Dynamical Systems Notes PDF

03.10 | dynamical systems Created @October 3, 2024 1:17 PM Tags 1. Introduction to Dynamical Systems Deterministic Systems: In dynamical systems, deterministic equations represent the time...

03.10 | dynamical systems Created @October 3, 2024 1:17 PM Tags 1. Introduction to Dynamical Systems Deterministic Systems: In dynamical systems, deterministic equations represent the time evolution of a system without any randomness. The system evolves according to a set rule, meaning no randomness is involved. Despite being deterministic, dynamical systems can exhibit unpredictable behavior over time, particularly when systems exhibit chaotic dynamics. Multidimensional Systems: In real-world applications, systems are multidimensional, meaning they have many interacting parameters (e.g., the temperature of the skin, pH of blood, etc.). These parameters influence each other and evolve over time. The equation dtdx(t)​describes the variation of one dimension over time, modeling how a particular state evolves. dx(t)dt\frac{dx(t)}{dt} A trajectory or orbit describes the path a system takes in its phase space, a space representing all possible states of the system. 2. Continuous vs. Discrete Dynamical Systems Continuous Dynamical Systems:dtdx(t)​=f(x) The system evolves continuously over time, without discrete steps. The behavior is modeled by a continuous function, often represented as: dx(t)dt=f(x)\frac{dx(t)}{dt} = f(x) Continuous systems often have solutions that are smooth and evolve gradually. 03.10 | dynamical systems 1  Discrete Dynamical Systems (Maps):xn+1​=M(xn​) In discrete systems, the system evolves in steps, where the time variable n is discrete (e.g., n=0,1,2,…), and the state changes in discrete time intervals. nn n=0,1,2,…n = 0, 1, 2, \dots A discrete dynamical system is expressed as: xn+1=M(xn)x_{n+1} = M(x_n) Intersection points (i.e., specific moments in time) are considered, reducing a continuous system to a discrete one. 3. Chaos and Sensitive Dependence on Initial Conditions Chaos in Dynamical Systems: Even in deterministic systems, chaos can arise, where small changes in initial conditions lead to vastly different outcomes over time, making long- term prediction impossible. Lyapunov Exponent: A system exhibits chaotic behavior if its Lyapunov exponent is positive, indicating exponential divergence between two initially close points. This means that even minute differences in initial conditions can lead to wildly divergent trajectories as time progresses. Example: Logistic Map:xn+1​=rxn​(1−xn​) A simple chaotic system is the logistic map: xn+1=rxn(1−xn)x_{n+1} = r x_n (1 - x_n) This function produces chaos for certain values of the parameter r ∈ (typically r [2.5,4]). rr r ∈[2.5,4]r \in [2.5, 4] 03.10 | dynamical systems 2  Bifurcations occur as r increases, where the system moves from a stable point to periodic oscillations and eventually to chaotic behavior. rr 4. Bifurcations and Attractors Bifurcations: As the parameter r in a system like the logistic map changes, the system may experience bifurcations, where the number of attractors (stable states) doubles, causing the system to transition between different types of behavior (e.g., from stable fixed points to periodic orbits). rr Period Doubling: At certain values of r, the system exhibits a period doubling bifurcation, where the number of periodic attractors doubles (e.g., 1 → 2 → 4 → 8...). rr Types of Attractors: Fixed Point: The system settles at a single state. Limit Cycle: The system oscillates in a periodic cycle between two or more states. Strange Attractors: In chaotic systems, attractors have a fractal structure and no repeating patterns. They are non-periodic and can appear in systems like the Lorenz system or the Henon map. 5. Stability of Fixed Points Fixed Points: A fixed point is a state where the system does not evolve anymore, i.e., dtdx​=0. dxdt=0\frac{dx}{dt} = 0 The stability of these points depends on the derivative of the system at the fixed point. 03.10 | dynamical systems 3  If the derivative at the fixed point is less than 1, the point is stable (attracts nearby points). If the derivative is greater than 1, the point is unstable (repels nearby points). If the derivative equals 1, the fixed point is meta-stable, meaning small perturbations may lead to instability. 6. Conservative vs. Dissipative Systems Conservative Systems: A system is conservative if it preserves the total "volume" in phase space over time (i.e., it does not lose energy or change the system's total state). Example: A harmonic oscillator without damping is conservative. Dissipative Systems: A system is dissipative if it loses "volume" in phase space over time (i.e., energy is dissipated). Example: A damped harmonic oscillator loses energy due to friction and is dissipative. Dissipative systems can have attractors where all orbits eventually converge to a single state or periodic behavior. 7. Chaotic Systems and Fractals Chaotic Systems exhibit sensitive dependence on initial conditions. For example, small differences in the initial state of a chaotic system can lead to exponentially diverging behaviors. Strange Attractors in chaotic systems have a fractal dimension and exhibit complex, irregular patterns. These attractors exhibit self-similarity at different scales, a key property of fractals. 8. Symbolic Dynamics and Phase Space Partitioning 03.10 | dynamical systems 4  Symbolic Dynamics: Symbolic dynamics involves dividing the phase space of a system into regions, each of which is assigned a symbol from an alphabet. The system's trajectory can then be described as a symbolic sequence where each symbol corresponds to a region of the phase space that the system visits. Phase Space: The phase space represents all possible states of the system. In chaotic systems, the phase space can be partitioned into symbolic regions, and the trajectory can be traced as a sequence of symbols. 9. Real-World Applications Human Health: In complex biological systems (e.g., heart rhythms or brain dynamics), the characteristic time could represent a time frame relevant to the system’s life cycle (e.g., the human lifespan). Geological Systems: In geology, systems evolve over much longer time scales, like millions of years, and have different dynamics compared to human-scale systems. Living Systems: Living organisms are inherently multidimensional systems with many interacting parameters, such as temperature, pressure, and biochemical states. 10. Summary and Reflections Deterministic Chaos: Chaos theory demonstrates that deterministic systems can exhibit unpredictable and complex behavior, especially when the system has many interacting parameters. Bifurcations and Attractors: The transition from stable states to periodic orbits and chaos is a key feature of dynamical systems. Strange attractors and sensitive dependence on initial conditions are hallmarks of chaotic systems. 03.10 | dynamical systems 5  Fractality in Chaos: Fractal dimensions describe the complexity of chaotic systems, where attractors exhibit self-similar patterns at different scales. Practical Implications: Chaos theory has wide applications in fields like biology, physics, economics, and engineering, where it helps understand complex, non-linear behaviors that evolve over time. Through the study of dynamical systems and chaos, we gain insight into the unpredictability of nature and patterns of complexity that emerge from seemingly simple deterministic rules. 03.10 | dynamical systems 6