By Wiesław M. Macek
Causality is the basic principle underlying relations in nature, as coined from the Latin noun causa (in Greek αἰτία). Aristotle (384–322 BC) developed the theory of four different concepts of efficient, final, material, and formal causes, but only the efficient cause was adopted in natural sciences. Namely, from the 17th century onwards, when Isaac Newton (1643–1727) invented differential equations, scientists were convinced that any trajectory, which is the effective solution of these equations, can be determined from physical laws, given initial conditions. In particular, the principle of causality allowed to calculate the exact orbits of celestial bodies for the two body problem (e.g., the motion of the Earth around the Sun) from the universal law of gravitation. However, curiously, the problem of three bodies (including the Moon) turned out to be much more difficult or even essentially impossible to solve exactly.
Only in the 20th century (in 1963) did Edward Lorenz (1917–2008) notice the importance of sensitivity to initial conditions in some nonlinear deterministic systems. Philosophically this means that in nonlinear systems the effect could be out of proportion to the cause. Consequently, a small cause can result in a dramatic event, popularized as ‘the butterfly effect’, and on the contrary, a bigger cause can produce tiny effects. This leads to the new scientific paradigm of chaos, which is non–periodic, long–term behavior in a deterministic system that exhibits sensitive dependence on initial conditions. Basically, we see that the principle of causality in nonlinear dynamical systems governing relations in nature needs rethinking: a simple cause may result in a complex behavior.
Nonlinear dynamical systems can be modeled well with difference iteration maps, i.e., using simple mathematical recursive rules (repeated for any discrete time step n = 0,1,2,…,∞). A well–known example is a one–dimensional logistic map xn+1 = a xn (1- xn), where xn describes relative (normalized) populations (e.g., of a species) during the n–th period. If the variable xn is small, the next population xn+1 will be larger, however, if xn is large, its growth will be reduced. Obviously, the resulting final asymptotic population depends on the control parameter, 0 ≤ a ≤ 4.0. Namely, using any calculator (with an x2 button), one can verify that for small a the population quickly diminishes to zero, or eventually, after bifurcation (a ≥1), settles to a steady state (another equilibrium fixed point). However, for a = 3.0, we obtain the same value after two iterations and hence we say that period 2 is born (the next period 4 appears at a ≈ 3.45). Figure 1 depicts successive periods emerging for a ≥ 3.5. We see that periods (8, 16, 32,64,…) come faster and faster, converging to a∞ ≈ 3.57, where long-term behavior becomes nonperiodic (infinite period, 2∞). Since the map clearly exhibits a period doubling route to chaos, this diagram has become the icon of deterministic chaos[1]. Surprisingly, however, suddenly (for a >a∞), at ac ≈ 3.83, period 3 (periodic window) is born again; this means that both chaos and order (nonperiodic and periodic solutions) are intertwined. Robert R. May (1870–1924) has proposed the logistic map as an idealized ecological model for the yearly variations in the population of an insect species with a more general plea that intuition may be enriched be seeing the wild things that simple nonlinear equations can do (May, 1976). In fact, this quadratic map can well describe how populations of consumers limit or enhance the resources in the market. In addition, strange chaotic attractors that are sensitive to initial conditions often exhibit a fractal structure.
Similar complex behavior has been noted for low-dimensional systems with a continuous time described by ordinary differential equations. Here Lyapunov exponents (λi) measures a divergence (positive) or convergence (negative) of neighboring trajectories. Hence a largest positive value (λ1>0) indicates that solutions separate exponentially fast and implies chaotic behavior (e.g., Macek, 2020, chapter 3). We have generalized the famous Lorenz system with four variables describing convection in a magnetized viscous fluid, appearing naturally in plasmas, where electrically charged particles interact with a magnetic field (Macek and Strumik, 2014). Figure 2 plots the largest Lyapunov exponent (λ1) for some chosen cases depending on the control parameters ω0 and r. Convergence of the asymptotic solutions of the model to equilibria described by fixed points (λ1 < 0) is shown in black, to periodic (limit cycles) solutions (λ1 = 0) – in violet/blue color (see the color bar for λ1 = 0), to chaotic (nonperiodic) solutions (λ1 > 0) – in a color from violet/blue to yellow, according to the color bar scale. The panel in the bottom–right part of plot shows an enlargement of the region bounded by black lines. Fine structures are shown in the inset. This proves that various kinds of complex behavior are very closely neighbored in the space of control parameters. Therefore, the obtained results could be important for explaining dynamical processes in solar sunspots, planetary and stellar fluid interiors, and possibly for plasmas in nuclear fusion devices.
Generally speaking, nonlinear differential equations or iterated discrete maps are useful models of some phenomena appearing naturally in the contexts in biology (e.g., animal populations), economics, including finance theory and social sciences. I have also argued that a simple but nonlinear law, within the theory of chaos and fractals, can describe a hidden order for the creation of the Cosmos, at the Planck epoch, when space (at a scale of 10−35 m) and time (10−43 s) originated (Macek, 2020, section 3.4). Summarizing, one can say that nonlinear systems exhibit complex behavior including deterministic chaos, strange attractors, and fractals can describe the real world. This means that nonlinear dynamics allow us to grasp how complex dynamical phenomena at various scales in nature relate to basic mathematical rules.
References
May, R. M. (1976), Simple mathematical models with very complicated dynamics, Nature, 261, 459–467.
Macek, W. M., Strumik, M. (2014), Hyperchaotic intermittent convection in a magnetized viscous fluid, Physical Review Letters, 112, 074502, 10.1103/PhysRevLett.112.074502.
Macek, W. M. (2020), The Origin of the World: Cosmos or Chaos? Wydawnictwo Naukowe UKSW, ISBN 978-83-8090-686-0, e-ISBN 978-83-8090-687-7.
[1] The bifurcation diagram of the logistic map is taken from https://hyperchaos.wordpress.com/2011/04/27/logistic-map-bifurcation-diagram/
Wiesław Marian Macek is a Full Professor at the Institute of Physical Sciences (Cardinal Stefan Wyszyński University, UKSW) and researcher at the Space Research Centre in Warsaw, Poland.
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