endogenous and exogenous forces. This very unstable nature of economic dynamics is clearly manifested by the unstable structures, such as unstable periodic orbits and chaotic saddles, inherent in chaotic economic systems. Recently, there is a surge of interest on the relevance of these unstable structures in economic dynamics. They showed that in addition to chaotic attractors, this model can possess coexisting attracting periodic orbits or simple attractors, which imply the emergence of transient chaotic motion chaotic saddles.
They applied straddle methods to numerically analyze this model in order to detect compact invariant sets which are responsible for the complexity of the transient motion, and concluded that chaotic saddles are prevalent in nonlinear economic models. These unstable periodic orbits not only look similar in shape to the chaotic attractor, there is a correspondence between the unstable periodic orbits and the chaotic attractor in terms of their statistical properties such as means, variances, Lyapunov exponents and probability density functions.
Their results indicate that both statistical and dynamical features of a chaotic attractor in complex economic systems are captured by just a few unstable periodic orbits, in agreement with the periodic orbit theory of dynamical systems of Auerbach et al. This monograph renders strong support for the conclusions, that unstable periodic orbits and chaotic saddles are essential elements of complex economic systems, of Lorenz and Nusse and Ishiyama and Saiki We demonstrated that intermittency is an intrinsic behavior of a chaotic economic system by analyzing in detail two examples of economic intermittency due to a local or a global bifurcation, namely, type-I intermittency and crisis-induced intermittency, respectively.
The former 7 Conclusion 79 is generated by a saddle-node bifurcation, the latter is generated by a crisis phenomenon such as the attractor merging crisis. For example, the anticipation of the turning points is fundamental for forecasting business-cycle recessions and recoveries for countries showing asymmetric cycle durations Garcia-Ferrer and Queralt Modeling of intermittency in nonlinear economic cycles can provide an estimate of the average duration of the contractionary phases of economic cycles and predict the turning points to expansionary phases.
The classical NBER model of leading economic indicators was built solely within a linear framework which is inadequate for predicting the complex behavior of business cycles. By combining the complex system approach such as chaotic theory developed in this paper and the intelligent system approach such as neural network , a superior performance for forecasting business cycle can be obtained relative to the classical model Jagric The techniques developed in this monograph can be readily applied to the study of chaos and complexity in management systems such as 80 7 Conclusion logistics and supply chain management Mosekilde and Larsen ; Sosnovtseva and Mosekilde , organizational dynamics and strategic management Senge ; Stacey , public policy and public administration Kiel In fact, economic dynamics is a result of complex interactions of economical, political, social, climate, environmental and technological systems.
Nonlinear models of solar cycles, climate, and ecological systems indicate that these natural systems exhibit chaotic behaviors. Chian et al. Numerical modeling based on complex systems approach may be useful for the development of these emissionstrading markets, by assisting society to better understand the complex coupled energy-climate-environment system and assist policymakers to identify and implement optimal policies for managing the risks related to climate change. The sensitive dependence of a dynamical system on small variations of its parameters can be used to control the chaotic behavior of a system by applying a small perturbation Ott, Grebogi and Yorke , which can be useful for stabilizing economic systems and optimizing management policies.
Lai and Grebogi showed that chaotic transient can be converted into sustained chaos by feedback control. There is 7 Conclusion 81 evidence of chaos control in laboratory and numerical experiments. Lopes and Chian showed that chaos in a coupled three-wave system, resulting from period-doubling bifurcations and type-I intermittency, can be controlled by applying a small wave with appropriate amplitude and phase.
Kaas used the chaos control technique to show that the government can in principle stabilize an unstable Walrasian equilibrium in a short time by varying income tax rates or government expenditures. In this monograph, we only considered economic systems which are of low-dimension and varying only in time, described by ordinary differential equations. In many areas of economics and management, we must deal with dynamical systems which are of high-dimension and varying both in space and time.
Some recent papers have demonstrated that nonlinear phenomena such as chaotic saddles, crisis, type-I intermittency and crisis-induced intermittency, observed in low-dimensional dynamical systems appear also in high-dimensional spatiotemporal dynamical systems Chian et al. Note, however, that uncertainty always plays a role in the economy, therefore a real economic system consists of both deterministic and stochastic dynamics Hommes Many of the traditional techniques being used by economists for modeling economic dynamics are based on linear approaches which are only valid near the equilibrium, and many of the tools being used by the investment professionals are based on the assumption that the asset returns have Gaussian distribution.
In reality, the economic dynamics is often highly nonlinear and far away from the equilibrium, and the asset returns are usually intermittent with typically non-Gaussian distributions. Although sensitivity analyses do test different scenarios, they are generally an exercise focused on asking how various assumptions about measurement error or other barriers to inference influence the observed findings.
By contrast, the modelling exercise of sweeping through a broad range of possible scenarios is more focused on understanding how the relationships within a complex system behave in an unexpected fashion. Although, as noted above, complex systems dynamic models will still need to be parameterized using observational or experimental epidemiological data, these data will need to be used creatively, quilting together data from disparate sources in order to create simulation models that best help us answer the key epidemiological questions of interest.
The shift from a dominant paradigm where we search for association in available data to the use of modelled data albeit informed by existing data sources to test scenarios is not insubstantial. Our central premise in this article is that complex systems dynamic models have much to offer epidemiology and it is time for epidemiology to consider adopting these methods as part of its toolkit. However, as we have noted in several places in this article, this recommendation is focused mainly on non-infectious disease epidemiology.
Indeed, infectious disease epidemiologists have used complex systems dynamic methods effectively to model the transmission of diseases from person to person. Complex systems dynamic models rest on modelling interactions and interrelations, and on understanding how these interactions contribute to the emergence of patterns in populations, be they in the form of interrelations among individuals or of dynamic feedback between states of a particular individual within environments that are also dynamically changing.
This approach is intuitively easier to understand when considering transmission of pathogenic organisms between individuals, providing clear links among persons within a model. Epidemiologists are less accustomed to modelling inter-individual relations when concerned with pathology that is not predicated on person-to-person transmission. Epidemiological inquiry focused on the role of the social environment in shaping individual health, or social epidemiology, is one of the most rapidly growing fields of epidemiology. For example, although there is evidence that social supports are protective of the risk of coronary death, 54 these social supports are typically modelled as properties of individuals even though they are, by definition, relational and exogenous properties of a particular micro- or macro-population.
Absent a clear conceptualization of the interrelations between individuals that shape health outcomes, the application of complex systems dynamic models to epidemiology will remain limited. Recent writing in epidemiology has drawn attention to a lifecourse perspective 55 which recognizes that disease production in the individual is not a static product of individual circumstance at any given time, but rather a product of circumstances over the lifecourse, possibly starting in utero and proceeding through an individual's life.
In many respects this more closely approximates the temporal nature of most disease processes that develop over considerable periods of time, including the complex interactions over time that lead to dynamic down- and up-regulation of regulatory systems. Although there is much that can be said on this topic that is beyond the scope of this article, complex systems dynamic models, allowing the incorporation of changing, dynamic processes and their interrelations provide a promising optimal analytic approach to considering lifecourse perspectives in epidemiology.
However, although lifecourse approaches have been well conceptualized, there is a substantial gulf between this conceptualization and our parameterization of the role of time in the determination of health and disease. Epidemiological studies remain largely short term, and even the few studies that have followed persons and populations over long periods of time seldom provide the richness of epidemiological detail that allow the reliable parameterization of changing temporal relations.
Empiric studies have demonstrated the importance of certain early-life influences on health later in life, but it is rare to have exposure measures throughout the lifecourse in a cohort study that can be used to tease apart the relative impacts of different exposures at different life stages.
We suggest that epidemiological thinking needs to broaden its conception of causes, and that such thinking may well be served by the adoption of complex systems dynamic models as part of our armamentarium. We have articulated a set of challenges that we argue has contributed to the slow diffusion of these methods within epidemiology.
The potential of these methods seems vast, and the challenges that we need to bridge to successfully adopt them in epidemiology commensurately daunting. Is there then a way forward? We are all slow adopters of novel methods, even when the barriers to adoption of new methods are much lower than they are here.
The additional challenges discussed here add to the inherent difficulties when a discipline faces new methodological approaches. We suggest, however, that in this case the potential offered by these methods is considerable, and that ultimately epidemiologists will identify ways to overcome these challenges and to adopt complex systems dynamic models in much the same way as regression models, once new and foreign to the field, quickly became lingua franca in epidemiology.
Although we remain concerned with looking for individual causes of individual diseases, and noting the mismatch between our methods, our outlook and the hunt for individual cause—disease relations, the chorus of epidemiological voices expressing concern is rising. Similarly, although some work in complex computational analytic approaches remains highly theoretical, other uses are solidly grounded in the use of real data.
The spread of these models throughout infectious disease epidemiological practice suggests that, in the right context, epidemiologists have also been able to turn to these unfamiliar methods to push parts of our discipline forward. Complex system models have contributed to the theoretical understanding of the spread of communicable diseases, as well as practical applications for predicting the effectiveness of different intervention strategies.
For example, one of the key theoretical results in infectious disease epidemiology is that there is a threshold for the density of susceptibles in a population that determines whether an epidemic will die off or spread through the population. Fox et al. These threshold results were derived from the use of systems dynamic models that assumed random mixing of the population. If the contact rate is higher in some subsets of the population than others, then the required immunity level in those populations may need to be higher than in others.
Complex systems dynamic models that explicitly model the social network structure have developed this idea further, showing how different network structures can lead to different results for the epidemiological threshold, 61 or even the non-existence of an epidemiological threshold. Recent modelling of the spread of infectious diseases has also drawn on extensive data sources and modern computing power to provide even more effective tools that can be practically used to predict the effectiveness of different real-world interventions.
It incorporates census data and travel patterns in the USA into a model that can handle several billion agents. This type of empirically based model can provide an important tool in efforts to prevent the spread of infections such as the H1N1 virus. Other empirically driven models have been used to model the spread of different types of infectious diseases, such as malaria and HIV. We close by presenting an example of how a complex systems dynamic model could provide practical information that could be used in evaluating potential interventions for improving population health.
Congruent with the example we have followed throughout the article, we consider policy interventions that aim to reduce obesity. Interventions to improve health outcomes can target causes at multiple levels: individual, neighbourhood, school district, city, state or national. They could target downstream causes that directly influence the target variable, or more upstream causes such as income or education levels, whose influence is felt indirectly. Evaluating direct, individual-level interventions is relatively easy: we can perform randomized controlled trials to determine whether individuals who receive the intervention are less likely to develop the health outcome of interest than individuals who do not receive the intervention.
In the case of upstream, group-level causal variables, this is more difficult. A randomized experiment would require identifying large numbers of groups neighbourhoods, cities, states, etc. In addition, upstream policies may be harder to evaluate because it may take longer for their health effects to be felt. An evaluation of health impacts over a limited time frame may not tell the full story if the immediate impacts are different than the long-run impacts.
To judge the relative effectiveness of different policy options, it therefore becomes crucial to have a model that can predict how the short-term effects will translate into long-run outcomes. This requires a model that can capture the complexity of the situation, combined with careful data analysis to ensure that the relationships in the model are an accurate depiction of the real world. We have developed a preliminary model of the influence of social and behavioural factors on obesity and cardiovascular disease, which we can use to illustrate the role that complex systems modelling could play in helping to evaluate upstream, group-level policies.
The time required for a policy to have an impact and the long-run persistence of these effects depend greatly on the pathways from the intervention to the outcome, and on the strength of feedback loops that occur along this pathway. We illustrate this by using simulations of the impact of investing in good food stores on body mass index BMI , under different assumptions about the importance of friend networks in influencing diets.
The model we use for these simulations cannot be described in detail here due to space limitations but the key features that are relevant for these simulations are summarized here. We are using an ABM with agents arranged in a grid divided into neighbourhoods, with social ties formed primarily between agents who live nearby. An agent's diet is determined by a combination of the availability of good food stores in her neighbourhood, her education level, the diet of her parents and friends and genetic predispositions.
This in turn influences BMI, which adjusts gradually to changes in diet. The policy that we simulate increases the level of investment in attracting good food stores in all neighbourhoods. These relatively extreme assumptions about the importance of friend networks are chosen to illustrate the point more clearly, but noticeable differences in the results can be seen with smaller variations in the importance of friend networks as well. With each set of assumptions, we then evaluate the difference between the average BMI at each point in time in the policy simulation against the average BMI in a counterfactual simulation where there is no such policy.
In both cases the policy is funded for 25 time units e. The results of these simulations are shown in Figure 1. In the case with weak social network effects, the impact of the policy is felt more quickly, and the maximum impact is stronger. However, the impacts are more persistent in the case with strong social network effects, and it takes longer for them to dissipate. Agent-based-modelling simulation of population changes in BMI subsequent to the implementation of a policy to attract better food stores to local neighbourhoods, stratified by populations characterized by strong and weak network ties.
This illustrates the importance both of understanding the strength of social network effects, and of having a model that can help illustrate how the network effect translates into policy relevant conclusions. Some progress has been made recently in evaluating the importance of friend networks in influencing the evolution of BMI over time, using an analysis of network data from the Framingham Heart Study. Just as crucial is the construction of complex models that can be used to identify key pieces of information that should be studied, and to translate what is learned from the data into conclusions about policy; in this case translating information about the strength of friend network effects into conclusions about the timing of the impacts of policy interventions.
There is precedent for the use of complex systems dynamic models for the purpose of understanding system behaviour and outcomes, for parameterizing these relations using real data and for deriving from these models insight that has practicable and immediate implications for populations. We intend this article to serve as both a challenge and as an encouragement. We suggest here that complex systems modelling approaches have the potential to integrate our growing knowledge about multilevel causes of health and their patterns of feedback and interaction, and to inform our knowledge about how specific policy interventions influence the health of populations.
It is important to note that we do not think that these approaches will necessarily be a panacea or that they will necessarily offer a solution to all the challenges epidemiology faces as we grapple with causal thinking. As in all statistical and computational models, the utility of models depends strongly on the quality of the data that are input into the models and the assumptions that inform the modelling effort.
We do think that complex systems dynamic models provide a promising approach that can augment our epidemiological armamentarium and push us forward both conceptually and methodologically. Time will tell whether widespread adoption of these methods will move the field substantially forward and, of course, the ultimate test will be whether or not the adoption of these methods will help us address important epidemiological questions and move us closer to improving the health of populations.
However, as more epidemiologists recognize that complexity is a compelling and essential aspect of population systems, we think that growth in the application of complexity approaches to epidemiology in coming years is near inevitable. In that light, clearly articulating the methods that can help us achieve that goal and the challenges we face in the adoption of these methods can suggest the barriers that need to be overcome and suggest a way forward.
HD to G. The growing recognition that the interrelation among factors at multiple levels that influence health and disease often involves dynamic feedback and changes over time challenges the dominant epidemiological approach to identifying causes. Complex systems dynamic models may provide one approach for epidemiologists to account for the complexity of disease causation in populations.
There are several challenges facing the discipline in incorporating these methods into non-infectious disease epidemiology. Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide. Sign In or Create an Account. Sign In. Advanced Search. Article Navigation. Close mobile search navigation Article Navigation. Volume Article Contents. The hunt for causes in epidemiology.
Complicating causes. A methodological shift: complex systems and complex systems dynamic analytic approaches. Challenges in the application of complex systems dynamic models to epidemiological questions. A way forward. Causal thinking and complex system approaches in epidemiology Sandro Galea. E-mail: sgalea umich. Oxford Academic.
Google Scholar. Matthew Riddle. George A Kaplan. Cite Citation. Permissions Icon Permissions. Abstract Identifying biological and behavioural causes of diseases has been one of the central concerns of epidemiology for the past half century. Agent-based modelling , dynamic systems modelling , epidemiology , regression. Open in new tab Download slide. Search ADS. What is a cause and how do we know one?
A grammar for pragmatic epidemiology. Directed acyclic graphs, sufficient causes and the properties of conditioning on a common effect. The contribution of social and behavioral research to an understanding of the distribution of disease: a multilevel approach. Google Preview.