Lyapunov exponents of dynamical systems and their applications, Georgy Osipenko says

Georgy Sergeevich Osipenko – Doctor of Physical and Mathematical Sciences (1990), Professor (1992), Professor of the Applied Mathematics Department of Moscow State University named after M.V. Lomonosov, Sevastopol Branch of Moscow State University.

What is a Morse Spectrum?

One of the basic concepts of the dynamical systems theory is the trajectory characteristic exponent, which Lyapunov introduced in 1892 in his doctoral thesis. The importance and relevance of research related to Lyapunov’s exponents cannot be overestimated. The Lyapunov trajectory exponent is a value that characterizes the change rate in the distance between close trajectories. The most simple indicators are considered for a periodic trajectory. The different exponents’ number does not exceed the space dimension, since the characteristic exponent depends on the direction in which close trajectories are considered. If the exponent is positive, then the trajectories move away from each other; the trajectories approach each other in the case of the negative exponent. If a periodic trajectory has some positive and some negative exponents, then the trajectories approach it in some directions and move away in others. The calculation of a periodic trajectory Lyapunov exponents presents no special difficulties. Nonlinear systems have often infinitely many periodic trajectories in a compact region, the periods of which are not limited, which is typical for chaotic regimes. We are unable to calculate the Lyapunov exponents of all these trajectories in this case.

There is one more problem because all the calculations are approximate. For example, a personal computer counts with an accuracy of 10 to the degree of -17. This leads to the fact that we operate not with real trajectories, but with pseudo trajectories, which are somewhat different from the trajectories. The Morse spectrum is the set of Lyapunov exponents for all periodic pseudotrajectories of the system. It is shown that the Morse spectrum consists of segments and it is extremely important to calculate it at least approximately for practice.

What is your proposed method of a dynamic system symbolic image?

We considered a discrete dynamical system generated by the image f. The system trajectories global structure is a rather complex object. A dynamical system symbolic image is a digraph, which is this system finite approximation. The symbolic image study allows one to get an idea of the system’s trajectories global structure. Graph theory has a well-developed theory and computation practice now, which allows obtaining information about the system dynamics by graph theory numerical methods.

The symbolic image is built as follows. Let there be a finite coverage of the study area by closed cells. A dynamic system symbolic image is a directed graph with nodes corresponding to cells, and nodes i and j are connected by an i?j arc if the f(i) cell image intersects j cell. Dynamic system trajectories correspond to paths on a symbolic image, which allows estimating the system’s trajectories global structure. The received information accuracy is determined by the maximum cell diameter. We have shown that the symbolic image paths set converge to the system trajectories set when the coating diameter tends to zero.

What is the reason for the introduction of the trajectories encoding and invariant measures?

Humanity has always used encoding. For example, we write “bucket”. This does not mean that we bear in mind these five letters in a particular order. We use this word to refer to a well-known object for water storing and moving. Thus, this sequence of five letters is the object encoding, but not the object itself. Moreover, this object encoding in France is significantly different from the encoding in China. There is another example: all computer objects and operations are encoded in binary characters. We have a coverage of the study area with a set of cells when constructing a symbolic image.

If each cell has its symbol, then the dynamical system trajectories can be encoded by the symbols of the cells through which the trajectory passes. Since the cells number is finite, the trajectories encoding is carried out using a character finite number. These symbols’ order is important in this case. Thus, the trajectory is encoded by the symbolic image nodes sequence, i.e. path on a directed graph.

A dynamical system invariant measure generates a probability distribution (flow) on the symbolic image arcs. This distribution’s main property is the Kirchhoff law: the incoming flow is equal to the outgoing one for each node. We have shown that every abstract flow is some invariant measure approximation. This allows us to consider the flow on the graph as an encoding of the dynamical system invariant measures.

Thus, the paths on the symbolic image encode the trajectories and the flows encode the dynamical system invariant measures. The study of a symbolic image paths space and flows space gives an idea of the system dynamics.

What is your proposed method for the Morse spectrum calculation?

We proposed the following scheme for the Morse spectrum calculation: transform the dynamic system into a symbolic image and transform the spectrum calculation problem into a graph theory problem. The dynamical system trajectories correspond to valid paths on the graph. In particular, periodic trajectories correspond to the graph periodic paths. For the Lyapunov exponent calculation, it is necessary to construct a graph clothing in the way that each node is associated with a certain number. The periodic path Lyapunov exponent is the specified clothing average value over the period in this case.

The periodic paths number on a graph is usually unlimited. However, it has been proved that it is sufficient to find the average values on the cycles to calculate the Morse spectrum; a cycle is a periodic path passing through different nodes. Since the nodes number is finite, the cycles number is finite also. It is sufficient to find extremal cycles exponents to calculate the Morse spectrum. There is a graph theory algorithm that calculates the extremal cycles average values, the algorithm allows finding the Morse spectrum approximately. It is necessary to build a symbolic image to cover a sufficiently small diameter to achieve the required accuracy.

You have explored the properties of biological processes with memory. How are they characterized within the dynamical systems formalism?

We have considered the problem of modeling the biological species quantity evolution. It is well known that the biological species volume dynamics have a periodic character. For example, the quantity of salmon going for spawning this year depends on what the quantity was in previous years. It seems that the species remembers its prehistory or has a memory. Such dynamics can be expressed by an equation with a lag when the future state of the system depends not only on the present state but also depends on the states in previous periods.

There is the Takens theorem, which gives a justification for applying equations with lag. It consists of the following. If there is a dynamical system of n dimensions unknown to us, but we can observe the z value depending on this system, then the z value in the present period is determined as a rule by the z values in the previous 2n + 1 time periods. Thus, by studying one observed value for a sufficiently long time, we can predict the observed value dynamics. Moreover, the observation periods number may be less than 2n + 1 in real conditions.

We have studied equations containing one and two lags. These equations contain coefficient of generation, coefficient of intraspecific competition, and so on. The environment influence is described as these equations chaotic small perturbation. The proposed models show rather complex dynamics, which depend significantly on the coefficient of generation. It turned out that with a large coefficient of generation (as in viruses), chaotic population dynamics are possible.

How can bifurcation theory describe the economic models’ dynamics?

We have studied the macroeconomic system dynamics, in which the national income, the interest rate, and the price level interact. Such an interaction is simulated by a discrete dynamical system in three-dimensional space. The system has a curve filled with fixed points that describe equilibria in the money, goods, and services market. It is shown that there exists a foliation transversal to this curve, each layer of which is invariant for the system. There are layers on which the equilibrium state is steady. The system dynamics changes from layer to layer, a bifurcation occurs. There are two bifurcation routes in this case. The first path follows the scheme: the fixed point loses stability, a stable invariant ellipse is born (Neimark-Sacker bifurcation), periodic hyperbolic orbits appear on the ellipse, which generates chaos through the transversal intersection of stable and unstable manifolds.

Another path leads to chaos through the period-doubling bifurcation. The economic system chaos means the forecasting impossibility. If chaos reaches significant values, then an economic crisis happens. We simulated the external environment impact on the system by small random uncontrolled perturbations also. The system does not preserve the equilibrium state and the layers’ invariance, which generates more complex dynamics in this case. It turned out that the interest rate perturbation has a significant effect on the system dynamics. Its small perturbations lead to the fact that the trajectory begins to shift along with the equilibrium states and first falls into an unstable equilibrium state, and then into a layer where chaos is observed, the magnitude of which can increase, reaching significant sizes.

Interview: Ivan Stepanyan