Hidden Order in Chaos: The Network-Analysis Approach To Dynamical Systems

769 Hidden Order in Chaos: The Network-Analysis Approach To Dynamical Systems Takashi Iba Faculty of Policy Management Keio University iba@sfc.keio.ac.jp In this paper, I present a technique for understanding a whole behavior of chaotic dynamical system, where state-transition networks of the systems are built and analyzed. With using this technique, I show that there is an order undetected yet, where state-transition networks in several chaotic dynamical systems are scale-free network, binary-tree network and so on. I anticipate that this study marks a step toward seeing well-known chaos in a new light. 1 Introduction Chaotic phenomena that are intrinsically complex and unpredictable exist everywhere in natural, technological, and social world. Deterministic chaos exhibits highly complex behavior, even though it is governed by a simple rule. In many cases, unfortunately, it is impossible to solve the systems analytically due to their nonlinearity; therefore a numerical simulation is commonly used as a powerful tool to explore the systems evolution in time. However, the nature of the system as a whole

770 is still unrevealed, since a numerical simulation shows only an instance of possible evolutions generated from a certain initial condition. In this paper, I will present a technique, building and analyzing state-transition networks for chaotic dynamical system, for understanding a whole behavior of the system. Also, I will show state-transition networks in several chaotic dynamical systems. 2 Method 2.1 How to build state-transition networks The technique I introduce in this paper follows a three-step procedure (Figure 1) [1]: First, a continuous unit interval of variables of the target system is subdivided into a finite number of subintervals, discrete states. In the term of numerical operation, it is carried out by rounding the value of the variables to d decimal places with round-off, round-up, or round-down function. Second, for each state, mapping from the state to its next state is carried out with using the map function. Third, a network is built, combining the all relation between two states together. Figure. 1: Building the state-transition network of a dynamical system. A unit interval is subdivided into subintervals, states. Then, for each state, mapping is carried out. Finally, a network is built with these states, combining the all relation between two states together. 2.2 Terminology Connecting different fields of the science of complexity [2-4] and network science [5-7], I here denote the correspondence of terminology in the figure 2-5. A node in state-transition networks represents a state of a variable in a one-dimensional dynamical system, or a composite state of all variables in a dynamical system whose dimension is greater than one (Figure 2). A link represents a transition from a state to another state in a dynamical system. Each link has a direction; therefore state-transition networks are directed networks.

771 Figure. 2: Elements of state-transition network and their correspondence with the terminology for dynamical systems: node and link. In-degree, which means the number of incoming links to a node, represents the number of possible previous states for mapping (Figure 3). On the other hand, out-degree, which means the number of outgoing links from a certain node, represents the number of possible next states for mapping. In deterministic systems, out-degree must be equal to 1, by definition. It means that the successor state for each state is uniquely determined. An authority, which is a node whose in-degree of an authority is very large, represents a state that yields many possible previous states. Figure. 3: Degree of nodes in state-transition network and their correspondence with the terminology for dynamical systems: in-degree, out-degree, and authority. A loop represents a fixed point, where the system repeats the same states (Figure 4). A cycle represents a periodic cycle, where the system oscillates between two or more states. The cycle that has two nodes is called period-2 cycle; the cycle that has four nodes is called period-4 cycle; and so forth. Both fixed point and periodic cycle is a kind of attractor, where the system eventually falls into after a long enough period of time.

772 Figure. 4: Cyclic structure in state-transition network and their correspondence with the terminology for dynamical systems: loop and cycle. Connected component represents a basin of attraction, a set of states that eventually flow into an attractor (Figure 5). Each basin of attraction, by definition, must have only one attractor, namely either fixed point or periodic cycle. Therefore, each connected component will have only one attractor, which is either a loop or a cycle. For building a state-transition network of a chaotic dynamical system, the whole behavior is often mapped into more than one connected components. Figure. 5: Component in state-transition network and their correspondence with the terminology for dynamical systems: connected component. 2.3 Drawing a Map of System s Behavior Building state-transition networks is analogous to draw a map of the whole behavior of a dynamical system. With the state-transition networks as a map, we can overview the landscape of behavior from bird s-eye view. Thus, the method is much different from numerical simulations, which is often used for understanding the behavior of dynamical systems. To clarify this difference, it may be helpful to use a metaphor of river. In the metaphor, a node represents a geographical point in the river, and a link represents a

773 connection from a point to another. The direction of a flow in the river is fixed. While there are no branches of the flow, however you can find confluences of two or more tributaries everywhere. To be exact, the network is not a usual river but dried-up river, where there are no flows on the riverbed. On the other hand, exploring a dynamical system with a numerical simulation of system s evolution in time means to observe an instance of flow starting from a point of the river. Determining a starting point and discharging water, you will see that the water flow downstream on the river network (Figure 6), and typical techniques of drawing a trajectory on phase space and bifurcation diagram are just to sketch out the instances. This type of study, building and analyzing state-transition networks, has been carried out for understanding the nature of cellular automata and random Boolean networks [8-12]. In the study for chaos, there are some pioneering studies [13-16], however the network analyses of chaotic dynamical systems have not been studied in much detail so far. Figure. 6: What a run of numerical simulation does in representation of state-transition networks. A run of numerical simulation traces an instance of flow on the state-transition network of the system. 3 State-Transition Networks of Chaotic Dynamical Systems I will show state-transition networks of several types of chaotic dynamical systems: one-dimensional maps, two-dimensional dissipative maps, and two-dimensional conservative maps. In this paper, round-up function is used for rounding the value; and the values of variables are rounded up to d decimal places. Note that the all following networks are visualized with the circular layout of yfiles [17] on the Cytoscape environment [18]. 3.1 One-dimensional maps For one-dimensional maps, the following maps are visualized and analyzed: the logistic map [19, 20], the sine map, the cubic map, the Gaussian map, the sine-circle map [21], the tent map, the binary shift map, and the cusp map. As a result, these maps can be categorized into four types.

774 First type is the state-transition network that is always a scale-free network [22, 23] independent of the control parameter. The maps belonging to this type are the logistic map, the sine map, and the cubic map. Figure 7-9 show state-transition networks in the logistic map x n +1 = 4µx n (1 x n ), sine map x n +1 = µ sin(πx n ), and the cubic map x n +1 = 27µx 2 n (1 x n ) 4 respectively with µ = 1.0 and d = 3 ; and figure 10 shows state-transition networks of the sine-circle map x n +1 = x n + Ω (K 2π )sin(2πx n ) (mod 1) with K = 2, Ω = 0.5 and d = 3. All of them are scale-free networks whose degree distribution follows a power. It means that there are quite many nodes having few links but also a few hubs that have an extraordinarily large number of links. Note that the detail mechanism why these networks are scale-free networks is discussed in my previous paper [1]. Figure. 7: Scale-free state-transition network in the logistic map. (A) Graph of the logistic mapping from x n with µ = 1.0, where the gray line is an orbit starting from the initial condition x 0 = 0.2. (B) State-transition network in the logistic map with µ = 1.0 and d = 3, which is a scale-free network.

775 Figure. 8: Scale-free state-transition network in the sine map. (A) Graph of the sine mapping from x n with µ = 1.0, where the gray line is an orbit starting from the initial condition x 0 = 0.2. (B) State-transition network in the sine map with µ = 1.0 and d = 3, which is a scale-free network. Figure. 9: Scale-free state-transition network in the cubic map. (A) Graph of the cubic mapping from x n with µ = 1.0, where the gray line is an orbit starting from the initial condition x 0 = 0.2. (B) State-transition network in the cubic map with µ = 1.0 and d = 3, which is a scale-free network. Figure. 10: Scale-free state-transition network in the sine-circle map. (A) Graph of the sine-circle mapping from x n with K = 2 and Ω = 0.5, where the gray line is an orbit starting from the initial condition x 0 = 0.2. (B) State-transition network in the sine-circle map with K = 2, Ω = 0.5 and d = 3, which is a scale-free network. Second type is the state-transition network that is a scale-free network only when the control parameter is set in a certain range. Figure 11 shows a state-transition network in the Gaussian map x n +1 = e bx n 2 + c with b = 3.0, c = 0.3, and d = 3,

776 which is a scale-free network. Figure 12, then, shows the cumulative in-degree distribution of the networks in the Gaussian map with the value of b varying from 1.0 to 15.0 by 2.0 and c = 0.3. The distribution follows a power law when b is small (e.g. 1.0 and 3.0) and when b is much larger; however it does not when b is intermediate value (e.g. 7.0 and 9.0), where no nodes have an exceptional in-degree. Figure. 11: Scale-free state-transition network in the Gaussian map. (A) Graph of the Gaussian mappings from x n with c = 0.3 and b = 3.0, where the gray line is an orbit starting from the initial condition x 0 = 0.2. (B) State-transition network in the Gaussian map with c = 0.3, b = 3.0, d = 3, which is a scale-free network. Figure. 12: Structural change of the state-transition networks in the Gaussian map. (A) The plot of the Gaussian map obtained by varying the value of b from 1.0 to 15.0 by 2.0 with c = 0.3 and d = 3. (B) The cumulative in-degree distribution of the Gaussian map with the same settings. Third type is the state-transition network that is a binary-tree-typed network, where each branch splits into two, then into two again, and so forth. Figure 13 shows the state-transition network in the tent map x n +1 = µ ( 1 2 x n 1 2)with µ = 1.0 and d = 3; and figure 14 shows the state-transition network in the binary shift map x n +1 = 2x (mod 1) with d = 3. The state-transition network in the piecewise linear

777 map tend to be this type of network, since all states of x have exactly the same in-degree due to the constant slope. Figure. 13: Binary-tree-typed state-transition network in the tent map. (A) Graph of the tent mappings from x n with µ = 1.0, where the gray line is an orbit starting from the initial condition x 0 = 0.2. (B) State-transition network in the tent map with µ = 1.0 and d = 3, where binary-tree sub-networks branch from the cycle. Figure. 14: Binary-tree-typed state-transition network in the binary shift map. (A) Graph of the binary shift mappings from x n with µ = 1.0, where the gray line is an orbit starting from the initial condition x 0 = 0.2. (B) State-transition network in the binary shift map with µ = 1.0 and d = 3, where binary-tree sub-networks branch from the cycle. Forth type is the state-transition network that is an irregular-tree-typed network. Figure 15 shows state-transition networks in the cusp map x n +1 = 1 4µ x n with µ = 1.0 and d = 3, where tree sub-networks consisting the node whose in-degree exhibits several values branch from the cycle.

778 Figure. 15: Irregular-State-transition network in the cusp map. (A) Graph of the cusp mappings from x n with a = 2.0, where the gray line is an orbit starting from the initial condition x 0 = 0.2. (B) State-transition network in the cusp map with a = 2.0 and d = 4, where tree sub-networks consisting the node whose in-degree exhibits several values branch from the cycle. 3.2 Two-dimensional dissipative maps For two-dimensional dissipative maps, which contract areas in phase space, the following maps are visualized and analyzed here: the delayed logistic map [24], Hénon map [25], and Lozi map [26]. Figure 16 shows state-transition networks in the delayed logistic map x n +1 = 4µx n (1 y n ) and y n +1 = x n with µ = 0.5675 and d = 2, which is a scale-free network. Figure 17 shows state-transition networks in the Hénon map x n +1 = 1 ax 2 n + y n and y n +1 = bx n with a = 1.4, b = 0.3, and d = 2. Figure 18 shows state-transition networks in the Lozi map x n +1 = 1 a x n + y n and y n +1 = bx n with a = 1.7, b = 0.5, and d = 2.

779 Figure. 16: Scale-free state-transition network in the delayed logistic map. (A) Attractor drawn by plotting successive points obtained by the delayed logistic mapping with µ = 0.5675. (B) State-transition network in the delayed logistic map with µ = 0.5675 and d = 2, which is a scale-free network. Figure. 17: State-transition network in the Hénon map. (A) Attractor drawn by plotting successive points obtained from Hénon mapping with a = 1.4 and b = 0.3. (B) State-transition network in the Hénon map with a = 1.4, b = 0.3, and d = 2. Figure. 18: State-transition network in the Lozi map. (A) Attractor drawn by plotting successive points obtained from Lozi mapping with a = 1.7 and b = 0.5. (B) State-transition network in the Lozi map with a = 1.7, b = 0.5, and d = 2. 3.3 Two-dimensional conservative maps For two-dimensional conservative maps, the following maps are visualized and analyzed: Hénon s area-preserving map [27], Chirikov standard map [28], the

780 Arnold s cat map [29], and the gingerbreadman map [30]. Figure 19 shows state-transition networks in the Hénon s area-preserving map x n +1 = x n cosα (y n x 2 n )sinα and y n +1 = x n sinα + (y n x 2 n ) cosα with cosα = 0.24 and d = 2. Figure 20 shows state-transition networks in the Chirikov standard map x n +1 = x n + y n +1 (mod 1) and y n +1 = y n (K 2π )sin 2πx n (mod 1) with K = 1.0 and d = 2. Figure 21 shows state-transition networks in the gingerbreadman map x n +1 = 1+ x n y n and y n +1 = x n with d = 2. Figure. 19: State-transition network in the Hénon area-preserving map. (A) The orbit drawn by plotting successive points obtained from Hénon area-preserving mapping with cosα = 0.24, starting from the point x 0 = 40π 25, y 0 = 40π 25. (B) State-transition network in the Hénon area-preserving map with cosα = 0.24 and d = 2. Figure. 20: State-transition network in the Chirikov standard map. (A) The orbit drawn by plotting successive points obtained from Chirikov standard mapping with K = 1, starting from the point x 0 = 0.57, y 0 = 0.16. (B) State-transition network in the Chirikov standard map with K = 1 and d = 2.

781 Figure. 21: Regular state-transition network in the Arnold s cat map. (A) The orbit drawn by plotting successive points obtained from Arnold s cat mapping with k = 1, starting from the point x 0 = 0.2, y 0 = 0.2. (B) State-transition network in the Arnold s cat map with k = 1 and d = 2, which is a regular network where all nodes have just one incoming link. Figure. 22: Regular state-transition network in the gingerbreadman map. (A) The orbit drawn by plotting successive points obtained from gingerbreadman mapping, starting from the point x 0 = 0.1, y 0 = 0. (B) State-transition network in the gingerbreadman map with d = 2, which is a regular network where all nodes have just one incoming link. 4 Conclusion In this paper, a technique was introduced to grasp the whole behavior of chaotic dynamical systems and the state-transition networks of several systems demonstrated the hidden order in chaos. This implies that our technique of mapping the transitions between states is useful to capture the nature of the system. Thanks to the recent development of computer power, it become to be easy to draw a map of a dynamical system as a whole with higher resolution, and the

782 development of the network science enables us to understand the nature of state transition in light of new concept of complex networks, like scale-free network. Thus we anticipate that my study opens up a new way for the network-analysis approach to dynamical systems, where one can understand complex systems with methods and tools developed in the network science. On the other hand, this study also provides a demonstration of analyzing a network embedded in temporal behavior, where more than one node cannot exist together at the same time, which have not been studied in much detail in the network science. Acknowledgement I would like to thank to K. Shimonishi for collaborating on the exploration for new ways of studying chaos, T. Malone and P. Gloor for providing a research environment at MIT Center for Collective Intelligence, and also the Cytoscape project for providing the useful tool for network analysis and visualization. Bibliography [1] Iba, T., 2010, Scale-Free Networks Hidden in Chaotic Dynamical Systems, arxiv:1007.4137v1 [nlin.cd] [2] Strogatz, S. H., 1994, Nonlinear Dynamics and Chaos, Westview Press (Cambridge, MA) [3] Hilborn, R. C., 2000, Chaos and Nonlinear Dynamics, 2 nd ed., Oxford University Press (Oxford) [4] Sprott, J. C., 2003, Chaos and Time-Series Analysis, Oxford University Press (New York, NY) [5] Newman, M., Barabasi, A. -L. and Watts, D. J., 2006, The Structure and Dynamics of Networks, Princeton University Press (Princeton, NJ) [6] Caldarelli, G., 2007, Scale-Free Networks, Oxford University Press (Oxford) [7] Newman, M. E. J., 2010, Networks, Oxford University Press (Oxford). [8] Wuensche, A. and Lesser, M., 1992, The Global Dynamics of Cellular Automata, Addison-Wesley (Reading, MA) [9] Wolfram, S., 1994, Cellular Automata and Complexity, Addison-Wesley (Reading, MA) [10] Wolfram, S., 2002, A New Kind of Science, Wolfram Media (Champaign, IL) [11] Kauffman, S. A., 1993, The Origins of Order, Oxford University Press (New York) [12] Kauffman, S. A., 1995, At Home in the Universe, Oxford University Press (New York) [13] Binder, P. -M. and Okamoto, N. H., 2003, Phys. Rev. E 68, 046206 [14] Grebogi, C., Ott,, E., and Yorke, J. A., 1988, Phys. Rev. A 38, 3688

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