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Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.
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Continuous-time systems have continuous input and output signals, with time measured continuously. These systems are generally defined by differential or algebraic equations. For instance, in an RC circuit, the relationship between input and output voltage is expressed through a differential equation derived from Ohm's law and the capacitor relation,
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Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
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A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
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First-order systems, such as RC circuits, are foundational in understanding dynamic systems due to their straightforward input-output relationship. Analyzing their responses to different input functions under zero initial conditions reveals significant insights into system behavior.
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Topological classification of driven-dissipative nonlinear systems.

Greta Villa1, Javier Del Pino1, Vincent Dumont2,3

  • 1Department of Physics, University of Konstanz, 78464 Konstanz, Germany.

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This study introduces a new framework for classifying topological properties in driven-dissipative nonlinear systems. It uses a graph index to reveal the topology of complex dynamics, with applications in advanced materials and computing.

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Area of Science:

  • Topological physics
  • Nonlinear dynamics
  • Condensed matter theory

Background:

  • Topology in physics reveals global features from local details, crucial for quantized transport and boundary effects in linear systems.
  • Characterizing open (non-Hermitian) and interacting systems extends topological physics beyond linear Hamiltonian models.

Purpose of the Study:

  • Establish a framework for topological classification of driven-dissipative nonlinear systems.
  • Define a graph index for Floquet semiclassical equations of motion.
  • Encode particle-hole nature of excitations in non-equilibrium stationary states.

Main Methods:

  • Developed a graph index based on the topology of vector flows.
  • Applied the index to analyze nonlinear resonator dynamics under forcing.
  • Investigated driven-dissipative phases, including damping responses and symmetry breaking.

Main Results:

  • Uncovered the topology of nonlinear resonator dynamics.
  • Characterized driven-dissipative topological phases, such as under- to overdamped responses.
  • Identified symmetry-broken phases linked to population inversion.

Conclusions:

  • Demonstrated a pervasive link between topology and nonlinear dynamics.
  • Framework has broad implications for interacting topological insulators, solitons, neuromorphic networks, and bosonic codes.