Systemic cycles of accumulation, spatial expansion and chaos: Adding a missing link

Goldsmiths, Giorgos Galanis; Koutny, Christian; Weber, Isabella Working Paper Systemic cycles of accumulation, spatial expansion and chaos: Adding a missing link Working Paper, No. 2022-04 Provided in Cooperation with: Department of Economics, University of Massachusetts Suggested Citation: Goldsmiths, Giorgos Galanis; Koutny, Christian; Weber, Isabella (2022) : Systemic cycles of accumulation, spatial expansion and chaos: Adding a missing link, Working Paper, No. 2022-04, University of Massachusetts, Department of Economics, Amherst, MA, https://doi.org/10.7275/h4hg-e505 This Version is available at: https://hdl.handle.net/10419/266994 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/3.0/ University of Massachusetts Amherst University of Massachusetts Amherst ScholarWorks@UMass Amherst ScholarWorks@UMass Amherst Economics Department Working Paper Series Economics 2022 Systemic Cycles of Accumulation, Spatial Expansion and Chaos: Systemic Cycles of Accumulation, Spatial Expansion and Chaos: Adding a Missing Link Adding a Missing Link Giorgos Galanis Goldsmiths, University of London Christian Koutny Kingston University, UK Isabella Weber Economics Department, University of Massachusetts Amherst Follow this and additional works at: https://scholarworks.umass.edu/econ_workingpaper Part of the Economic History Commons, Economic Theory Commons, Growth and Development Commons, Macroeconomics Commons, and the Political Economy Commons Recommended Citation Recommended Citation Galanis, Giorgos; Koutny, Christian; and Weber, Isabella, "Systemic Cycles of Accumulation, Spatial Expansion and Chaos: Adding a Missing Link" (2022). Economics Department Working Paper Series . 321. https://doi.org/10.7275/h4hg-e505 This Article is brought to you for free and open access by the Economics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Economics Department Working Paper Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. 1 Systemic Cycles of Accumulation, Spatial Expansion and Chaos: Adding a Missing Link Giorgos Galanis, Institute of Management Studies, Goldsmiths, University of London, United Kingdom [email protected].uk Christian Koutny, School of Economics, Kingston University London, United Kingdom mailto:[email protected] Isabella M. Weber, Department of Economics, University of Massachusetts Amherst, United States of America (corresponding author) [email protected]du Abstract We re-examine the Systemic Cycles of Accumulation (SCA) of Arrighi (2010) and Arrighi and Silver (1999) which provide a framework for the analysis of the cyclical patterns of geographical expansion of trade and production and the related shifts of hegemonic power within the world capitalist system. Within the SCA framework, the last stage of a hegemonic cycle is characterized by what is called ‘systemic chaos’, however the drivers of these chaotic dynamics have not been explicitly analyzed. This article fills this gap by providing a link between the accumulation process, the spatio-temporal fix, and systemic chaos, in three steps. First, we show that the accumulation process can be mathematically described by what is known as a logistic map. Second, we show that the different stages of the accumulation processes in Arrighi (2010) correspond to different values of the parameter of the logistic map which in our framework captures barriers to accumulation. Third, drawing on wellknown properties of the logistic map in dynamic systems, we demonstrate that as this parameter crosses certain thresholds, the accumulation process becomes more complex exhibiting nontrivial and chaotic dynamics. In this way, our approach provides the missing detailed understanding of how systemic chaos is an outcome of the contradictory dynamics of capital accumulation itself while being based on the key insights of the SCA framework of hegemonic cycles. Keywords: systemic cycles of accumulation, hegemony, spatial fix, chaos, switching crisis Acknowledgements We would like to thank Duncan Foley, Ali Khan, Karsten Köhler and Gregor Semieniuk as well as the participants of the London Political Economy workshop in November 2020 for helpful comments and encouragement. All remaining mistakes are our own. 2 1. Introduction The ongoing pandemic and the US-China tensions along with the inability of global capitalism to fully recover from the 2008 crisis by restoring previous levels of accumulation, raises the question of the stability of the world system with new urgency. Are we at the verge of an Asian century that would involve a wide-ranging reorganization of the geography of capitalism? If so, how can we conceptualize the process of hegemonic transition? The immanent instability of the present juncture can be characterized by what is known as chaos in physics and mathematics and corresponds to behavior of systems which appear to be unpredictable or random. The notion of ‘chaos’ has been introduced to the study of world systems by Abu-Lughod (1989) who compared world systems with weather systems; while Rosenau (1990) applied the concept of chaos to changes in world politics. Arrighi (2010) and Arrighi and Silver (1999), building on the world systems literature, introduce the concept of ‘systemic chaos’ to capture changes in the hegemonic structures and the related spatial organization of trade and production within the world system. In this article we revisit the notion of systemic chaos in the last stage of cycles of hegemony from a theoretical perspective. Our focus is on the phase of hegemonic transition that corresponds to the ongoing geographic reorganization of the world system. We demonstrate that the link between capital accumulation, barriers to accumulation and systemic chaos can be established consistently within the framework of Arrighi and Silver. Both from a mathematical, and a geographical political economy viewpoint, what is most interesting is not the observation that there are chaotic dynamics as such, but to develop an understanding of the specific forces and their interaction which drive the chaotic behavior of a system (which can be physical, social, political, economic etc). This has been a key focus in the research of Rosenau (1990), Arrighi (2010), as well as Arrighi and Silver (1999) who highlight that their ‘investigation has sought clues as to what [the] underlying patterns [of the chaotic world] might be in the present turbulence by uncovering the underlying patterns in comparable past instances of systemic change’ (Arrighi and Silver, 1999, 22). Their analytical framework known as Systemic Cycles of Accumulation (SCA) shows a repeated pattern in the dynamics of capital accumulation where an initial phase is characterized by stable accumulation centered in a specific geographic location which is followed by a transition characterized by a spatial relocation of that center which exhibits turbulent dynamics. The increasing geographical expansion and intensification is on one hand able to overcome existing barriers to capital accumulation but at the same time increases the instability of the accumulation process. The end of a SCA overlaps with the end of a given hegemonic center and the birth of a new one starting with stable expansion. This cyclical behavior is shown in the graph from Arrighi (2010) reproduced below. 3 Figure 1 Figure 1: This is figure 3.10: “Metamorphosis Model of Systemic Cycles of Accumulation” from Arrighi (2010). As Arrighi (2004, 2007) also argues, the SCA framework shares some of the key insights of Harvey’s (1981, 2003), theory of ‘spatio-temporal fixes’ and ‘switching crises’ according to which spatial expansion and geographic reorganization can temporarily overcome crises which are the outcome of barriers to accumulation. But this spatial expansion at the same time creates new crisis tendencies. The fact that the spatio-temporal fix does not provide a permanent solution provides a useful intuition regarding why according to the SCA approach, the turbulent dynamics of accumulation are persistent. However, it remains open what the exact dynamic is, that forges the path from stability, via turbulence to chaos. This leads to the more general question of whether the theoretical foundations of this missing link are consistent with the general theoretical foundations of the SCA framework. Our article focuses on building this link between chaos and the preceding phases of a systemic cycle of accumulation. We show that the foundations of the links between the different phases are indeed consistent with the underlining theory of SCA where each cycle is composed of two phases. The first phase (A-phase) is a stable expansion while the second one (B-phase) is characterized by turbulence and geographic shifts. The B-phase also corresponds to a period of expansion of financial activities across sectors and space where financial markets become the dominant source for profits. 1 This serves to overcome acute limits of accumulation that eventually emerge in the A-phase. Our analysis is based on the observation that the accumulation process presented graphically by Arrighi (2010) as illustrated in the figure above can be captured by what is known as a logistic map. The logistic map represents the evolution of a variable over time, where the specific dynamics of the process depend on the value of a (bifurcation) parameter. In our framework derived from Arrighi and Silver (1999), the bifurcation parameter corresponds to 1 Arrighi (2010) also refers to this financial expansion as ‘financialization’. 4 barriers to capital accumulation. More specifically, the dynamics of the bifurcation parameter are such that, as the capitalist system overcomes barriers to accumulation through geographic expansion, the value of the parameter increases. As the value of the parameter grows, the dynamic behavior of the process changes. Based on this we can identify various phases each of which corresponds to a range of values for the bifurcation parameter. As a result, we can show that the barriers to accumulation and the related push to overcome these, can be systematically linked to the phases in the SCA. The first two phases that we observe perfectly fit the ‘A’ and ‘B’ phases of the SCA framework. The last phase characterized by chaotic dynamics – as discussed in Arrighi (2010) and Arrighi and Silver (1999), is more difficult to locate in this framework. We argue that this chaotic phase, ‘endogenously’ evolves from the dynamics of turbulence: capital continues to further extend the financial and spatial expansion phase to overcome further barriers to accumulation. In this process, turbulence eventually gives way to chaos. Thus, turbulence and chaos should be seen as two sub-phases within the B-phase of financial and geographic expansion. In the phase of turbulence, the role of the spatio-temporal fix becomes more prominent and a switching crisis is set in motion. Nonetheless, the hegemon remains initially dominant as the financial center of the capitalist system. In the chaos phase these two processes come into conflict preparing the geographic shift in hegemony. As a result, our observation is not only confirming the consistency between the rise of systemic chaos and the accumulation process in SCA but also reformulates the B-phase as being composed of two distinct sub-phases that prepare the hegemonic shift. Our work makes two key contributions to economic geography and political economy literatures: First and foremost, we strengthen the systemic link between SCA, spatio-temporal fixes, switching crises, and the mathematical concept of chaos through the use of a logistic map. Arrighi (2007) points to the synergies between the SCA framework and Harvey’s spatial analysis. This paper aims to provide a step towards a systematic integration of these two foundational contributions to geographic political economy. Our approach combines a formal analysis with Geographical Political Economy insights where spatial and economic processes co-evolve over time (Sheppard, 2011). In doing so, our paper is methodologically related to Bergmann et al. (2009) who link complex dynamics with political economy analysis while our work is aligned with what Barnes and Sheppard (2010) call ‘engaged pluralism’. Second, we address two critiques raised against the SCA framework. Pollin (1996) argues that the SCA framework does not systematically account for the sources and movements of the profit rate. In this paper, we highlight the link between the SCA dynamics and the classical economics’ analysis regarding limits to accumulation due to a tendency of the rate of profit to fall. We show that in the SCA framework new modes of governance emerge in a cyclical fashion that allow to push the limits of accumulation up to the point where they give rise to systemic chaos related to hegemonic transitions and spatial reconfigurations. As such, the movement of the rate of profit is linked back to the hegemonic and geographical shifts in the world system. 5 The structure of the rest of the article is as follows. The next section presents the key points which define the SCA framework and its connection to the spatio-temporal fix, and introduces the logistic map and its bifurcation parameter, relating the latter to the spatial extent of accumulation and its limits. We argue that the logistic map with this geographical bifurcation parameter, is an appropriate representation of the general intuition regarding the (over)accumulation dynamics in the SCA. In Section 3, we show how the increase of the value of the bifurcation parameter to overcome accumulation limits corresponding to spatial shifts in production and trade activities, leads to different phases which correspond to the different phases in SCA, including the chaotic one. The last section provides a concluding discussion. 2. Systemic Cycles of Accumulation 2.1 General Framework In the Long Twentieth Century Arrighi (2010) presents an analysis of hegemonic cycles in the capitalist world system. For Arrighi, ‘[t]he concept of “world hegemony” … refers specifically to the power of a state to exercise functions of leadership and governance over a system of sovereign states’ (ibid., 28). This power is derived at least in parts by the capacity of the hegemonic state to lead the world system in a way that is perceived by other states as being not only in the interest of the hegemon but also conducive to a more general interest (Arrighi, 2007, 149). SCA drive hegemonic cycles and rest on Braudel’s (1984) notion of reoccurring financial expansions 2 . In such reoccurring financial expansions, profits which are initially ploughed back into the further expansion of trade and production but eventually cannot be invested without jeopardising profit margins, lead capitalist agencies to hold larger portions of their incoming cash flows in liquid form. Arrighi argues that over the last 700 years the capitalist world system has experienced four different SCA each of which corresponds to a different world hegemon and a different geographic location of the core of the capitalist world system – the Italian city states, the Netherlands, Great Britain and most recently the United States. SCA start with an initial material expansion and endogenously lead to financial expansion where large sums of capital are transferred to the rising global center of accumulation, providing the economic foundation for the hegemonic transition. Importantly, each of the Dutch, British, and US hegemonies have been the outcome of a competitive struggle for world leadership. This struggle creates the conditions for chaos and hegemonic crises, gives rise to new SCA and new hegemons, and also creates an ever-greater global concentration and consolidation of economic and political power. Each cycle exceeds the previous one in terms of the mass of accumulated capital and geographical size as well as complexity of organization and integration of previously externalized processes. 2 Also see Arrighi (2001). 6 According to Harvey (1981) geographical expansions of financial and commercial capital provide a “spatial-fix” 3 allowing to temporarily overcome economic crises related to low profitability. Importantly, Harvey (1981, 2003) does suggest that the spatial fix is linked to the permanent process of capital accumulation and hence is not just a spontaneous solution to acute crises related to the accumulation process. The link between the spatial-fix and finance is spelled out in The New Imperialism, where Harvey (2003) includes the temporal aspect of the spatial-fix and coins the term “spatio-temporal fix”. A switching crisis constitutes the acute form of the spatio-temporal fix where capital is devalued in one place and future income thus starts to flow from other geographic regions (Harvey, 2003, 121-122). There are clear overlaps between the SCA, spatio-temporal fixes and switching crises. In fact, Arrighi (2007) points to the link between Harvey’s concept and SCA. However, he refrains from a systematic incorporation of his and Harvey’s conceptualizations. We aim to do so here, and show that by consolidating them, the transition from tranquillity to chaos, can also be supported coherently within this expanded framework. Arrighi (2007, 25) takes inspiration from Adam Smith’s concept of the extent of the market as limiting the degree of the social division of labor 4 and as he points out, rather than seeing the economic independent of the political, Smith had built his analysis on Hobbes’ insight that wealth and power are inseparable. Arrighi (2005a, 2005b) also stresses similarities to Arendt (1958) who links the accumulation over and above what can be invested profitably to a spatial-expansion of capitalism. There, “superfluous men” and “superfluous capital” search for employment abroad as a solution to the internal contradictions that created that surplus. The idea of such an ‘overaccumulation crisis’ has roots both in Smith and Marx 5 and is fundamental in SCA. The fact that “[o]nly the unlimited accumulation of power could bring about the unlimited accumulation of capital” (Arendt, 1958, 137) highlights the significance of political organization for the expansion of capital. Due to the key role of accumulation in SCA each of the different phases of an SCA should be understood in relation to the dynamics of the circuit of capital; the process from a monetary investment via production and trade to a monetary revenue that exceeds the initial investment yielding a profit for the capitalist. In SCA this general formula is reinterpreted as a general theory of the organization of society and the evolution of the world capitalist system as a whole. Thereby, the circuit of capital serves as a lens to analyze the historical development of 3 Although Harvey only spells out a proper definition of his treatment of the term “fix” in The New Imperialism (Harvey, 2003), he clearly means the abstract definition of the term “fix” here, where internal contradictions of capital are fixed through an external solution. 4 See Weber (2019) for a detailed analysis of the dynamic co-evolution of the extent of the market and the division of labor in Smith. 5 According to Smith there is a general tendency for profits to fall when too much capital is competing as a result of general overaccumulation. This drives up wages and squeezes profits. Another reason for falling profits is overproduction: Marx suggests that when wages are too low, aggregate demand is falling short of supply pushing down profits. For a discussion on the tendency of the profit rate to fall see Foley (1986) and Fine and Saad-Filho (2016) among others. 7 the capitalist system while linking to Marx’s theory of crisis and falling rate of profit. Importantly, each systemic cycle of accumulation corresponds to a specific political, geographic and economic configuration of the world capitalist system. In each cycle, the initial phase of material expansion eventually culminates in a conflict between a falling rate and a growing mass of profit as the limits of accumulation under a given political and economic configuration are approached (Arrighi, 2010, 232-233). This generates a drive towards an intensified spatio-temporal fix as well as a switching crisis signalling a hegemonic transition. But rather than leading immediately to a collapse of global hegemony, a financial expansion creates a belle époque where the declining global economic center switches to financing an economic expansion in a different geographical location. Now, profitability is temporarily restored and the limits to accumulation of capital are pushed back further. With the framework we establish in this work we are able to explicitly show how the SCA framework effectively demonstrates that the spatial and institutional barriers of the world market are constantly overcome through geographical expansion and reorganization of the system as such, as a result of dynamics originating from the accumulation of capital itself. 2.2 The role of chaos The phase of SCA in which a spatial relocation and intense financial expansion pushes the accumulation barriers, signals the beginning of the transition from one hegemon to the next. This transition is characterized by what Arrighi calls systemic chaos. More specifically ‘“[c]haos” and “systemic chaos,” […] refer to a situation of total and apparently irremediable lack of organization’ (Arrighi, 2010, 31). Furthermore, he provides a summary of possible channels through which systemic chaos can emerge: “It is a situation that arises because conflict escalates beyond the threshold within which it calls forth powerful countervailing tendencies, or because a new set of rules and norms of behavior is imposed on, or grows from within, an older set of rules and norms without displacing it, or because of a combination of these two circumstances.” Simply put, it is a situation of escalating conflicts and a collapse of predominant rules and norms. Arrighi highlights the role of systemic chaos to hegemonic transition: “As systemic chaos increases, the demand for ‘order’ – the old order, a new order, any order! – tends to become more and more general among rulers, or among subjects, or both” (Arrighi, 2010, 31). Note that the quote above provides both a definition of (systemic) chaos through its characteristic of “total and irremediable lack of organization” and also describes the possible channels (conflict and lack of rules and norms) that can lead to chaos and through this to a hegemonic transition and a geographic reorganization of the world system. Even though the relationship between (systemic) chaos and the notion of chaos in mathematics and physics is not explicitly discussed by Arrighi (2010) or Arrighi and Silver (1999), the property of chaotic systems to appear as random or unpredictable can be related to what they refer to as a “lack of organization”. More specifically the characteristic of apparent unpredictability is sufficient to prevent a system from being organized. Given the role of accumulation within 14 Figure 5: Figures 3.8 “Local Turbulence” and 3.9 “Systemic Turbulence” from Arrighi (2010). In order to show the correspondence between the behavior captured in Figure 5 and the dynamics generated by the logistic map as 𝑟 increases, we need to check whether an increase of the value of the bifurcation parameter in the logistic map will first lead to local turbulence and then to systemic turbulence corresponding to the two graphs of Figure 5. The graph below portrays the case of 𝑟 = 2.95 (which is higher than the previous value). Figure 6 Figure 6: Local turbulence, r=2.95 As we can see, a further increase of 𝑟 as the barriers to accumulation grow in size and complexity progresses results in turbulent fluctuations that are, however, decreasing over time. This captures exactly the key insights of the first graph of Figure 5 where fluctuations eventually disappear. Importantly, the average value of the level of capital accumulation 𝑥 (or M in Figures 2 and 5) is almost double compared to the initial extended expansion phase 20 40 60 80 100 120 140 0.645 0.650 0.655 0.660 0.665 0.670 0.675 0.680 15 mapped in Figure 4. This highlights the fact that even though local turbulence emerges, the accumulation rate is higher than before, hence this type of “local instability” does not yet provide a serious challenge to the accumulation path or the prevailing hegemon. “If the surpluses of capital and labour power exist within a given territory … and cannot be absorbed internally … then they must be sent elsewhere to find fresh terrain for their profitable realization if they are not devalued. … [But] … [t]he problem of overaccumulation is alleviated only in the short term.” (Harvey, 2003, 116-117). As the geographical expansion progresses further, financialization deepens and capital becomes ever more mobile further intensifying interstate competition. A higher 𝑟 leads to a behavior which captures the key insights of the systemic turbulence, where fluctuations are not temporary but are persistent over time with no sign of fading. The graph below is for 𝑟 = 3.3. Figure 7 Figure 7: Systemic turbulence, r=3.3 Again, we observe an increase regarding the average value for the level of capital accumulation 𝑥 but also the variance of the fluctuations is much higher compared even to the maximum variance of the previous example. These two examples of values for r demonstrate the correspondence between the accumulation dynamics derived through increasing values of the bifurcation parameter of the logistic map and the insights of the SCA process shown in Figure 5. This demonstrates the correspondence between the SCA dynamics and the logistic map. 4.3 Chaos 20 40 60 80 100 120 140 0.2 0.4 0.6 0.8 16 For Arrighi and Silver (1999, 21) hegemonic transition at the end of an SCA is through “a process of radical reorganization of the modern world system that changes substantively the nature of the system’s components, the way in which these components relate to one another, and the way in which the system operates and reproduces itself.” Based on Abu-Lughod (1989, 369) in such a situation “the ‘same-cause-yields-same-effects’ logic that underlies our thinking…is ill-equipped to apprehend this kind of change, and we should instead draw inspiration from ‘chaos theory’” (ibid.). While the relationship between chaotic dynamics and hegemonic transition is mentioned in Arrighi (2010), the concrete link is not analyzed. Furthermore, it is not clear whether a possible link would fit with the SCA framework. Despite the systemic turbulences that arise when a continuous expansion reaches a certain threshold, the limits to accumulation are pushed even further as capital encounters further barriers arising from the contradictions set in motion by its previous solution to the previous barriers. Considering that “[t]he entire credit system … rests on the necessity of expanding and leaping over the barriers to circulation and the sphere of exchange” (Marx, 1973; 416), credit facilitates the geographical expansion of capital, speculation is fuelling more speculation and the limits to accumulation are pushed once again. This corresponds to a further increase in 𝑟. However, if 𝑟 is increased even more (here 𝑟 = 4), the accumulation process takes a chaotic form which is characterized by what looks like randomness (Figure 8). Small changes can lead to very different values of 𝑥. This means that it is impossible to predict 𝑥𝑡+1 even if all the previous values of 𝑥 up to some point in time t are known. This corresponds to Arrighi’s characterization of “systemic chaos” as a total lack of organization (Arrighi, 2010, 31). Figure 8 Figure 8: Chaos, r=4 20 40 60 80 100 120 140 0.2 0.4 0.6 0.8 1.0 17 This shows the link between extended expansion and chaos when the SCA is represented by a logistic map. Thus, the use of a logistic map to express the accumulation process underlying SCA along with the theoretical understanding of the fact that spatio-temporal fixes (and switching crises) are only a temporary solution hence there is a continuous need for pushing towards higher accumulation limits. This also means a continuous increase of the value of bifurcation parameter. As chaos appears after the bifurcation parameter crosses a threshold, it means that chaos is an endogenously created sub-phase following the preceding sequence of stable expansion, local and systemic turbulence. Our analysis leads to a reformulation of the original SCA graph where the accumulation process is divided into two phases A and B. Using our insight, the B-phase includes two subphases. The first corresponds to the original insights of Arrighi (2010) where a stable extended expansion is followed by local and then systemic turbulence; while the second subphase corresponds to chaos and means that the hegemon has lost its power and the transition is under way. The different parts of the B-phase can be presented through the (well known) bifurcation diagram of the logistic map below which in this case shows the behavior of the accumulation process for different values of the limits of accumulation bifurcation parameter. Figure 9 Figure 9: Bifurcation diagram of the logistic map The line in the diagram above shows the values that 𝑥 can take for given values of 𝑟. As we can note, the initial part corresponds to the second part of the logistic, hence the B-phase. Furthermore, we note qualitative changes as 𝑥 initially takes one value which is increasing with 𝑟, then two, then four, then eight and then what seems to be infinitely many values (this corresponds to chaos). 18 To be more precise for 𝑟 between 3 and roughly 3.45 there is a bifurcation which corresponds to oscillations between two points of the type shown in the graphs representing the turbulent dynamics. The divergence of the two lines in this sub-phase represents the divergence of the points between which 𝑥 oscillates. Put differently, as 𝑟 increases the variations also increase. The second sub-phase roughly corresponds to 𝑟 >3.56 where chaotic dynamics appear and is represented by the blue surface. The two subphases are connected by an intermediate phase where 𝑥 oscillates between four and eight points respectively 10 . This intermediate phase is short and while the oscillations are non-cyclical they are not formally chaotic. In the last stage, when 𝑟 crosses a new threshold, it becomes impossible to predict the value of 𝑥 and we get chaos which according to SCA signals a hegemonic transition. As Silver and Arrighi (1999, 21-22) point out drawing on Henri Poincaré who coined the term “bifurcation”, “the order that will eventually emerge out of the present turbulence… is not inscribed in the parameters of the order that has broken down. But … there is an order within chaos”. Chaos is itself the result of an order but the order that follows is not predetermined by that chaos. Cycles of accumulation follow a pattern but the characteristics of the new hegemon cannot be predicted from the preceding one. 5 Conclusion The aim of this paper has been to show that by expressing the dynamics of the global accumulation process at the basis of the SCA through the use of a logistic map, we are able to explicitly place the endogenous emergence of chaos within the SCA phases. Our starting point has been the well-known fact that for low values of its bifurcation parameter the logistic map corresponds to a logistic (sigmoid) graph. Hence, given that the accumulation process underlying the SCA is described by a sigmoid, as the logistic map, for certain parameter values is also a sigmoid, this is a suitable formal representaion. Our analysis was based on three further key observations. The first is related to two observations the bifurcation parameter. On the one hand higher values of the bifurcation parameter allow for higher levels of accumulation, while on the other there is a constant need for increasing its value, as new accumulation barriers appear. This means that for a capitalist economy where the accumulation process is described by a logistic curve there will be an endogenous push to increase the limits of accumulation and the value of the bifurcation parameter. This increase corresponds to geographic expansion and intensification, hence shows that the parameter has a spatial interpretation. The second observation is related to the exact mapping of the different stages of the SCA with the various dynamics portrayed by the logistic map as the bifurcation parameter crosses 10 Even if it is not visible here, we know from bifurcation theory that after eight points, the oscillations are between sixteen points. 19 certain threshold values. Not only as the parameter increases, we are able to replicate the key insights of the SCA but chaos also follows the turbulent last phase of the SCA. Third, in order for the hegemon of the world capitalist system to be able to overcome accumulation limits, a geographical expansion and organizational complexification takes place. This provides a spatio-temporal fix to the accumulation barriers, however this fix is only temporary which means that there is a continuous need for an increase of the bifurcation parameter, which highlights how the process that leads to different phases of the SCA framework is endogenous within the capitalist accumulation process. Hence our analysis shows that exactly because the logistic map can represent the SCA and the spatio-temporal fixes are only temporal, chaos is actually inherent in the accumulation process. Through this approach we have ‘extended’ the (B-phase of the) SCA framework by explicitly including the chaotic dynamics which are related to hegemonic transition as a ‘natural’ next step in the accumulation process. The theoretical framework we established consists of a consolidation of Harvey’s (1981, 2003) spatio-temporal fix and switching crisis concepts with Arrighi’s (2010) and Arrighi and Silver’s (1999) framework of Systemic Cycles of Accumulation. Adding the missing link in this paper leads to new research questions. Possible extensions of the logistic map include for example the explicit incorporation of the different fixes to overaccumulation crises and/or allowing for the explicit role of hegemony and power in stabilizing the accumulation process. 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