| Summary: | This paper focuses on the relationship between a non-autonomous discrete dynamical system (NDDS) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>H</mi><mo>,</mo><msub><mi>f</mi><mrow><mn>1</mn><mo>,</mo><mo>∞</mo></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula> and its induced set-valued discrete dynamical systems <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="script">K</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>,</mo><msub><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mrow><mn>1</mn><mo>,</mo><mo>∞</mo></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula>. Specifically, it explores the chaotic properties of these systems. The main finding is that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mrow><mn>1</mn><mo>,</mo><mo>∞</mo></mrow></msub></semantics></math></inline-formula> is Devaney chaotic if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mrow><mn>1</mn><mo>,</mo><mo>∞</mo></mrow></msub></semantics></math></inline-formula> is Devaney chaotic in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>w</mi><mi>e</mi></msup></semantics></math></inline-formula>-topology. The paper also provides similar conclusions for weak mixing, mixing, mild mixing, chain-transitivity, and chain-mixing in non-autonomous set-valued discrete dynamical systems (NSDDSs). Additionally, the paper proves that weak mixing implies sensitivity in NSDDSs.
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